AD-AIIS 432 YALE UNIV NEW HAVEN CT DEPT OF COMPUTER SCIENCE F/S 12/1LOCAL-MESH, LOCAL-ORDER. ADAPTIVE FINITE ELEMENT METHODS WITH A--ETC(U)DEC 81 A WEISER N00014-76-C 0277

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YALE UNIVERSITYSDEPARTMENT OF COMPUTER SCIENCE

82 04 'RO 127

Local-Mesh, Local-Order, Adaptive Finite Element

Methods with A Posteriori Error Estimators

for Elliptic Partial Differential Equations

Alan Weiser"

Technical Report #213

December, 1981

IDepartment of Computer Science, Yale University, New Haven, CT 06520

This research was supported in part by ONR Grant N00014-76-C-0277, !SU-ONRGrant FI.N00014-80-C-0076, the Department of Energy (DOE Grant DE-AC02-77ET53053),The National Science Foundation (Graduate Fellowship), and Yale University.

" i o V.IM.

A3STACT

Local-esho Local-Order, Adaptive Finite ElementMethods with A Posteriori Error Estimatorsfor Elliptic Partial Differential Equations

Alan Veiser

Yale University. 1981

The traditional error estimates for the finite element solution of

elliptic partial differential equations are a priori, and little

information is available from them about the actual error in a specific

approximation to the solution. us, in the traditional finite element

method, the choice of discretization i eft to the user, who must base

his decision on experience with similar equations.

- In recent years, locally-computable a posteriori error estimators

have been developed, which apply to the actual errors committed by the

finite element method for a given discretization. These estimators lead

to algorithms in which the computer itself adaptively decides how and

when to generate discretizations. So far, for two-dimensional problems.

the computer-generated discretizations have tended to use either local

mesh refinement, or local order refinement, but not both.

\n this thesis, we present a new class of local-mesh, local-order,

square finite elements which can easily accomodate computer-chosen - ,-_

I, I .io, IIl~

I4 .4I

[~~0 i ... .. ... .. .__ . .. 1- I

- discretizations. We present several now locally-computable a posteriori

error estimators which, under reasonable assumptions, asymptotically

yield upper bounds on the actual errors omlitted, and algorithms in

which the computer uses the error estimators to adaptively produce

sequences of local-mesh, local-order discretizations. We present

theoretical and numerical results indicating the ex c d and actual

behavior of our methods.

I

Table of -Cote.ts .

CAPR1: Introduotion ....... . .. . . 1

CHAPTR2: Local mosh refinement .. . .. . . . 6

2.1 Introduction . . . . . . . . . ..... 62.2 Terminology . . ........... . 72.3 The 1-irregular rule and some of its consequences . . 112.4 Alternative mesh refinement rules . . . . . . . 172.5 A mesh data structure . . . . . . . . . . . 21

CHAPR 3: Local order refinement ... ....... 25

3.1 Introduction ...... .............. 253.2 The Lagrange case ..... ............. 263.3 The Hermite case . . . ............. 29

3.4 A mesh data structure for local order refinement . 29

CHAPTER 4: A posteriori error estimators , . , ..... 33

4.1 Introduction . . . ......... . 334.2 Terminology ...... .............. . 354.3 A Dirichlet a posteriori error estimator ..... 404.4 A Neumann a posteriori error estimator ...... 424.5 Properties of the Neumann error estimator ..... 444.6 Alternative error estimators . . . .... . S1

4.6.1 Ignoring irregular corners . .. .. 514.6.2 Using the matrix for a local Poisson problem . 524.6.3 Using a smaller matrix . . . . . 54.6.4 Using the residuals and Jumps in normal derivatives

574.7 Extensions . . . . . . . . . . . . . . . 604.8 Summary . . . . . . . . . . . . . . . . 64

CHAPTER 5: Heuristic refinement criteria . . . . . . . . 67

5.1 Introduotion . . . . . . . . . . . . . . 675.2 A model of work . ........... 685.3 A model of predicted error behavior ... ... 705.4 A symetized model of predicted error behavior . . . 765.5 Choosing the refinement cutoff . . . . . . . . 785.6 Drawbacks ... . . . . . . . . . . .. 79

-iv-

5.7 Expected error behavior . . . . . . . . . . . 815.7.1 A smooth solution . . . . . . . . . . 825.7.2 A point singularity . . . . . . . . . . 835.7.3 A line singularity . . . . . . . . . . 84

CfhAFM 6: Computational aspects . . • . . . . . 85

6.1 Introduction . . . . . . . . . . . ... 856.2 Assembling the linear system . . . . . . . .. 876.3 Solving the linear system . . . . . . . . . . 90

6.3.1 A smooth solution . . . . . . . . . . 906.3.2 A point singularity . . . . . . . . . . 926.3.3 A line singularity . . . 9 . . . . . . 946.3.4 Disoussion . . . . . . . . . . . . 95

6.4 Computing the error indicators 96

C1APTM 7: Numerical results . . . . ..... . . . 101

7.1 Introduotion .. ............ 1017.2 The problem set ............. 1017.3 Error behavior . . . . . . . . . . . . . 1047.4 Error estimator behavior . . . . . . . . . . 1077.5 Error estimator convergence . . . . . . . . . 108

List of Figures

2-1: A bisection-type loeally-refined noes . . . . . . 6

2-2: Lagrange versus tonsor-product basis functions .. . . 11

2-3: Configurations of eleensts in su3p(B(V)) .. . .. . 13

2-4: f(E) .. . . . . . ..... 14

2-5: Minimum and alnum N(Y) for interior vertices .... 15

2-6: 1 in 1(L+1) . . ... . . 16

2-7: SinSularity at P - (2/3.2/3). IlX - 4 . . . . . . . 19

2-8: A modified basis function ... . . ..... . 19

2-9: LUX - 4: alternative nosh refinmeont rules . ., . . . 22

2-10: Nodes in ICT and IVURT .. ...... ... 23

3-1: A side basis function ............ 28

3-2: Losal-order basis funetions .......... 30

3-3: Basis functions mapped to vortiees ........ 31

4-1: Ignoring the irregular crners ......... 52

5-1: Early order refinement penalty . . . ...... 81

6-1: An element with one irregular corner ....... 89

6-2: Nested dissection ordering with a smootk solution . . . 91

6-3: Reosrsive ordering vith a point singularity . . . . . 93

6-4: Nested-disseotion-type ordering with a line singularity . 95

7-1: Problem - tru erro . . ..... . ., . 111

7-2: Sample grid for problem 1 (method +, residuals and Jumps) 112

-vi-

7-3: Problem1 ,_.approxhiito error , . * . . . . . , 113

7-4: Problem 1 - residuals ad Jumps .... . . . . . 114

7-5: Problem 2 - tru error ,. . . * . * 115

7-6: Sample $rid for problem 2 (method . residuals sad Juva) 116

7-7: Problem 2 - approximate error ..... . . . o . 117

7-8: Problem 2 - residualssadJws * * * . * *. 118

7-9: Problem 3 - true error . . . * . , ..... 119

7-10: Sample grid for problem S (method 2. residuals asd Jumps) 120

7-11: Sample grid for problem 3 (method . residuals and Jumps) 121

7-12: Problem 3 - approximate error . . . . . . . . 122

7-13: Problem 3 - residuals sad Jumps .. .. .. . . . 123

7-141 Problem 1 - approzimate error . . . . . . . . . 124

7-15: Problem 1 - residuals sad Jumps . * . , . . . . 125

7-16: Problem 2 - approximate error . .. . .. . 126

7-17: Problem 2 - residuals andJups . ...... * a 127

7-18: Problem 3 - approximate error . . . . . . . . 128

7-19: Problem 3 - residualsaadJumps . .... . . . . 129

List of Tables

7-1: Test problems S . . . 102

7-2: Refinement metkeds 0 . 0 0 . 0 . 0 0 * a 0 0 103

7-3: Iffiaiezy sunary .. a 0 o 0 00 0 0 * 107

7-4: Error estimator behavioz 0 110

List of "yols

0, so 7

Il, z E Iy, 1 8

M 8

admissible mesh 8

(iW)regular vertex 9

k-irregular mesh 9

S, BI (piecovise bilinear) 10

1-irregular rule 11

f( ) 12

k26

Pi(x), pi(z) pa(y) 26

S, BI (local-order) 27

k 28

% (.,.k), 1111 is36

a* 36

-ix-

.,'

7 W!dl

Symbol -_.lu

SE 37

SE ' SE-S97 SI (.), I(.) 3E

N(E), ya) 39

up U, 0. U 39

, e 39

y 40

h 40

FE(.), [aaU/n], FE(-) 42

E 43

*I 44

koax 47

5o

~so

S"E(., I11 , (E) 52

! ._ 53

~ss-E -

55

so AE 60

-P., P.5 65

-: .... ________ --. . '- '- .-. ..... . . ... .. . . ..

z 69

MO.) p(z) (p.). v(p(z)) (WE) 69

m(z) , p(z) 70

C(;), G(z.P) 71

4B , 471

(h) ()72

4E 73

PCUT 7

~CUT 78BE,(B' 86

IE, IE (B,) 86

G(E) . ws is t 8

vL 89a ,

w 99p.

CN.FlR 1

Introduotion

The traditional error estimates for the finite element solution of

elliptic partial differential equations are a priori, and little

information is available from them about the actual error in a specific

approximation to the solution. Thus, in the traditional finite element

method, the choice of discretization is left to the user, who must base

his decision on experience with similar equations.

In recent years, locally-computable a posteriori error estimators

(e.g., Babutka and Rheinboldt [8]) have been developed, which apply to

the actual errors committed by the finite element method for a given

discretization. These estimators lead to algorithms in which the

computer itself adaptively decides how and when to generate discreti-

zations. So far, for two-dimensional problems, the computer-generated

discretizations have tended to use either local mesh refinement (e.g.,

Babulka and Rheinboldt [10], Bank and Sherman [13]), or local order

refinement (Babulka, Szabo, and Katz [111), but not both.

In this thesis, we present a new class of local-meeh, local-order,

1

-.. -4, vv ~mU~ d bud e vie

2

square finite elements which can easily accomodate computer-chosen

discretizations. We present several new locally-computable a posteriori

error estimators which, under reasonable assumptions, asymptotically

yield upper bounds on the actual errors comitted, and algorithms in

which the computer uses the error estimators to adaptively produce

sequences of local-mesah, local-order discretizations. We present

theoretical and numerical results indicating the expected and actual

behavior of our methods.

We begin in Chapter 2 by considering bisection-type local mesh

refinement with square elements. We present the class of f1-irregular"

meshes, a variant of a class of meshes suggested by Bank and

Sherman [14], such that any bisection-type locally-refined mesh can be

converted into a 1-irregular mesh with no more than a constant times as

many elements, and the resulting finite element matrix (with bilinear

basis functions) has no more than a constant number of nonzeroes in any

row. Conversely, we show that there are simple bisection-type

locally-refined meshes with essentially dense finite element matrices.

We discuss simple data structures for handling 1-irregular meshes.

In Chapter 3, we extend the finite element spaces considered in

Chapter 2 to include basis functions with locally-varying polynomial

orders.

In Chapter 4, we develop new a posteriori error estimators. We

present a new error estimator which has two main advantages over the

3

estimator presented in Babulka and Rheinboldt [8]:

1. Under reasonable assumptions, the new error estimator is

asymptotically an upper bound on the norm of the true error.

2. The new error estimator can be computed locally in each

element. (The error estimator in [8] must be computed over

more complicated local regions.)

Under suitable assumptions, we show that, like the estimator in [8], the

new error estimator is no larger than a constant times the norm of the

actual error. We also present several cheaper error estimators, and

discuss their properties. Under suitable assumptions, Babulka and

Miller [4] have recently shown that the error estimator used in their

code (with piecewise bilinear basis functions) converses to the norm of

the true error. We have not yet proven convergence for our error

estimators.

In Chapter 5, we develop several heuristic models of error behavior

which lead to adaptive refinement procedures (of. Brandt [16], Babulka

and Rheinboldt [71). We present the asymptotic expected error and work

behavior for three problem classes consisting of problems with:

1. smooth solutions.

2. solutions with point singularities.

3. solutions with line singularities.

Except in the presence of line singularities, the expected error

converges to zero faster than inverse polynomially with respect to work.

4

In Chapter 6, we discuss some of the computational aspects of the

methods presented in Chapters 2-5. We consider efficient assembly of

the finite element system (of. Bisenstat and Schultz [181, Weiser,

Eisenstat, and Schultz [31]), and efficient construction of the error

estimators. We present the asymptotic complexity of nested-dissection-

type Gaussian elimination for the three classes of problems considered

in Chapter 5. The operation counts and storage for the problems with

sinsularities are linear (optimal-order) in the number of elements N, in

contrast to the operation counts and storage for uniform meshes, which

are proportional to N3 1 2 and N los(N) respectively (e.g., George [231).

In Chapter 7, we present numerical results obtained using prototype

codes. We present the resulting error and error estimator behavior for

problems from the three classes considered in Chapter 5. The error

estimators are usually accurate to within a factor of two. In some

cases, the error estimators appear to converge to the norm of the true

error as the mesh size decreases.

I an deeply grateful to my advisor, Professor Randolph E. Bank, for

suggesting the topic of this research and working closely with me on its

realization. I am honored to know him and to have worked with him. I

would like to thank my other advisors, Professors Martin N. Schultz and

Stanley C. Eisenstat, for helping to guide my research, and for their

unfailing intelligence, integrity, and support.

Many other past and present members of the numerical analysis group

at Yale have contributed much to my enjoyment of graduate school and my

appreciation and understanding of numerical analysis. They are (in

order of appearance at Yale) Andy Sherman, John W. Lewis, inti Chandra,

Rob Schreiber, lack Perry, Dave Fyfe, Trond Steihaug. Craig Douglas,

Howie Elman, Zen Jackson, Tony Chan, John Kindle, and Denren Zhu.

I an grateful to many other members-at-large in the numerical

analysis comunity. I = especially grateful to Professor Mary

F. Wheeler, for her encouragement and guidance when I was an

undergraduate, and Professor Ivo Babulka, for his excellent and

influential work.

I a grateful to the Office of Naval Research CONK Grant

N00014-76-C-0277, FSU-ONR Grant Fl .N00014-8O-C-0076). the Department of

Energy (DOE Grant DE-ACO2-77ET5353), the Notional Science Foundation

(Graduate Fellowship), and Yale University for their financial support.

Most of all, I -grateful to my family, and my loving wife Mary

Ann.

CRLAiPTm 2

Looal mesh refinement

2.1 Introduction

In this chapter, we consider bisection-type local mesh refinement

with square elements. A saiple mesh of this type is shown in Figure

2-1.

I I I I lI I I l II I I I I-- 0----- -.. 0--0

SI I I III II I -- o---.I I I I I I

0- .-- 0--- •-V--OI I II I Il I II I II I II I I0-- 0- 0

Figure 2-1: A bisection-type lovally-refined mesh

This type of mesh refinement has been intensively investigated in

recent years, mainly by Babulka and Rheinboldt and their students

t6

7

(e.g. [4]). The data structures and computational algorithms required

to implement the finite element method with general noshes of this type

are fairly complicated (see Rheinboldt and Mesztenyi [251 and

Gannon [21]). Thus, Bank and Sherman [14] and Van Rosendale [30]

suggest using restricted classes of meshes which can be implemented with

much simpler data structures.

In Section 2.2, we present some standard terminology for dealing

with bisection-type local mesh refinement, and define the class of

"l-irregular" meshes, a variant of a class of meshes suggested by Bank

and Sherman [14]. In Section 2.3, we consider a variant of a mesh

refinement rule suggested by Bank and Sherman which can convert any

bisection-type locally-refined mesh into a 1-irregslar mesh with no more

than a constant times as many elements, such that the resulting finite

element matrix (with bilinear basis functions) has no more than a

constant number of nonzeroes in any row. Conversely, in Section 2.4. we

show that there are simple bisection-type locally-refined noshes such

that the resulting finite element matrices are essentially dense. We

also consider alternative mesh refinement rules. In Section 2.5, we

present data structure details for handling 1-irregular meshes.

2.2 Temiznology

We now oonsider bisection-type local mesh refinement with square

elements. For simplicity, we take the domain to be the unit square,

B := C0,1]X[01.] = ((Z.y): 01z.l, 0lyll),

!1

with boundary 8Q. The square region B - [xE, xe+h]X[yzyi+hh] is called

an element. The elements

[xE xE+hE/2 lX [y+/ 2 ,y+yh], [£+h/2oE+h£]X[y+hI/2,y£+N],

axExe+hE/2]X[yE.yE+hE/2]. [zx+hE/2,z£+hEX[yEye+h/21

are called the sons of E, and E is called the father of its sons. A

o M is defined to be an unordered set of elements. An element in K

is called refined if its four sons are in N. and unrelied otherwise.

The class of bisection-type locally-refined noshes with square

elements, or admissible noshes (Babulka and Iheinboldt [8. 101), is

recursively defined by the following two rules:

1. M) is an admissible mesh.

2. If a mesh N is an admissible mesh, and E is an unrefined

element in N, then the mesh (N, the sons of El is an

admissible mesh.

The level L of R is defined to be the number of generations between

Q and B. Thus, in an admissible mesh on B, hE - (1/2)L . A noizhbor of

£ is defined to be another element at the same level as E that shares a

side with B.

A corner of an unrefined element is called a vertex. A vertex

which it a corner of each unrefined element it touches is called a

9

xauar vertex. All other vertices are called igula . The

.irULILity iadex (Babulka and Rheinboldt [101) of a mesh is defined to

be the maximum number of irregular vertices on a side of an unrefined

element. We ocall a mesh with irregularity index I k a k-ieulaz nesh.

A (regular) vertex on 80 is called a boundary vertex. All other

vertices are called interior.

In figures depicting meshes,

0 denotes a regular verteg.

o denotes a vertex (r,'gplsr t irregular),

* denotes an irregular vertex,

(v) at a vertex denotes function value.

For example, Figure 2-1 depicts an admissible 2-irregular mesh with 4

refined elements, 13 unrefined elements (one labelled K), 12 regular

boundary vertices, 5 regular interior vertices, and 6 irregular vertices

(one labelled V).

This is the name used by Babuaka and Rheinboldt [10). Bank and

Sherman [141 use the name "green vortex". Gannon [211 uses the name

"inactive vertex".

10

Let M be an admissible mesh. A function S is called viecewise

bilinear on M if, on any unrefined element E in M,

S(x.y) s(zE.yE) (xB+hZ-x)(yE+hE-y) /E

+ g(zhEyE) (xE ) (yE+hey) S h

2+ S(xEoyE+hE) (xE+h -x)(y-YE) /I

+ S(zE+hEyE+h) (xxE)(yyE) / 2

Let the finite element space S (= S(M)) be the space of continuous

piecewise bilinear functions on M. Let the regular vertices for I be

Vl.....VN . Let B (x,y) (= B(VI)(xy)) be the function in S which

satisfies the Lagrange conditions

BI(VJ) - ij f1 if I j

0 if I #,

for J'1,...,N. BI is well-defined using induction on elemnet level.

Moreover, (BI) is a basis for S, since each BI is in S. the BIIs are

linearly independent, and each function in S is a linear combination of

the Bi'a.

The suipozt of B., supp(B1 ), is defined to be the union of all

unrefined elements E such that BI is not identically zero in R. In

general, the support of BI may be non-convex or non-simply-connected.

Alternately, the standard tensor-product basis functions {1I may be

11

defined for S, so that each supp(p_(V)) is a rectangle. Most of the

considerations applying to (BI ] also apply to (1). However, supp(B(V))

is always contained in supp(p(V)) (see Figure 2-2), and we know of no

computational advantage for (1i). Thus, we restrict our attention to

(B).

(o -0 . .(0) O(0) -O -()IIJ I III I

(0)-(0)(1/2) I (0)(1/4)(1/2)I I I I III I(0)(1/2)(1)----() (0)(1/2)(1)- -(0)I I I I III I I I II

(0)---(o) ---- (0) ()--()-(o)

B(1/2,1/2) D(1/2.1/2)

Figure 2-2: Lagrange versus tensor-product basis functions

2.3 The 1-irregular rule and some of its consequences

Given an admissible mesh, we force possible additional refinement

by applying, as many times as possible, the following variant of a rule

of Bank and Sherman [14):

refine any unrefined element with more than one (2.1)

irregular vertex on one of its sides.

We call rule (2.1) the 1-irruuu Lg". The 1-irregular rule must

terminate, since it never increases the maximum element level in the

mesh. After the 1-irregular rule terminates, the resulting mesh is

1-irregular. Conversely, if any refinement forced by the 1-irregular

12

rule is omitted, the resulting mesh is not 1-irregular. Thus, the

resulting mesh has the fewest elements of any 1-irreSular mesh

containing the original mesh.

The resulting configurations (up to symetry) of the support of

B(V) for interior V are depicted in Figure 2-3. Because of the

1-irregular rule, further refinement of an element in one of cases I-VI

produces another of cases I-VI, possibly at the next finer level.

Each unrefined element E in the support of B(V) satisfies either

A : V is a corner vertex of B,

or

B : V is not a corner vertex of K, but

V is a corner vertex of the father of E, and

V is next to an irregular corner vertex of E.

Thus, it is simple to determine which basis functions are nonzero

in B. Figure 2-4 depicts a sample B in its father, f(B), and numbers

the 6 relevant vertices. B3 and B4 are nonzero in B (V4 is regular

since f(E) is refined). B2 (B1 ) is nonzero in B if V2 is regular

(irregular). B$ (B6) is nonzero in B if V5 is regular (irregular).

There are exactly four basis functions nonzero in K. Since the

element stiffness matrices are small (44), and the mappings from local

to global basis functions are simple, the resulting finite element

system is straightforward to assemble.

13

.. .. 0 .. .. 00--0----- 0

I IBI II A I A I o-o---o A II I I IBlAI I

IAIAI I A IAII I I I I I0-- 0--0------ 0

I l BI I A Io----o A I o--o A I

IBIA I I IAI IIII :o--o-.--V--o-.--o IV: o--V- -o

I A o-o---o o-o A II IB I IBI I----- 0 0- 0- 0

0- 0-- 0

IBI I iBI Io-o A I o-o A IIAI I IAI I

V o-V--o--o VI ?ooV-o---oIA I A I B o- AI BO----O-- O O'--

Figure 2-3: Configurations of elements in supp(B(V))

14

2 3

I ElI I I

Figure 2-4: f(E)

Furthermore, the basis functions are locally independent:

in each unrefined element E, (2.2)

(BIlE : E in supp(BI)) are linearly independent,

where "1E denotes restriction to element E. We can show that (2.2)

holds if and only if

no irregular vertex touches unrefined elements (2.3)

at three different levels.

For example, in the mesh shown in Figure 2-1, irregular vertex V touches

unrefined elements at levels 1, 2, and 3, and the restrictions to

element E of the five basis functions nonzero in E are linearly

dependent.

Let N(V) be the number of basis functions with support intersecting

supp(B(V)), where V is a regular vertex in a 1-irregular mesh. N(V) is

the number of nonzeroes in the row of the finite element matrix

corresponding to B(V). The minimum possible N(V) for an interior vertex

is 5, and the maximum possible N(V) is 13 (Figure 2-5). N(V) is 9 for

interior vertices in a locally uniform mesh. N(V) for boundary vertices

can be as small as 4. N(V) is 6 for boundary side vertices in a locally

uniform mesh.

O--O- -0O--o-- OI I I I I I

0_-_.. &--o---. I 0- II I I I I I I I I I-0-- -.-- V-- .-- 0 0- -V-- -- 0

I I I I I I I 0-0-. I Io--o--o I .--- o---o o.-o--o.--oI I I0__OO- ... _O O 0-0 0__

N(V) = 5 N(V) = 13 N(V) = 13

Figure 2-5: Minimum and maximum N(V) for interior vertices

Many refinement sequences arising in practice naturally yield

1-irregular meshes, so the 1-irregular rule introduces no additional

refinement. We now show that, given an arbitrary admissible mesh, and

applying the 1-irregular rule as many times as possible,

the number of elements forced to refine by the (2.4)

1-irregular rule is no more than eight times the number

of refined elements in the original mesh.

Hence the number of basis functions introduced by the 1-irregular rule

is no more than a constant times the number of original basis functions.

Suppose there are LMAX levels in the mesh. Let

16

O(L) : (original refined elements at level L),

R(L) := (elements at level L refined by the 1-irregular rule).

It suffices to show that each element E in R(L) shares a corner vertex

with at least one element F in O(L). We do this using induction from

Lx LMAX back down to 0. There are clearly no elements in R(LMAX).

Suppose the statement holds for all levels with index greater than L.

Since E is in R(L), the 1-irregular rule forces E to refine because G, a

neighbor of E, has a son, H, which is refined at level L+l, and shares a

side with E. If H is in O(L+l), then G-F is in O(L), and (2.4) holds.

Otherwise, H is in R(L+1). By induction, there is an element I in

O(L+1) which shares a corner vertex with H. Then the father F of I is

in O(L) and shares a corner vertex with E (Figure 2-6). So. (2.4) holds

in all cases.*

I I I I Io-F--o -- oIl lI I I0--0------0I Im I Io--G--o ] II I I I0-0 00- -- o

Figure 2-6: H in R(L+1)

We do not know if (2.4) is sharp. The largest ratio of enforced

refinement to original refinement we have encountered is 5.75.

. -I - - - ' " "- . . . . . . .. ...

17

In summary, given any admissible mesh, the 1-irregular rule forces

additional refinement such that

the resulting 1-irregular mesh has no more than a (2.5)

constant times the original number of basis functions,

and

the resulting element stiffness matrices are of (2.6)

bounded size (4X4), and simple to assemble,

and

the number of nonzeroes in any row of the resulting (2.7)

finite element system is bounded independent of the mesh.

2.4 Alternative mesh refinement rules

In this section, we show that a refinement rule like the

1-irregular rule is needed for properties (2.6) and (2.7). We also

consider alternative mesh refinement rules.

Suppose, because of a sharp point singularity, that a mesh results

from refining only elements which contain the point P - (2/3,2/3).

Since

2/3 = I - 1/2 + 1/4 - 1/8 + 1/16-...,

the mesh is formed by successively refining the alternately upper right

or lower left son of the smallest refined element. If there are

LMAX >) I levels in the mesh, the active vertices are the ten boundary

vertices and the LMAX interior vertices

VL PL PL PL L (-1/212 L-lI...., XA"L L . . . ..L .L II -M II "O

ris

Let element E be the smallest element containing P. For any VL , it cas

be shown by induction on I that B(VL ) is nonzero at the points (P1 _I,P 1 )

and (Pl,PI_1 ) if and only if I > L. Setting I - LKAX, since

(PI.X_,PIAX) and (PIf,Pj/X_l) are corners of E, E must be in the

support of each B(VL). Thus, the element stiffness matrix for E, and

the resulting finite element system, are essentially dense. Since the

mesh is 2-irregular, this example disproves the conjecture (Babulka and

Miller [4], page 17) that (2.7) holds for k-irresular meshes with k

bounded.

Assembly of the finite element sytem is fairly complicated (e.g.,

Gannon [21], Rheinboldt and Mesztenyi [251). If Gaussian elimination is

used to solve the system, the solution time is asymptotically

proportional to LEA.. By contrast, if the i-irregular rule is applied,

assembly is simple. If the vertices are ordered according to the

maximum levels of the elements in their supports, solution time is

asymptotically proportional to LMAX (see Section 6.3).

In the previous example, (2.6) and (2.7) can be insured without

enforcing additional refinement, by using modified basis functions which

are identically zero in the element containing the singularity. Figure

2-8 depicts the modified basis function 1(V1 ). When there are Jam sharp

point sinsularities at

P1 = (9/16 + (2/3)116, 10116 + (2/3)/16)

and

P 2 (9116 + (2/3)116. 13/16 + (2/3)/16),

19

I I I I V : regularS I I I interior vertex

-- 0

(0) -V0- ..... (0)

I iI I II IiI I iII I

II I

-VoFigure 2-7: Singularity at P = (2/3,2/3), IJ(AX = 4

(0) ..... (-1)- (0)- (0)II I iII I 01I :(

I (o)-()-()-(o)I 1 lo1 I

(.S)-(O)-(o) III I I I(0)- (1)(.5) -(0)--(-1)

II II.I III III III III III I

(0) (0) (0)

Figure 2-8: A nodified basis function

however, the support of the basis function for V1 must asymptotically

intersect the supports of roughly half the basis functions in S, since

20

it can only be zero at one point along the line segment from (1/2.1/2)

to (1/2,1). Thus, with aU rule for forming basis functions, it may be

advantageous (purely from matrix sparsity considerations) to force

additional refinement.

We briefly mention alternatives to the 1-irresular rule. Another

rule which insures properties (2.5). (2.6), and (2.7) (when applied as

many times as possible) is:

refine all unrefined side- and corner- neighbors of any (2.8)

element which has grandsons (sons of a son).

Rule (2.8) is equivalent to a condition in Van Rosendale [301, page 59.

With this rule, only cases I-V in Figure 2-3 are possible, and (2.5),

(2.6), and (2.7) hold for both (Ut) and %fi). However, rule (2.8) may

force more additional refinement, and hoe require more work, than the

1-irregular rule, and is no easier to implement.

Conversely, any rule which may force less additional refinement

than the 1-irreSular rule does so only when neighboring elements with

size ratios at least 4:1 occur. For example, if we apply the rule

if an irregular vertex touches unrefined elements at three (2.9)

different levels, refine the element at the middle level,

as many times as possible (refining elements with lowest level first),

the resulting mesh will satisfy (2.6). and nay have fewer elements then

21

the resulting 1-irregular mesh. Unfortunately, (2.5) and (2.7) may not

hold: for a problem with a sharp point singularity at (1/2+s,1/2+s), as

c approaches zero, the number of nonzeroes in the row of the finite

element system for B(1/2,1/2) may be arbitrarily large.

Figure 2-9 depicts the results of applying the 1-irregular rule,

rule (2.8), and rule (2.9) to the mesh in Figure 2-7.

2.5 A mesh data structure

The data structure we use to represent 1-irregular meshes is a

variant of a data structure presented in Bank and Sherman [141. The

data structure has two main components: a refined-element tree IRCT,

stored in a 4XNIRCT array, and a vertex list IVET, stored in a 2XNIVUT

array. For each refined element 9, there is a 4X4 node in IRCT

containing pointers to the father of E, to any of the four sons of E

which are refined, and to the nine vertices which are corners of the

sons of E. For each interior vertex V which bisects an element side,

there is a 2XI node in IVUT containing pointers to the elements with

sides bisected by V. Other types of vertices have other information

stored in IVIUT. Figure 2-10 depicts a sample refined element E, its

node in IRCT, and a sample node in IVURT.

The neighbors of any element can be easily found. For instance, in

Figure 2-10, the pointer location for the bottom neighbor of S1 is in

the node for E. while the pointer location for the right neighbor of S1

is in the node for the right neighbor E' of E. The node for K' (if it

- .----- !~~.

22

0-----------------o 0-------o----o--o

0- I.a-0 ----.--. 0III I 0 0 -.- 0II0

.-r 0-o- 0 0 ---0

0- -0------O0 0-0-------0-the 1-irregular rule rule (2.8)

0-

0I -- 0- 0rul i (29

Fiur 2-9 l-; 4: atr tiems rinm true

23

I f(E) I - - I C

Is IS I I V V I v I-0 1--- w Ef I_ 1 2 1 WIEs .I I w2

1s1 2I vi W 2 1 1V3 W2 V2 - 0 - -

I so I s1 S2 I S3

element E, with sons Sop ... S3 node for E in IRCT

I E lI I ElII---I III E I I null I

node for W. in IVERT node for W1 in IV MTE' refined E' not refined

Figure 2-10: Nodes in IRCT and IVEUT

exists) is pointed to in the node for W1 in IVUT.

Moreover, suppose the entire mesh is rotated 90, 180, or 270

degrees. Then IV T and the top row of IRCT remain the same, and the

last three rows of each node in IRCT are rotated. For example, if the

mesh is rotated 90 degrees counterclockwise, the node for E becomes

If(E)l - - I C

I V1 I v2 v3 vo

ueW1 W2 W3 dWe n eI S 2 S 3 1S O

Thus, the neighbors of any son of E can be found with the same procedure

used to find the neighbors of $T aae long as indices into the last

three rows of IRCT are incremented by the appropriate value sood 4.

24

This mesh data structure makes it relatively easy to construct the

finite element system for 9, to evaluate the app:xoximate solution. and

to refine the nesh. More data structure considerations are in Dank and

Sherman [141.

CNAPMIR3

Local order refinement

3.1 Introduction

Traditional finite element methods for elliptic problems employ

finite element spaces with low-order (e.g.. bilinear) basis functions.

These spaces are appropriate when the solutions to the elliptic problems

are not smooth. When the solutions are smooth, standard approximation

theory indicates that high-order piecewise polynomial basis functions

can yield much faster convergence. Thus, it is advantageous to be able

to use both low-order and high-order basis functions.

Local-order finite element spaces have received recent attention by

Babu'lka, Katz, and Szabo [3, 11, 12] and Babulka and Dorr [2). Full

local-order, local-mesh methods in more than one dimension have not yet

been considered.

In this chapter, we extend the finite element spaces with

1-irregular meshes considered in Chapter 2 to include basis functions

with locally-varying polynomial orders. In Section 3.2, we consider the

C case, and in Section 3.3, we briefly consider spaces of smoother

25

26

functions. In Section 3.4, we present simple data structures for the

spaces considered.

3.2 The Lagrange ease

Let M be a given 1-irregular mesh. In this section, we construct a

space S (= S(M)) of C0 piecewise polynomials such that, in each

unrefined element E in X,

smooth functions are locally approximated to order k 2 2. (3.1)

Our local order refinement follows the basic approach of

Taylor [29] and Babulka, Katz, and Szabo [3]. On [0,1], let the

polynomials (pi(x): i=l...,kmax) satisfy

Pl(X) = -x,

p2(x) =x,

and for i > 2,

Pi(x) is a polynomial of order exactly i,

Pi(c ) e P(1) - 0.

Since (Pl ..... ap )is a basis for the polynomials of order k > 1. the p

are called (C0) hierahc v9. lm23Jla.

On element E, Pi(z) and p (y) are defined by mapping [0,1J onto

(e,,Eehul and t[yy+h,]:

27

D .p.(x) DJpi ((-xE)/hE) hE-j,

and

DJpE(y) := DJp M((y-yE)/hE) hE-,

j=O,...,kmax-1.

Let pi(t) be the Legendre polynomial on [0,1] of order i. Since

Pi(t) = 1, and

Opi p 0O iJ,

a convenient choice for [pi is

= f; Pig1 (t dt i 2 2. (3.2)

We define S to be the span of three types of basis functions

(B (x,y)): vertex, side, and element.

The vertex basis functions are the piecewise bilinear basis

functions from Chapter 2, with B I(V) M 61i for all regular vertices V1.

The side basis functions are products of piecewise linear functions

in one direction with quadratic or higher-order hierarchic polynomials

in the other direction. Each side a of an unrefined element, not

strictly contained in a side of another unrefined element, has side

basis functions with hierarchic polynomials of order up to

28

(3.3)k : max(k E s contains a side of unrefined element E).

A basis function for side s in Figure 3-1 is h()pE(y).

II all

II B s-....II all

hlx)y.

h(x)

Figure 3-1: A side basis fuhction

The element basis functions for an unrefined element B are the

functions

Element basis functions for E are nonzero only in E, and only occur when

k 2 5.

Since the mesh is 1-irregular, IL2c basis function B. for S is of

the form p (x) Ey(y) in each unrefined element E, where Ex is either E

or f(E), and Ey is either E or f(E). Moreover, (3.3) and (3.4) imply

that in each element E, any monomial z y with i+u-I I is a linear

combination of restrictions to E of basis functions, so (3.1) holds.

-! - .... T

29

Figure 3-2 depicts the values of the basis functions in an element

in a sample local-order finite element. Note that many of the basis

functions are of order greater than kE = 3.

3.3 The Hermits case

The construction in the last section extends immediately to spaces

of globally Ca- 1 functions for any integer a 1 1. The hierarchic

polynomials satisfy Hermite interpolation conditions on their first a

derivatives at 0 and 1, so kE 2 2a. Vertex, side, and element basis

functions are defined as in the C case. There are

2a vertex basis functions for each regular vertex,

(kE-4a)(kE-4a+l)/2 element basis functions for element E

(none if kE I 4a),

and

a (ks-2a+1-i) side basis functions for side s.

and condition (3.1) holds as before.

3.4 k soah data struoture for local order refinement

In this section, we extend the mesh data structure from Section 2.5

to handle local-order finite element spaces.

For each unrefined element E, we store -kE in IRCT, in the location

in the node for f(E) which will contain a pointer to the node for E if E

is refined.

30

mesh:

O---...0-----0- ,0

I I I I

o-------- k-4 II El

I k-3 I k=3 I0----0--0 0

I Ik- I I0----II I I II I I I0--0-0- 0

basis functions in element E :

E E *1P3 (2)p 2 (y)E E f( ] )Ipl~x)P(y~l ' . , f(E(y)p2xp

0

0-E1 E

B 1P4(x)P1 (y P

p E BX~ E ( ()

Figure 3-2: Lool-order basis functions

Let the function NDV map the basis functions for S to the regular

31

vertices in the mesh, such that

1. MBV maps each vertex basis function to its vertex.

2. MRV maps each side basis function for side a to the

last-created regular vertex at an end of a.

3. RE3V maps each element basis function for element E to the

center vertex of f(E).

A sample mesh indicating MBV is shown in Figure 3-3.

~Iv-S-v--- V---S- V V :vertex basis functions

S E S E S : side basis functions

V--S---V--S--S E S E : element basis functionsI I I IS E S E II

I4 I : basis functions sappedV -S--V---S-- V- ----- S_ba V to vertex VI IjSE sE II I IV -S--V--S- - S E S

S E S E I

[V -s---v--s-= v-..

Figure 3-3: Basis functions mapped to vertices

We order the vertices and basis functions for S so that the basis

functions mapped by NOV to vertex V areIV

BMVB(IV) ' 8 MVB(IV)+1 . , BMVD(IV+l)-1

32

and we associate a nuamber IBFT(IBF) with each basis function BIBF

encoding the spatial relation to the vertex NDV(BB F) (o.S., left, lower

right, etc.).

With the data structure from Section 2.5, and two additional arrays

containing MVB and IBFT, we can easily determine the basis functions

which are nonzero in a given unrefined element.

As in Chapter 2, this mesh data structure makes it relatively easy

to construct the finite element system for S. to evaluate the

approximate solution, and to refine the mesh.

C(APERU 4

A posteriori error estimators

4.1 Introduotion

The traditional error estimates for the finite element method for

elliptic problems are a priori (e.g., Ciarlet (17]), predicting the

expected rate of convergence of the error to zero, but not saying much

about the actual error for a given finite element space.

In recent years, locally-computable a posteriori error estimators

* have been developed, mainly by Babulka and Rheinboldt, which estimate

the actual error for a given space. The landmark paper in this area is

by Babu'ka and Rheinboldt [81, giving a method of constructing a

locally-computable error estimator which is within a multiplicative

constant of the norm of the actual error. Further results for

one-dimensional problems are given by Babulka and Rheinboldt [6, 7. 51

and Reinhardt [24]. Further results for two-dimensional problems are

given by Babulka and Miller [4]: under suitable assumptions, they prove

that the error estimator used in their code (with piecewise bilinear

basis functions) converges to the norm of the actual error as the mesh

size goes to zero.

33

34

In Section 4.2, we present some terminology for discussing

a posteriori error estimators for two-dimensional Neumann problems. In

Section 4.3, we briefly describe the error estimator presented in [8].

In Section 4.4, we present a new error estimator which has two main

advantages over the estimator in [8]:

1. Under reasonable assumptions, the new error estimator is

asymptotically an upper bound on the norm of the true error.

2. The new error estimator can be computed locally in each

element. (The one in [81 must be computed over more

complicated local regions.)

In Section 4.5, we prove the upper-bound property of the new error

estimator, and, under suitable assumptions, we show that the now error

estimator is no larger than a multiplicative constant times the norm of

the actual error. In Section 4.6, we present several cheaper

alternative error estimators. In Section 4.7, we extend the estimators

to handle more general boundary conditions and differential operators.

In section 4.8, we summarize the results in this chapter.

In Chapter 7, we present numerical evidence that, in some cases,

our error estimators converge to the norm of the actual error as the

nosh size goes to zero. We have not been able to prove this

convergence. Currently, we can only prove convergence for our error

estimators for one-dimensional problems [15].

35

4.2 Terminology

Consider the model Neumann problem

Lu := - D (a(x,y)Dxu) - D (a(xy)D yu) + b(x,y)u f(x,y) (4.1)

in the interior of f0,

au/8n = 0 on 80,

where D , D , and a/8n denote derivatives in x, y, and the outwardx y

normal to 0 respectively, a(xy) and b(xy) are in L(0), f(xy) is in

2L (0), 0 < a . a(xzy) i a, and 0 < b S b(x,y) S b. Variants of (4.1)

are discussed in Section 4.7.

Let w be an open subset of 11 with piecewise smooth boundary aw.

For a non-negative integer k, and a function v in CO(w), we define the

norm

11V112 != ko J-O (D• 'Di-Jv(x'Y))2 dx dy.

Let BkW) denote the completion of Z(w) with respect to the norm

11.11 iksw) is equivalent tok,w

(v in L2(w) : IVikw <.1,

if derivatives are defined in the distributional sense (e.g..

Ciarlet [171, page 114). Let k(s) denote the completion of the

functions in C(w) with compact support in a with respect to the norm

11'11 When k is omitted, k = 0 is assumed. For v,w in 0(a)k,.1

36

(= L2(w)), let

(vw) : f1 v(x,y) w(x,y) dx dy.

For v,w in H 1w), let

a (vw) - (aD xv,D w) + (aDy vD w) + (bv,w)

If w is the interior of Q, a (u,v) is obtained by multiplying (4.1) by a

function v, and integrating over w using Green's theorem. Let the

n.eraz lllvill := a (vv). By the assumptions on a(x,y) and0

b(x,y), there exists C > 0 independent of v, such that

C 1 Ilvll S. IIIvll 5.C 11vll', (4.2)

for all v in Hl(w).

On the boundary 8w, for v and w in L2 (w), let

v'w> ae: w Lv(o) w(a) do,

where do is arc length. We define the norm

SC denotes a generic constant, not necessarily the same in each

instance.

Si6

37

Iv12 : <v,va V)

Let a/an denote differentiation in the direction of the outward normal

to o, with the limit in the differentiation taken over points inside w.

For brevity, we sometimes replace v by its closure in names for

spaces, norms, and inner products: for example, I1VIlk,E stands for

"v''k,i nterior(E) . When the subscript w is omitted in norms and inner

products, w = interior(Q) is assumed.

Given a 1-irreSular mesh N with unrefined elements (E), we choose a

finite element subspace S (= S(M)) of ff1(Q) as in Chapters 2 and 3. For

simplicity, we assume S is based on C hierarchic polynomials, as in

Section 3.2. We also choose a larger finite element space S ( S(M))

containing S, constructed in the same manner as S, such that in each

unrefined element E,

smooth functions are locally approximated to order kF ) kE

Since S contains S, and the basis functions (B1) are hierarchic, the

basis

(B1 : BI in SI

for S contains the basis

(B1 B1 in S]

38

for S. For each unrefined element E, we define SE to be the space with

basis functions

(B 11 E in supp(B1) B I in )

where (B I E in supp(B1) are linearly independent because of (2.2).1 EI

(3.3), and (3.4), and we define S E to be the space with basis functions

(B I 1 E in supp(B 1), B I in S).

We define S CSEto be the span of the functions

(B IE: E in supp(B1) B1 in 8,. B1 not in S).

For any function

V B 'S I B IB

in SE , we define

Yv Ia vIBI IE(4.3)

in SE Then v-1E (v) is in 'E7 ad for any function

V BI i vi B I

in S, the function

I(v) B I as v B I

is in S, and is equal to I (v) in E.

39

Let N(E) denote the set of unrefined elements which intersect with

E at more than one point. For example, in Figure 3-2. N(E) is the set of

labelled elements.

For convenience in discussing functions in the product space

M (a) (v: v in H (E) for all E),

let

a(vw) E E aE(vw) for v and w in HM

and

IIvI12 V E l E for v in HM(a).

The weak form of (4.1),

a(uv) = (fv) for all v in ]Jm)

has a unique solution u in H1 () (e.g., Ciarlet [173, page 19). We

impose the weak form of (4.1) in S to get a finite elental arnroximation

U in S. where

a(Uv) = (fv) for all v in S.

The ±uu for S is e :- u-U. Similarly, U in S is required to satisfy

A(lv) = (f'v) for all v in S.

1i

40

and the corresponding error for S is e :- u-U.

In this chapter, we consider ways of estimating the norm of the

error 1llelll. Let

a := U-U.

We assume the saturatio condition that there exists 0 y< 1 such that

lll~l l lll lll.(4.4)

If u is sufficiently smooth, then y goes to zero as h goes to zero.

Thus, since

we concentrate our attention on estimating IIIlt.

4.3 A Diriehlet a posteriori error estimator

Traditionally, the error estimates for the finite element method

have been a priori, predicting the expected rate of convergence of

IIleill to zero (usually as h :- max(h.) goes to zero), but not saying

much about the actual error for a given S.

In [8], Babulka and Rheinboldt present a posteriori error

estimators constructed basically as follows. Lot (B) be a subset of

the basis functions for S which form a partition of unity on 9. For

each 3, let :supp(B and let e be the solution in fO(Q) of the

local Dirichlet problem

41

a0 (e,v) = (f'v)O a. (U.,v) for all v in R10(0")

Under suitable assumptions, Babulka and lheinboldt show that there exist

C1 > 0 and C > 0 independent of S such that2

cI l 2 Illel111.1112.

In practice, since H0(Q ) is infinite-dimensional, e is not

computable. The most straightforward computable approximation of a3 is

@J in

S := (p(xy) : p in S, supp(p) contained in 03)

defined by the equations

a (siov) (f'v) - a (U,v) for all v in S (4.5)

Then

c m i2 C2 llll (4.6)

The quantity (TIlls1112 )112 is called an e*zor eiMas .

Each a in (4.5) is a projection of a in the energy norm onto a set

of functions, so the right hand inequality in (4.6) arises naturally,

while the left hand inequality is harder to prove.

42

4.4 A Neumann a posteriori otter estimater

The motivation for a Neumann a posteriori error estimator is that

it is more Important to have an upper bound on ilsli than a lover

bound. We construct an error sinulator a, such that a is the projection

of s onto S with respect to the energy norm. Then 1iJJ S1 111;111

arises naturally. Following Babulka and Rheinboldt [10], we call

I I an error estimator, and for each element E, we call 1116-111 E an

error indicator.

To motivate our construction of a, suppose u is in 12(E). By

Green's theorem on element E,

ao(e,v) - FLIV) :- (fv)E + <aau/n,v> - a(U,v) (4.7)

for all v in H1(E). For a given v, we can compute every quantity in

(4.7) except au/On on the boundary of E. Suppose E shares side a with

element E'. On a, a possible approximation to aau/SnIE is the average

value of aU/OfnE on E and E', where n. is the outward normal from E.

Since

LSj~1j3 - 'Is '

we define the approximation

[aOU/On] IE : 1/2 ( aU/Dnl - aOU/nl, , a not in 8a. (4.8)

10, B *in Ol

43

Note

[aOU/n]I E - [aBU/8n] EP

The resulting approximation to FE(v) is

FE(v) :- (f'v)E + <[a8U/8n],v)SE - aE(U,v). (4.9)

Since aE(.,.) is elliptic, we can define an error simulator for e in E

by replacing FE with FE on the right hand side of (4.7). However, as

approaches zero, the finite element matrices for the error simulator

approach an indefinite matrix corresponding to the case when b(x,y) = 0.

Thus, we formulate error simulator equations which are

automatically consistent when b(x,y) = 0 and aE(',') is indefinite. We

approximate s by the function aE in SE defined by the equations

aE(E*V) = FE(v-IE(v)) for all v in SE (4.10)

where IE(v) is defined by (4.3). Since a E(.,) is elliptic, aE exists

uniquely. Since v need not vanish on DE, (4.10) is a Neumann problem.

Moreover, since IIE is in SE

IE(1) 1,

so the consistency condition aE( EO) - 0 is automatically satisfied.

Equations (4.10) are equivalent to choosing aE in SE such that

44

aE E tE(v) for v in

for v in S3

The corresponding matrix equation is A , f for the coefficient vector

of ;E where

A:= A I 2 r:- I . f:I 2

the coefficients of the basis functions for SE are in ;1 and the

coefficients of the basis functions for SE-SE are in 2

We define the value of the error simlator ; at (xy) in Q to be

8(zy) :- a (Xy),

where E is an unrefined element containing (xy). When there is no

danger of confusion, we refer to a3 as a. Note that a is in %(1), but

not necessarily in Hl(Q).

4.5 Properties of the Neumama error estimator

The most important property of ; is that

Thmorn. A: IIslll 11 ll;Ill.

Proof: Suppose v is in S. Since (aOU/ni - - (aOU/nJi, and

[aU/nJ - 0 on 80,

45

E <[aaU/anv> DE = 0.

Thus,

B F(v') - E V - aB(U'V)) (f,v) - (U'v).

Since I(v) (page 38) is in S.

E FE(IE(v)) - E FE(I(v)) " (f,I(v)) - a(Ui(v)) 0. (4.11)

Thus,

a(sv) - a(ev) = (fv) - a(Uv) = E E(v-IE(v)) = a(;,v), (4.12)

and, in particular,

ll s l 2 = a(ss) = a e : l l l l l l l l

By (4.4). we have

Corollary 4.2: (1-T)Illl I1 1.

This is a jIokl inequality. We cannot generally expect

IllolllE S C I1l;lllE : suppose u is smooth, S contains bilinears,

hb nh in region R in O. and hE h >) h away from R. In region R, U is

a finite element approximation to the solution to

L(u.) = f(xy) in the interior of R, (4.13)

ul U on aR.

46

Let E be an element in the interior of R. Since the error simulator

equations in E are local, they are the same for (4.13) as for (4.1).

Since (Theorem 4.6) 1I1;11E is bounded by localE de by ocalquantities behaving

like IIIU-uRIIIE , and the boundary error in (4.13) pollutes the

solution throughout R, we expect

E >> III 11 E

(In practice, this may be acceptable, since the error is bigger away

from R due to the coarse mesh, and the small size of the error simulator

is in accord with the fact that no more mesh refinement should occur.)

By the triangle inequality and (4.12),

Ililill - Ilillll S Il-sill - inf(IIl-flll: f in 5),

so we can get an a posteriori upper bound for IIlll-IIlllI by

measuring the difference in the energy norm between ; and any function

in S. For instance, by averaging ;, we can construct an approximation

ave to 6 in S: s-s approximates the size of the Jumps in a across

element boundaries. If

AJ veJ JlJ

with y ( 1. then

Of course, the Jumps in a can be made as small as desired by adding

47

penalty terms to equations (4.10). but the resulting equations are

non-local.

We now prove a local a priori upper bound for IIIs Ill' in terms of

quantities depending on e. We assume there exists a constant kmax such

that

kE S. kaiax independent of S. (4.14)

We start with a few lemmas.

Leoma 4.3: 1111E(S)IIIESC IllEIIIE.

Proof: Suppose v is in S. Since the dimension of S E is finite andI

is linear,

WbIE(v).IE (0))E I C1 (bv,v) E

where C 1is independent of v and hE . (Otherwise, there would exist w

in S Esuch that (bww)E - 0 and WbIE w)IE (w))E 0 0. which leads to a

contradiction.) Similarly, since the dimension of BE is finite, X is

linear, and IEM =1,

(aD I (v),Dx1E(0) + (aDyIE(v),Dy1 (0)

C 2 ( (aD xvD v) E + (a y vD yv)1 E

where C 2 is independent of v and h3E Thus,

48

1102

2 2IIz~lE _B mICl2) IIvl E .

Let v=

Lemma 4.4: l-1R;)l ,idini~ j~: 1 E 7E(E) 1 aE C hi' 'E"'E

Proof: The trace inequality

- E)2' <- 1 E1,)(.5IE- E a - hIlE_ chEE-E EIE + E I EEA h 14

follows directly from Theorem 3.10 in Agnon [1].

Supposev is in E For 0 S xny < 1, let

v,(x,y) := v(x, + hE z, YE + hE Y) "

a (x,y) =E(v)(xE + hE x, YE + hE y).

Since the dimension of SE is finite, I E is linear. and IE( - 1,

11 12 SC(Iv112 + I 12

and thus

1IT-1(V)112l2 vII2tiV*-*I(v)t ES h 1.3t,,l

where C is independent of v and hE . The result follows by letting

v = 'B in (4.15) and using (4.2) and Loima 4.3.

vjft J.5: If u is in S2(3),

.. .. .... . .. --- . . ... .

rI

49

laae/nlaB SchE112 1 1Il1l. + 2 11.112, ./

Proof: Since u is in 92 (E), e is in 92 (B). The result follows directly

from Theorem 3.10 in Agmon [1].

The following theorem shows that the error is large in local

regions where the error indicators are large (and u is sufficiently

smooth).

Theorem 4.6: If u is in H2 (E' ) for all E' in N(W), then

l 1,2 1,12 I12.

IIElllB Ci E'sN(E) Ill, eE + ' l ll2,E'

Proof: For any v in 'E

a E(EV) -(fv-E(v))E + <([aU/n]'V-IE(V)>,E - aE(Uv-IE(v))

- FE(v-IE(v)) - (au/an'v-I,(v)>,E + ( [aaU/Dn ] .v-iE(v)>)B

SaE(e,v-IE(V)) - ([abo/8n],V-IE(v))BE

Let v = " By the Cauchy-Sohwarz inequality and the definition of

[-],

i t ~ ~~~labe/DnIDB s ;-+e > o

EsIN(E) SEP /8n1.E, )

By Lma 4.3 and Lema 4.4,

- w

50

-1/2 (4.16)I'I EIE l l + C hE 1 E E'sN(E) OaBe/OnI " 4

By Lema 4.5, noting that (since the mesh is 1-irregular)

hE1 /2 hE - 1 /2 C for all E' in N(E),

Il;ElllE C IllelllE+

11. 2 1l 2 )1/2.

E'sN(E) Iell + hi 2,E'

The result follows by noting that E is in N(E) and using (4.2).

If u is in H2 (E), we expect the right hand side of Theorem 4.6 to

converge to zero at the same rate as 11le11 If the coefficients

a(xy) and b(x,y) are sufficiently smooth, u is in B2 (C) (e.g.,

Ciarlet (17], page 138).

We now prove a slobal a priori upper bound for 1l1J11. We assume

that u is in 92 (E) for all E, and we assume the saturation condition

that there exists 0 j 7 ( - such that

B"B la8 a/nla ) S y Jlell. (4.17)

If u is sufficiently smooth, then 7 goes to zero if h goes to zero.

Theorn4.,1: There exists C independent of S and S such that

JJJahJiJ C 111o1JJ.

51

Proof: It follows from Theorem 3.10 in Agnon [1 and (4.14) that

* -1 85/OIOE .C 1 1.1d2.. + 2 11.112,

-1 2

C -1 IIsII81.9

Then, using ahe/On - ae/Sn + abc/On in Equation (4.16),

III;EIIIE 1 C 1110lllE + C hEl12 L E'sN(E) la./OnlB' +

C I sN() IIEll,

Squaring, summing over E, using the triangle inequality, and noting that

the maximum number of N(.)'s in whioh any element appears is bounded,

JJJ;iJJ SC 111e1i1 + C 118111 + C h lsaho/Sna12. 11/2.

The Theorem follows using (4.2), noting that 111*111 11 0lel, and

using (4.17).

4.6 Alternative *mz estimaters

In practice, there are several alternativ, error estimators that

are loss costly to compute than IlleIhl.

4.6.1 Ipnring irregular •0aeon

To can compute an error simulator in eloent 3 by replacing ZE(v)

with an analogous function I (v) which iaterpolates v locally in R. The

resulting error simulator, in effet, ignores the irregularity of the

corners of , and its system is slightly cheaper to assemble than the

52

system A v. f. In practice, the resulting error estimator appears to

be at least as accurate as 11lla111. However, Equation (4.11) in the

proof of Theorem 4.1 break do"e when v Is a side basis function for a

side s containing an irregular vertex (Figure 4-1): 1 (v), the analogue

to I(v), is not continuous across s. Thus, we do not consider this

alternative error estimator further.

(0) (0)- (0) (0)

)-( *() I(0)-(O) -- I0

I 11 ()

Figure 4-1: Ignoring the irregular corners

4.6.2 Using the matrix for a local Poisson problem

We can form and solve error simulator equations using a matrix

corresponding to an associated local Poisson problem. For v and w in

jj( ):- jL(E) ( (DxvDxw) E + (D yvDw) E)

where jj(E) Uinf(a(z,y) in E).

thtInstead of choosing ;E in 8Eby (4.10), we can choose aRin SEsuch

53

5I ( E v) - FE(v) for all v in S SE (4.18)

Lo for all v in SE #

and

(0E,1)K 0. (4.19)

(We impose (4.19) because &,(1.1) 0.) Ve construct the error

simulator a and the error indicators (111111 from se in the same way

we constructed a and (111;111E ) from ;E "

Equations (4.18) correspond to the matrix equation A f for the

coefficient vector v of a (xy), where

A, A EvA I I T v -II

I A I I _v I I f2 I

the coefficients of the basis functions for SE are in , and the

coefficients of the basis functions for S-7S are in _2 . Since

11Is1J111 III6+c1 R for any constant C. we can set

reorderinS the unknowns if necessary, so that A is positive definite.

For fixed kE , A depends only on the reSularity of the ceorners of R, and

not on h.

54

Ij. .u. 4.8: If a(x,y) is Lipschitz-continuous, then

II 1 E X ! ! E (1+o(h)) I111

as h oes to zero.

Proof: By (4.10) and (4.18), since QE(vJv) 5 aE(VV) for all v in SE

II IE -&.E(6,8) .. ..18ll 1llE 1_I1; I1E lllsll E III;III E .

Similarly,

2

-E--E8)1116111 18.*)II81E A E(s, =E( I E ll~l E •

and if a(x,y) is Lipschitz-continuous in E, so that

a(x,y) I #(E) + O(hE) in E, then by (4.19),

11111 E I C hE 1 I E11, ,

and

I11 1 1E (1+o(hE)) IIIllE

as hE goes to zero.

Since mesh refinement occurs in a very regular fashion, we can

precompute the factorization of A. and only incur the cost of a

backsolve plus computation of the right hand side for each element. If

the hierarchic polynomials (pi) are chosen by (3.2), then since

0 pip; 0 , i j J , max(ij) > 2,

'S

A is sparse. In practice, we replace the quantity A(E) with the more

convenient quantity a(xE+/2,yE-2). Then

Il;IlS IES (1+O(hF)) 1115111E (1(h ) 111;II .

4.6.3 Using a smaller matrix

We can also compute aE in SE-SE by the equations

AE(aE'v) - gE(v) for all v in SE-SE . (4.20)

constructing the error simulator a and the error indicators (Il1111 )

from *E in the same way we constructed a and (III;I) from a-,

Equations (4.20) correspond to the matrix equation A2 v 2 f2 * or

v = f, for the coefficient vector v of 's. where

We now show

Thoren 4,9: There exists a constant C independent of h such that

Proof: Th. left hand inequality follows from

l- Tj = ;Tf - T - £(s,5) j ilSlIlI l!ll.

11,11E vvAYi~~)jIIal 1811

56

The following approach to the right hand inequality is due to

Eisenstat (19]. Since A is positive definite, using block Gaussian

elimination, A4 Y2 - f2 , where

T -1A4:= (A2-A 3 A1 A3 )

is symmetric and positive definite. Then

2 T -TIlIsillI f 2 A4' f2E 24 2

Similarly, A2 is positive definite, and

2 T -TIIIEIII = f2 2 f2

Since A4 and A2 are finite-dimensional,

2 vT AT vT A2TC :=maxv v v A v<

v0

In particular, let v = f

For example, if E has no irregular corners, a(E) 1, SE contains

bilinears, and S-SE contains the biquadratic basis functions

(XE+hEx) (YE+hEY)(YYE) / h 3

(x-zE) ( h (Y-Y) / 3

Eh ., (IRE) (y +h y, / h

(z +h

E....) (..zE) (Y.YE) ..

57

then

6 0 0 0 o 0 0 0I 0 4 -1 -2 I I 1 1 -1 -1

A =10-1 4-1 I/6, A3 = -1 1 1-1 1112,1 0 -2 -1 4 I-1 -1 1 1

113 0 2 0 I1 0 13 0 2 1

A=1 2 0 13 0 1/90,I 0 2 0 13 I

and C (11/6)1/2 ; 1.354.

4.6.4 UsinS the residuals and Jumps in normal derivatives

We may choose not to solve any linear systems.

By Green's theorem, for v in SE7SE

a(Uv) = (LUv)E + (aaU/8n,v>8E

so

'E(sE.v) = (f-LU.v)B + ([a8U/an]-aaU/8n,v>OE.

In particular,

II; 1112 - (f-LU, )E + ([aU/On]-S8U/8n,sE >E S

and so

I-

58

111-8E IIE C ~1 (h E'-&(E)) hif-Lull E + (4.21)

C 2 (hEA(E)) l[a8U/8n]-aaU/anl.aE

where

C 1 hEA E)) sup Ilvhl /h1lvhllveS+ E-

C 2 (hEA(E)) :sup IVI aE -IIII

8+

and S+ :=(v :v v v0(x,y)+C, v0 in SE7SE ,v # 0, and (v1l)E 01.

For example, if E has no irregular corners, S E contains bilinears.

and SE-SE contains the biquadratic basis functions on page 56, then S+

is the span of

hE 3 (xehE~x) (yE+hEy) (yyE - 1/12,

hE (xKE) (yhgy (YY - 1/12,

*~~~~ h 3 (zE.Ex)( ye) Ehy) - 1/12,

h E3 (xEhE-x) (x-zE) (yyE - 1/12,

* I and

1/2C I(hE4j(E)) - hE / (22 ACE))

C2(hE..A(E)) - C 3 hE /(11 &CE)) )1/

Babulka and Rheinboldt [9) suggest error indicators when S contains

piecewise bilinear basis functions which amount to approximating

11,1,12 h 2/(24jL(E)) 11f-Lull 2 + (4.22)

8E

for interior elements. Since 1/22 : 1.1 X 1/24 and 3/11 :1.6 X 1/6,

the two schemes are in rough agreement.

If the mesh is uniform, S contains piecewise bilinear basis

functions, S contains piceewise biquadratic basis functions, u is

sufficiently smooth, and h goes to zero, then combining Corollary 4.2,

Theorem 4.8, Theorem 4.9, and (4.21),

... 112 -2 j (12A(E)) 1 1/2 hllf-LU1 +

(2jk(E))- -/2 h1/2, I[aSUf n3-aOU/OnI ) 2,

where y goes to zero. Similar results hold for higher-order elements

and nonuniform grids.

In numerical tests, we have observed that quantities of the form

C (h'j,()) lfLUI1 + C2 (h 1 9 aj(E)) I[aOU/On]-a8U/On 12

or

E (C I(h,,a(E)) hlf-Lull5 + C2(hR,gj(E)) l[aOU/Dn1-a&U/DnI, )2

only seem to approach 11l@1112 to within a multiplicative constant

60

depending on u. In Chapter 7, we present numerical results using the

error estimator

, :: E 2 1/2

where

2 K(k )(h2/(E) hf-Lull 2 + 4hE//j(E) i[aU/8n]-aU/8n12 ) ,(4.23)

=E E E E O

and

K(kE ) := (kE(kE+l)(kE+2 ))-l. (4.24)

E is chosen to extend formula (4.22). and to perform well on the

problem

D2u - D2 u = f(x,y) in 0 , u = 0 on 80,

x y

when u = sin(nx)sin(ny), k E and hE are uniform, and h goes to zero.

4.7 Extensions

If b(x,y) = 0 in 0, in order to insure nonsingular linear systems,

we impose the Neumann consistency conditions (u,1) = 0, (U,1) = 0, and

(,1)- E = 0. We still have

II1vlllE .i c Ilvlll,E for all v in H (E).

and globally,

1 0 1 C 1 11

61

Since (;E'1 )K 0 implies 11;E'E I C h "'l

LE I13 C 111;111E

for h sufficiently small. (Similarly, 1 ,9I~ C '1 ";1 1E . Thus'

Theorem 4.6 and Theorem 4.7 hold for h sufficiently small.

If a E (-,) is of the form

a(v~w) - (az,y) D Zv,D 2) E + (a~zry) D yvD yw) E +

(bx,y) vDV) E + (o(x~y) Dxv~w) E + (dxy) D yv~ w)E

then, under suitable assumptions, a112 .. ca(, and

liii j C a E(.es) as h goes to zero (see Schatz [271). Thus, Theorem

4.6 and Theorem 4.7 hold for h sufficiently small.

For problems with non-homogeneous Neumann boundary conditions

a~uISn - g(z,y) on 80 . £ in H (80),

if we set

a(u,v) - (f,,) + (s,v> for all v in Elm0,

a(U,v) C f,) + <gSv0 for all v in S.

and

[a#U/8a1 :-U~,y) if a is on 0.

62

then we obtain the same results as before.

For problems with homogeneous Dirichlet boundary conditions

u = 0 on fl,

1the solution u in H (0) satisfies

a(uv) = (fv) for all v in Hl0(Q).0

Similarly, the equations for U, ;, a, and a are now posed on subspaces

of KI(N). Since all the functions of interest now vanish on a0, the0

other results follow as before. Problems with non-homogeneous Dirichlet

boundary conditions which are exactl satisfied are handled similarly.

When Dirichlet boundary conditions are not exactly satisfied by U,in element E touching 80, we define a E in SE by the equations

aE(;EV) = FE(-IE(v)) for all v in SE which vanish on W0,

E on 80.

We define a from aE as before. Let ab be any function in S such that

Equation (4.19) is no longer needed in elements touching 80.

63

;0 "" -b vanishes on 80, a let 10 -;b B y the ame proof asfor Theorem 4.1,

Iol0111 . 111;0111,

so

111-11 1le l 11 II;0111 + 111;bill

Similarly, under the assumptions of Theorem 4.6,111 21612 )112

lllSlll~ S C ' |t'sN(E) 1 ]lll t, + h1 , l 2,. , +/

lll;blll n

and under the assumptions of Theorem 4.7.

; " 11 B, x

where (B } is the basis for S, we choose

;b B I b I I

where

Sb: span(B B I in S. BI(x'y) 0 0, (zy) in 80).

Since the area of supp(b) is O(h), we expect IlbiII - o(1ll'IIl) as h

64

goes to zero. Since a is nonzero on 80, s should be in S rather thr-

s -S when E touches 80.SESE

Analogous results hold for a and s.

As in Babulka and fleinboldt [10], on an element side s of E which

touches 801, we replace the term

(kE) 4 hE/A(E) I[aaU/an]-&aU/anl25

with the term

3a(E)/h 1.12E a

in formula (4.23).

4.8 Sugary

The error indcators discussed in this chapter are:

Ill;s III * where E is in SE and

a(e(;e'v) f s (v) for all v in SE-SE

for allyv in S E

IllEslIIE , where a is in S8 ( Ie'l)E - 0. and

S(B$E'v) P FEWv for all v in %SE

0 for all v in SE

65

111R IIIE .vhor o E is in SE-SE and

-4E( E V) F FE(v) for all v in k-S,

and s where

2 2221F SM= ) h-E/jL(E Ijf I2 + 4 hB/A(E) I[aOU/On]-a8U/nI

The corresponding error eiatou are

t - E s111;1,12 )1/2.

1118111 - B I'3"'1 )1/2,

I~~l~~lLI 2 ( ~ ~ l l 1/2

and

- ' 2 )1/2.

We define the corresponding relative erors in the estimators

eBabulka and Rheinboldt call (the relative error + 1) the "efficiency

index" [101 or the "effectivity index* [9).

66

and

We have shown, for sufficiently smooth u, that as h goes to zero,

(14y) F1 1111S lili~C1111

and

where the positive constants C, C, and C are independent of h, and If u

is sufficiently smooth, y goes to zero if h goes to zero.

Heuristic refinement criteria

5.1 Introduetioa

The guiding principle in choosing a finite element space is to

minimize the norm of the error as a function of work. However, there is

not usually enough information available to choose an appropriate space

a priori. Instead, it is necessary to choose a sequence of spaces

adativel . Each succeeding space in the sequence is chosen on the

basis of criteria computable from the preceding space, with the criteria

tending to satisfy the guiding principle when a heuristic model of error

behavior holds.

When the polynomial orders of the basis functions are uniform, the

resulting refinement algorithms tend to approximately equalize the sizes

of the errors in the elements in the next space. In finite element

methods, for example, this heuristic refinement criterion is used in

Fried and Yang [20], Babulka and Rheinboldt [8], and Bank and

Sherman [13]. When the polynomial orders of the basis functions vary

locally, the heuristic refinement criteria must be somewhat more

complicated (e.g., Brandt [16]).

67

68

In Sections 5.2 - 5.5, we develop two heuristic models of error

behavior for local-mesh, local-order finite element spaces, and an

adaptive refinement algorithm which uses the models. In Section 5.6, we

discuss drawbacks inherent in the models. In Section 5.7, we present

the asymptotic expected error and work behavior for three problem

classes consisting of problems with:

1. smooth solutions.

2. solutions with point singularities.

3. solutions with line singularities.

Except in the presence of line singularities, with appropriate local

mesh and local order refinement, the expected error converges to zero

faster than inverse polynomially with respect to work.

5.2 A model of work

The guiding principle in choosing a "good" finite element space S

can be stated as

minimize 111oll as a function of W, (5.1)

where Ille1l is the error in the energy norm, and W is the work.

However, there is not usually enough information available to choose an

appropriate space a priori. Thus, we must choose a sequence of nested

spaces

S contained in S contained in ... Sn

with finite element solutions U1 ... ,Un , taking S - Sn (for instance,

because the work limit is reached). We choose each Si+l adaptivelz , on

the basis of information obtained from Si Since the spaces are

0-

69

nested, the errors (ei) and work counts (WI satisfy

Ills0 lI I Ill21 ... 2 IlianlIl

and

W1 < W2 < ... ( Wn

The work counts can represent either storage or computer time. If

the work counts represents storage, V V . If the work countsn

represents time, V - n and if

wi S T wi+1 , 0 J y < 1 , I 5 i 5 n, (5.2)

then W I (l-y)- i Wn . Thus, V is usually proportional to Wn . Since it

is possible that n = i+l, in the remainder of this chapter, we take as

the "local guiding principle" a local version of (5.1):

given Si , choose Si+ 1 to minimize tilei+lIIl (5.3)

as a function of Wi+I

(subject to the possible enforcement of (5.2)).

Let S be the current finite element space S in the sequence, with

mesh M, unrefined elements (R), and error e. Let z :- (,y) denote a

point in 0. Let h(), p(z), and w(p(z)) be piecewise constant functions

defined by

70

'x(z) hE the local mesh size,

p(z) PE kE-l = the local polynomial degree,

w(p(z)) WE = the local work count,

if z is in element E. We make the heuristic assumption that the work W

for S is

W = EwE (5.4)

This is reasonable if V is the dimension of S, or (since the mesh is

1-irregular) the storage for the stiffness matrix for S, or the number

of arithmetic operations needed to solve the finite element system for S

when the work is proportional to the number of elements in the mesh (see

Section 6.3). It is not reasonable if the work grows faster than the

number of elements: then, W depends on S globally as well as locally.

5.3 A model of predicted error behavior

In this section, we construct a model of predicted error behavior

based on standard a priori error estimates, and derive heuristic

refinement criteria which tend to satisfy (5.3) when the model holds.

Let m(z) and ;(z) be piecewise constant functions defined by

m(z) : max( k : Hull kE -

p(z) := min(p(z),m(z)-l)

'- I i °- . . o .. ..

71

if z is in element E. m indicates the maximum local smoothness of u in

E, and p indicates the expected convergence rate in the energy norm

using polynomials of degree p (order kE). Standard a priori error

estimates (e.g., Ciarlet [17], Theorem 3.1.5 and proofs of Theorems

3.2.1 and 3.2.2) yield

2 C(p(z)) a(z) G(zp(z)) h(z) dz, (5.5)

where a(z) is the coefficient in (4.1), and

G~z~) :'S ( DJ Dp+'11 u(z) ) 2G(Z'1=J0 x Y2)

is in L2 (0) by the definition of p.

Following Brandt [16], Chapter 8, we make the crucial assumption

that, for heuristic purposes, (5.5) holds in each element, with

eauality:

* IeIll : E :4E f C(j(z)) a(s) G(zp(z)) h(z)ZP(z) dz, (5.6)

so that

111e1112 C B

We also assume, for heuristic purposes, that

a and G are approximately constant in each E (5.7)

and

72

C, G, and V are (formally) differentiable in h and p. (5.8)

Assumptions (5.4). (5.6), (5.7), and (5.8) comprise our first heuristic

model of predicted error behavior. The form of (5.6) is important, but

not the specific values of C(p) and G(z,p). We make assumption (5.8)

only for convenience: all derivatives with respect to h and p can be

replaced by divided differences.

In the following paragraphs, using (5.4), (5.6), (5.7), and (5.8),

we estimate the marginal efficiency of refinement in each element E,

obtaining

Xh) =- (/O/hE) / (OW/ahE) (5.9)

for mesh refinement (h-refinement), and

I - (E() at/p) / (OW/apE) (5.10)

for order refinement (p-refinement). Let

X E:mar( X(h) X (p )

E: E E

p - - - -

73

We use these quantities in the following adaptive refinement algorithm:

1. Find the unrefined element 3 with the largest 1K "

2. Determine the refinement cutoff X CUT < XK (e.g., as in

Section 5.5).

3. For each unrefined element E with 2E I XCUT

a. If Xi X ) refine the mesh in E.

b. If )< X(p), refine the order in E.

4. Take Si+ to be the resulting finite element space.

Mb (p)We now determine 'i and kP). First, we estimate p in each

unrefined element E. Suppose the order in E has never been refined, and

(from a previous finite element space in the sequence) we have an

approximation 4f(E) to the square of the error in f(E) before f(E) was

refined. By (5.6) and (5.7),

4f(E) tE 2

so we estimate

p = ( ln(Cf(1 )) - ln(QB) ) / ln(4) - 1.

On the other hand, suppose the order in B has been refined, and we have

an approximation CE to the square of the error in element 9 with kS one

less than its current value. If

74

4E' tE iPC=T(.1

for some constant pCUT < 1 (e.g., as in Section 5.4), there may still be

higher derivatives of u in E which can be exploited, so we estimate

- PE " Otherwise, we retain our previous estimate of p. In order for

our estimate of p to be logically consistent, we restrict it to the

interval (0,PE

Using our estimate of p, by (5.7) and (5.9), we estimate

[h) : 2 2 a C G h2p)/8h / 2 w h-2)lDh} (5.12)

=- P2 (a C Gh 2p) h- / ((-2) wh - 3)

-(P / w ) C

in element E.

There are two cases to consider in estimating )p). If p 2 m-l,

refining the order will not change the error at all (in (5.6)). Thus,

i) = 0. On the other hand, if p ( m-l, by (5.7) and (5.10),

(P) = ((E) / w,

where w' :- aw/Op. If u(z) C sin(Ox) sin(Oy) in E, where 0 is the

approximate "frequency" of u in E, then

1luf 12 C 0 2(p+l) ,

p+l, Ie

and by (5.6), since C(p) (l - for Taylor's theorem,

75

-2 2 2(p~)24E C2 (p0 h * 0 l h2 ~(.3

=C 2 (0 h) 2 (p~l (p 1) 2

Since e is independent of h and p. by (5.13),

e p12h . (5.14)

Using the crude bound a(pf)/dp I p p 1/2,

O(C /ap: 4F (2 WnO h) -p)

so we estimate

() Z 4 t: (-2nWeh)+ p) /1w' if p m-1 (.5

0 ~if p2.-i.

In practice, we estimate 4E by the squared error indicator ttIIE

(or ai!i~ I~II or 62) in formulae (5.12), (5.14). (5.15). and

the formulae f or ;

Expressions (5.12) and (5.15) indicate that X.tends to decrease as

E Is refined. Thus, the adaptive refinement algorithm tends to

approximately equalize U.). When the order is uniform, this reduce* to

"asymptotic equidistribution" of the errors amo&$ the elements (eog.,

Fried and Yang [201).

76

5.4 A symetrized model of predicted error behavior

In this section, we construct a second model of predicted error

behavior based on syumetrized a priori error estimates.

In considering combined h and p refinement, Babulka and Dorr [21

present an a priori error estimate of the form

I1e11 2 Cknax) G(z,m(z)) h(z) 2P(z)P(z)-2(m(z)-l)dz.(5.16)

Since C and G are independent of p, this estimate is more symmetric in h

and p than (5.5). Moreover, it indicates the manner in which

convergence occurs when p > w-1. Theorems and numerical evidence in

[11] show the exponent in p is optimal in the presence of singularities

of u not located at mesh nodes, and that if all the singularities of u

are located at mesh nodes, the exponent of p can double. The effect of

this doubling will be considered at the end of this section.

We make the heuristic assumption that (5.16) holds in eanh element,

with eauallit:

eillE (5.17)

:= C(knax) fialz) G(z,m(z)) h(z) 2(z) plz)-2(m(z)-l) dz.

Assumptions (5.4), (5.7), (5.8), and (5.17) comprise our second

heuristic model of predicted error behavior. The resulting estimates of

) , ( p) , and (X can be used in the refinement algorithm in

Section 5.3.

4eL

77

In our second model, if the order in unrefined element 9 has never

been refined, then by (5.17) and (5.7), we estimate

In (l(f(E)) - n(Q) )/ln(4) - 1.

On the other hand, if the order in B has been refined, then if < in-I,

E- h (P/(P-1))

else if p -,

4E/ 4 : (P/(P-1))-2(n-)

Since the function (p/(p-1))-2(p-l) equals 1/4 at 2, and is monotone

decreasing with limit e- 2 as k goes to * we decide whether to increase

our estimate of p on the basis of (5.11), with pCUT ° 2 In order for

our estimate of p to be logically consistent, we restrict it to the

interval (DIP1 ].

As in Section 5.3, we estimate

(h) w ) .18)

If p < in-, we estimate

P) (-2 In(h) + 2 (m-l)lp) I w'. (S.1)

Otherwise, we estimate

(P) (" 2 (mrl)/p) /'.(5.20)

IE

78

In practice, we approximate CE by the squared error indicator

I11;1 E (or 1111 1 _ or 82) in formulae (5.18), (5.19),

(5.20), and the formulae for p.

Suppose the exponent of p in (5.17) is doubled. Then our estimate

(h)(p)of X is exactly as before, and our estimate of XB changes by a

factor of 2 when p r a-i, and by a small factor when p < m-1 and hE is

small. The main effect is to encourage order refinement when a is small

and p > m-1. But then (p)E is usually much smaller than X , so the

practical effect on refinement behavior is slight. In general, the form

of (5.17) is more important than the values of the exponents.

5.5 Choosing the refinement cutoff

A simple choice for XCUT is

X U C X

for some constant 0 < C < 1, where

u= ax( ).

If X is too large, (5.2) may not hold. If X is too small, weCUTCUmay refine an element with small XE when it may be more efficient to

refine an element with large XE twice. We can avoid this situation by

letting XCUT approximate the largest XE for a once-refined element.

This quantity can be estimated by the marginal efficiency of refinement

for element R after it has been refined once:

79

If (h), and the order in B has never been refined, we can estimate

~ 2

C U T = ( p K / vX ) C 2 / f B

If )X = and the order in E has been refined, then by (5.13) and

(5.12) (or (5.17) in our second model), we can estimate

4 -(p +I) h)CUT

If -x - 4 P), then in our first model, by (5.13) and (5.15), we can

estimate

x UT X(P 0 33)2 / (p3+1) ( w,( 3j+1)Iw'(p+1)

and in our second model, by (5.17)-(5.20), we can estimate

XEST- (p) 2 -2 1)( )G w'(3 +1)

5.6 Drawbacks

The heuristic assumptions leading to our estimates of X) may be

too crude to yield useful refinement information. We base an alternate

refinement algorithm solely on M h) and p:

8o

1. Find the unrefined element 2 with the largest (h)

2. Determine X < Xh)CUT 9

3. For each unrefined element E with XE 2 CUT

a. If p > p, refine the mesh in E.

b. If p 5 p, refine the order in E.

4. Take S to be the resulting finite element space.i+l

If the order in element E is refined, p is estimated by (5.11).y o ll(P1)12 11e12

Then if 4E and 4E behave similarly to IIle(p-l)lII and IIletII, p

tends to be one more than its (usually) optimal value in-i. Conversely,

E e l E , then p tends to be less than m-1. The latter

situation arises most frequently when ks > kE on the sides of E. Thus,

in our current code, if k 2 kE+l on any three sides of E, or k 2 kE+2s E

on any side of E, then the next time E is refined, the order is refined,

regardless of the size of p.

Lack of de-refinement penalizes early order refinement. For

example, suppose h j 1/4 and k 2 3 are desired near the origin. If the

order is refined first, then k 2 3 everywhere (Figure 5-1).

The resulting finite element space may not satisfy (5.3). even if

the heuristic models of error behavior hold. For instance, if

Ai) XiP), it may still be more efficient to refine K in h if the

resulting decrease in is greater with mesh refinement than with order

81

0. 0 0II II I I II I I I I k-3 I k-3 II I I I I I I

k-2 I I k-3 0 oI I I I I I II I I I I k-3 I k-3 II I I I I I I

0 0 0 O0

order refinement first

o 0 O 0-0 0 - CI I I I I I I II I I k-2 I k-2 I I k-2 I k-2 II I I I I I I II k-2 I o -I I I I TIIIII I I k-2 I k-2 1 I k-3 I k-2I I I I I I I I00 O -0 O 0 0 0

mesh refinement first

Figure 5-1: Early order refinement penalty

refinement, and if the other 's are very small.

Rigorous models are preferable to the heuristic models used in this

chapter. A rigorous approach for o&-dimensional problems is developed

in Babulka and Rheinboldt [7).

5.7 Expected error behavior

In this section, we model the asymptotic behavior of error with

respect to work for three problem classes consisting of problems with:

1. smooth solutions.

2. solutions with point singularities.

3. solutions with line singularities.

82

We use the heuristic model from Section 5.3. We expect similar behavior

using the heuristic model from Section 5.4.

5.7.1 A smooth solution

Suppose u is smooth everywhere in L, and h(z) and p(z) are uniform.

By assumption (5.6),

~C(p) h2

and by assumption (5.4),

p A p h72 , ft > 2.

Minimizing 4 while holding W fixed, 2 p/h - 2/h T = 0 and

((C'/C) + 2 ln(h))t + X P/p W = 0 imply

h Z-/2

and

C e- ¥ p ,

where

:= p + C'/C.

Thus, it is best to fix the mesh and only refine the order. Suppose

C(p) = A? , A > 1.

Since C'/C - ln(A), then A e = A e- P-ln(A) < 1, so

= (A e-Y) p

is exponentially decreasing in p. But V = pp e is polynomial in p. so

if there exists A < * such that C(p) S Ap for sufficiently large p. then

is exponentially decreasing with respect to W.

*~*~ -

83

5.7.2 A point siagularity

Suppose u has an isolated singularity at a point a in 0, such that

near a, u has l+a derivatives in L2 , and u is smooth away from a.

Suppose there are unrefined elements at levels 1 to L in the mesh for

SL, and at each level only the elements which touch the point a are

originally refined. Suppose p(z) uniformly equals L. Then

C~)2+2aCE C ha) + if E touches a

and

C(p) hi 2 if E does not touch a.

By the proof of (2.4), there are no more than a constant number of

elements at each level, so

C (p) (( 2-L) 2+2a + L-l ( 2 -I)2+P)

C1 (L) 2- L

and

V 2 = 1

- C2 LP +l.

If CIL) AL for some A 2 as L goes to, then q is exponentially

decreasing with respect to W. Babulka and Dorr [2] make a similar

84

conjecture, and under suitable assumptions, prove that convergence is

faster than polynomial.

5.7.3 A line singularity

Suppose u has a line singularity at a line a in 0, such that near

2a, u has 1+a derivatives in L , and u is smooth away from a. To fix

ideas, suppose p(z) is uniform, a traverses 0 from top to bottom, and

there are elements at levels 1 to L in the mesh for SL # where at each

level the elements which touch the line a are present in SL Then

2+2atE C(a) hi if E touches a

and

2+2p

CE Clp) hE if E does not touch a.

There are at least 2L elements with level L touching a, so

SCl (a) 2L (2-L 2+2a

and

W C2(a) 2

Then

C3 (a) W-1-2a

regzdlest of how p is chosen. Convergence cannot be faster than

polynomial, since the complexity of geometrically approximating a

asymptotically dominates the cost.

AD-AI15 432 'ALE UNIV NEW HAVEN CT DEPT OF COMPUTER SCIENCE F/A 12/1LOCALMESH. LOCALORDER, ADAPTIVE FINITE ELEMENT METHODS WITH A--ETC(U)DEC 81 A WEISER N0001 -76-C-0277

UNCLASSIFIED TR-213 NL2fl/lllllllll

EI//I/EE///EEIEEE/I//I///EEIE//EEE//IL

lii. 128 11.liiU; mm*3

1111.2 '*.40 Jl -6IIIIIj8

MICROCOPY RESOL011ON TEST CHARTNATIONAL 1ALPAlT At11 0 AN[PARPI11-

RAFIPU 6

Computational aspoets

6.1 Introduotiom

In this chapter, we discuss some of the computational aspects of

the methods presented in Chapters 2-5. Since we have not yet

implemented all the procedures discussed, detailed operation counts are

omitted.

In Section 6.2, we consider the efficient assembly of the finite

element system (of. Eisenstat and Schultz [18], Weiser, Kisenstat, and

Schultz [31]). In Section 6.3, we investigate the complexity of sparse

direct elimination with nested-dissection-type ordering of the unknowns,

for the three classes of problems discussed in Section 5.7. The

operation counts and storage for the problems with singularities are

linear (optimal-order) in the number of elements. In Section 6.4, we

consider the efficient computation of the a posteriori error estimators.

We now outline the computational tasks to be performed.

There are four computational steps to perform for each finite

element space S in the sequence (Si ).

i$

nl

86

1. Form the linear system A U = f, where

fB1) is the basis for S,

U(xy) = .u3 r(x,y),

A.A E E (I

11 E -BE

I = E f : (f'BI)"- L E II

After initializing A and f to zero, for each E,

a. Assemble A and fE. the element stiffness matrix and

right hand side for the basis (IE) , where (I is the

basis which would result if E had no irregular corners.

b. Obtain AE and fE from A and ?E, by changing variables

from (B to (B

c. Add A and f to A and f.

2. Solve A U - f.

3. For each B, compute Il'IllE , IIsI6_lE, Il!is ,or &E*

4. Depending on the error indicators computed in Step 3, either

stop, or adaptively choose Si+1 , and go to Step 1.

Each integral fj v(z,y) dx dy is approximated by

G(E) G)(E) t=E ws wt v(x sy), (6.1)8- I

87

whete (w,,,z) and (w tlyd are the weights and points for the G(E-point

Gauss-Legendre integration rules on Cxj+h1 and [yE9y1*h3 ]

respectively. Choices for G(E) are specified in Chapter 7.

6.2 Assembling the linear system

Consider Step l.a (Section 6). Fot simplicity. suppose

E (v~w) - (a(14") VMw)

the bilinear form for a weighted least squares problem, since

considerations for general a~% are almost identical.

Since (i1)is a subset of

sparse versions of the tensor-product assembly procedures in Eisenstat

and Schultz [18] and Weiser, Bisenstat, and Schultz [31] are

appropriate. If

B3- i(KP~)

B - p (z)p (y),I J n

then, depending on whether a(z,y) and f(z,y) are non-separable,

separable, or constant, Aand f3 can be computed using the formulae

88

(a~zy B1,J0

(w p (z )p (x )> a (x ) ~ #w~

3 2 1 W5 0I~P x 3 t(t,)ny)) aytut

(f W().5

The quantities in angle brackets 0> can be precomputed independent of E.

The quantities in squatse brackets 11 should be computed as intermediate

sims in each B. Since

NB ' - ~- -EAlla (i~s)(j'n) Ain(Jm (jA,m)(i.n) MA(Jon)(im)

only entries in A with i I .1 and a I a should be computed explicitly.

Consider Step 1.b (Section 6). For simplicity, we discuss Step 1.b

only for element E in Flours 6-1, when kE is uniform. Other cases are

handled analogously.

Figure 6-1t An element with one irregular corner

The [B I) nonzero in B have the values

E a

E W f(E) Y

Suppose

f(E), ns.8(2.3)U (y) - (v). (6.2)

Since refinement is regular, the values (vL can be precomputed

EEindependent of E, and Aand f can be obtained from the formulae

90

E -EA(i,m)(J,n) A(i,m)(J,n) i02. J02,

V,L A ) i-2, J02,

EA(jn)(im) i2, J=2

and

fE ?Em)i(i'm) in

The resulting computation is relatively expensive when kE is small.

When a(x,y) is separable, fever multiplies are required if Step l.b is

omitted, and AE and fE are formed directly.

6.3 Solving the linear system

Consider Step 2 (Section 6), solving the linear system A U - f. In

this section, we investigate the complexity of sparse direct elimination

with nested-dissection-type ordering of the unknowns for the three

classes of problems discussed in discussed in Section 5.7.

6,3.1 A smooth solutioa

As in Section 5.7, suppose U is smooth everywhere in Q. Suppose k

and h are uniform, and there are N :_ h72 elements. There are

(k-4)(k-3)/2 element basis functions nonzero in each E. In sparse

A

91

Gaussian elimination, if the element basis functions are ordered first,

their elimination induces no fill-in (i.e., they are zstnaticall

condensed 1281). This initial elimiation requires N((k-4)(k-3)/2)3/16

multiplies and :NC(k-4)(k-3)/2) 2/2 storage. Although this cost

dominates when k )) N. in solving problems to usual accuracy.

elimination of the element basis functions Is relatively inexpensive.

There are : (2k-3)N remaining basis functions Mk-2 side basis functions

associated with each distinct element side, and I vertex basis function

associated with each distinct vertex).

Consider a nested dissection ordering of the side and vertex basis

functions, In conjunction with sparse elimination of A (e.g., George and

Lin [22]). A sample nested dissection ordering is shown in Figure 6-2.

0- c--a 0 00

I _ I I I I I I I "I0-0 0 -- '- -0----O 0I 1 1

O-0 0 c- o 0 0--c--0

c-o-0 0

0 0 I r I a I I 'I4 3

I----- 0-0'--- -- __7_

92

1-4 correspond to basis functions on uniform submeshes with N14

elements: in each group the unknown& are recursively ordered using

nested dissection on the submesh. Since group 5 separates the unknowns

in the first four groups, the number of multiplies to eliminate the

unknowns in groups 1-5 is. to dominant-order terms, bounded by U(N),

where

1(N) - 4 M(N/4) + C (kNI/2 )3 .

The rightmost term is a bound on the number of multiplies needed to

densely eliminate the unknowns in group 5. The solution to this

recurrence relation is O(k3N/ 2) (e.g., Rose and Whitten [26). Theorem

1), as is the operation count to eliminate the remaining unknowns in

group 6. Thus, the expected operation count, neglecting static

condensation cost, is O(k3N3 / 2). Similarly, the expected storage is

O(k 2Nlog(k 2N)).

6.3.2 A point singularity

As in Section 5.7. suppose u has an isolated singularity at the

point ar(0.0). Suppose there are unrefined elements at levels 1 to L in

the mash, and at each level only the elements which touch a are present.

Suppose k is uniform.

There are N : 3L unrefined elements in the mesh. After static

condensation, : (2k-3)N side and vortex basis functions remain.

Consider a recursive ordering of the side and vertex basis

93

functions, in conjunction with sparse elimination of A. A sample

ordering is shown in Figure 6-3.

3 0

I I II I II I I

I 2 II I I

0------0C-

I II I I II Ii ' I I

0 I

I II I I0-o-0---o I I00-0 I I o o-o--o---- 0

Figure 6-3: Recursive ordering with a point singularity

The unknowns are ordered according to the groups numbered 1-3. Group 1

corresponds to basis funotions on a submesh with N-3 elements: the

unknowns in group 1 are recursively ordered on the submesh. The number

of multiplies to eliminate the unknowns in groups 1 and 2 is, to

dominant-order terms, bounded by 1(N), where

N(N) - I(N-3) + C k3 .

The rightmost term is a bound on the number of multiplies needed to

densely eliminate the unknowns in group 2. The solution to this

recurrence relation is 0(WN), and the operation count to eliminate the

remaining unknowns in group $ Is O(W3 ). Thus, the expeoted operation

94

count, neglecting static condensation cost, is O(k3N). Similarly, the

expected storage is O(k 2N).

6.3.3 A line singularity

As in Section 5.7, suppose u has a line singularity along the line

a - ((O,y)). Suppose there are unrefined elements at levels 1 to L in

the mesh, where at each level the elements which touch the line a are

present. Suppose k is uniform.

- LThere are N - 3(2 ) unrefined elements in the mesh. After static

condensation, ; (2k-3)N side and vertex basis functions remain.

Consider a nested-dissection-type ordering of the side and vertex

basis functions, in conjunction with sparse elimination of A. A sample

ordering is shown in Figure 6-4. The unknowns are ordered according to

the groups numbered 1-4. Groups 1 and 2 correspond to basis functions

on submeshes with (N-2)/2 elements: in each group the unknowns are

recursively ordered on the submesh. Since group 3 separates the

unknowns in the first two groups, the number of multiplies to eliminate

the unknowns in groups 1-3 is, to dominant-order terms, bounded by U(N),

where

31(N) = 2 1(N/2) + C (klogN)

The rightmost term is a bound on the number of multiplies needed to

densely eliminate the unknowns in group 3. The solution to this

recurrence relation is O(k 3N) (e.g., Rose and Whitten [261). and the

iT ~ a 2

95

---- 0o-4---o0-0-0 1o-o-ol I I 1

0-0-0---- 0o--0 I Ioo-o--o I I

0-0-0 I Io-o-o---o------ o-3 --

-0-0 1,-0-0 i : r -.0

0-0-0-0 "o-o-o 2 I0-0-0 - - -

0-0 --00-0-0--'0

0-0 1 Alio -oo--o---- 0-- .. 0

Figure 6-4: Nested-dissection-type ordering with a line singularity

operation count to eliminate the remaining unknowns in group 4 is

O((klogN) 3). Thus, the expected operation count, neglecting static

condensation cost, is O(k3N). Similarly, the expected storage is

2O(k N).

6.3.4 Discussion

The operation counts and storage for the problems with

singularities are optimal-order in the number of elements N. We expect

similar operation counts for more general problems with point or line

singularities. The optimality for the line singularity problem depends

on the fact that we can systematically find sets of O(N1 13-) elements

(a ) 0) which separate the remaining elements into two or more disjoint

sets of equal size.

96

We do not know an efficient (i.e., O(N)), automatic, way of finding

orderings such as those presented. See George and Liu [22], Sections

7.2 and 8.2, for work in this direction.

In the example problems, the orderings of the interior unknowns

correspond to the reverse of the order in which the edges associated

with the unknowns are created. This correspondence does not always

hold. (For example, suppose u has a point singularity at (1/2,0).)

6.4 Computing the error indicators

Consider Step 3 (Section 6), the computation of the error

indicators iia-III E IIiI E , l lil1l , or sE * Recall

FE(v) :- (f'v)E + <[aO nl,v> - aElUv).

First, suppose E has no irregular corners. Then the possible quantities

to be computed in Step 3 are

1. (a (B,B)), where (B) is the basis for SE which would

result if E had no irregular corners. These quantities can

be computed as in Step l.a.

2. [(f,BI)E : in Si-SE). These quantities can be computed

as in Step l.a.

3. [E(UaBI) BI in 1j-SE First, convert the representation

U(XY)l - I Us BI(X'Y)

* t

97

into the form

For example, for element E in Figure 6-1, when k is uniform,

if

U(x'y)I E 2 #2 m U(i,m) PEtx) pE(y) +

p2U E pm(E) (y),m U(2,m) P2E Pm

then

Ui)-{ U (j,) 1 2,

ilL=L(m) vLm (Ni,L) i=2

where

Other cases are handled analogously.

Then, for 111;111IE 0 compute IaE (U. Il with a matrix-vector

multiply. For Isj Eor 11811 , compute (U(x,,Yt) by

the formula

U(xs#yt) m P3(yt) I i Pi(x) 8 (im) (6.3)

98

and then use the formula

I ~ ~(a(x,y) U,Bi) <w (WtnYt

[ p ( px)> [ U(xsyt)a(xs.yt)

4. {(<[aU/8u IuiI>E• In a preprocessing step, compute and

store the coefficients of the one-dimensional polynomials

U/n on each element side. Then, for example, on the left

side of element E, compute (a8U/an(x EYt )IE and

(a8U/an(xE.yt)IE,), where element E', if it exists, shares

part of the left side of E. Form

((sOg/OnJ IBI)aE(left) -

t <Wtpm (yt)/2>[aaU/an(xEYt)HE -aU/Dn(x Eyt) E#]

for each B I pi(x)pm(y). Other cases are handleda

analogously.

5. E ,1 , or 111-1112 . Suppose the system for

The storage required can be reduced by intermixing the preprocessing

step with error indicator computation.

99

the coefficients of aBis A v f. Compute a lower

triangular matrix L such that L L T- A, and solve L y - f.

(For and 'KL is independent of hE. Then compute

8' E

6. 11fLUII2 . for E First compute (Ux# sin :6.3),

lefthsieo compute

~~~~' (a1>(aU8(xsyt)~sp 8K+aOJO(Etyl)

No. s~aupposn as n ifrreuar corerors simplc ity heol

co deometin fgr 6-1.8' Ihn > SE are unin, for m exml, other

SNce refineme is airegular cvaler~.p can bemprcomputwedol

independent of E, and the right hand side components have values

100

whenB is in SE-SE and i 2, and

f(i~m) L P-k+1 wmP LP £Li vp,L PE'(iL) )

when k < a n. k and 1 2. The rest of the computations are unchanged.

CRAP=l 7

Numerisal results

7.1 Introduction

In this chapter, we present numerical results obtained using

prototype codes written in FORTRAN. which implement some of the methods

discussed in Chapters 2 - 5. The problem set, containing one problem

from each of the three classes considered in Section 5.7, is specified

in Section 7.2. In Section 7.3, we present the error behavior, which is

in fair accord with Section 5.7. In Section 7.4, we present the error

estimator behavior: the error estimators are usually accurate to within

a factor of two. In some oases, some of the estimators appear to

converge to the norm of the true error. In Section 7.5, we present more

numerical evidence of error estimator convergence.

7.2 The problem set

Prototype codes were written in FORTRAN, implementing some of the

methods discussed in Chapters 2 - 5. Numerical tests were conducted to

investigate the behavior of the methods.

The test problem set consisted of three Poisson problems with

101

, oW 1 , -

102

Dirichlet boundary conditions,

D2u - D2u - f(zy) in the interior of A,x y

u(xy) = g(xy) on 80,

with solutions specified in Table 7-1.

II Problem Solution class II Solution IIII II II III

II 1 smooth II u(.y) - cos(e +Xy) IIII II II IIII I I - - I I----**--IIII II II 12IIII 2 point II u(xy) r sin(012) , IIII II I IIII Isingularity II - r cos(e) . y - r sin(C) IIII II II IIII II II

II 3 line II u(x.y) - 0 t -1/2 II

II IIsingularity II sin(wt) -1/2 < t ( 1/2 IIII II II It1I 1/2 S t * IIII II II

S II II t = Sx-y-1 IIII II II II

Table 7-1: Test problems

The solution to Problem 1 is in fk(f) for all k. The solution to

Problem 2 is the fundamental esenfunction for a crack problem with

interior angle 2w; this function is in B312-*(G) for any a ) 0. but not

in 3312(a). The solution to Problem 3 is a smeared shock wave with

shock-width 1/5 in z; this function is in N2(0). but not in 22 +(Q) for

103

any s ) 0.

We tested the seven refinement methods specified in Table 7-2. All

adaptive sequences of finite element spaces started out with 2X2 meshes.

The adaptive sequences for methods + and X started out with ks- 2.

Method Ih IIk_II I

U uniform (1/2,...,1/32) II - 2II II II IK - 1/2 II adaptive (Section 5.4)2 adapv ( o 52 adaptive (Section 5.3) II - 2

3 adaptive (Section 5.3) II - 3

II 4 IIadaptive (Section 5.3) II - 4 I+ adapte a v (+ adaptive II adaptive (Section 5.6)

II I adaptive II adaptive (Section 5.4) I

Table 7-2: Refinement methods

We used three heuristic refinement procedures, based on the

formulae in Chapter 5. In the "true error" procedure, we set

= 11181o112 In the "approximate error" procedure, we set

M= I!,I!I , with S composed of piecewise polynomials one order

higher than S. In the "residuals and Jumps" procedure, we set 4B 2

For smplioity, we got xiS - X1/2. To avoid ill-conditioned linear

systems, we restricted k1 . 7. Our tests were run in single precision

on a DEC-System 2060 computer.

104

G(E)-point Gauss-Lesendre quadrature rules (6.1) were used to

approximate integrals in each element E. In the original assembly

phase, G(E) = kmaxE , where

kmaxE : max(kE, : El is in N(E)).

In the error indicator phase, G(E) - kmazE+l. In the evaluation of

111111E , G(E) - kmaxE+3 . With these values, for with smooth

solutions, the errors due to numerical quadrature are asymptotically

negligible (e.g., Ciarlet [17], Section 4.1).

7.3 Error behavior

In this section, the resulting errors for the adaptive sequences of

finite element spaces are depicted on 1og10-1og10 scales. The

x-coordinate of each data point is the number of nonzeroes in the

stiffness matrix , and the y-coordinate is 11le101. Since the true work

generally grows faster in k than the number of nonzeroes, the measure of

work is biased in favor of high-order methods.

Figure 7-1 presents the errors due to applying the "true error"

Our prototype codes should be made more efficient before computer

time is used as the measure of work.

105

procedure to Problem 1. Sines u(xvy) is smooth, nonuniform meeh

refinement does not improve the asymptotic oonversence rate. The slopes

hk-i)srran ~7of the lines correspond to Oh - ) error and O(h 2k4 ) work. Since k

increases for methods K, +, and I, these methods achieve faster than

polynomial convergence. Methods + and X perform some initial mesh

rfinesent (see Figure 7-2, with k. labelled in each element), so they

are somewhat less efficient than method K.

Figures 7-3 and 7-4 present the errors due to ippiyJ.u the

"approximate error" and "residuals and Jumps" jroedu-1s to Problem 1.

The errors are similar to the ones shown in 1-1,

Figure 7-5 presents the errors due to appl-ing the "true error"

procedure to Problem 2. The slope of the line for method U corresponds

to an O(h 1 / 2 c) onvergence rate and an O(h -2 ) work count. Method K has

almost the same line, with slope corresponding to an 0(k -1 ) convergence

4rate and an 0(k ) work count. The slopes of the lines for methods 2. 3,

With the work count proportional to the number of unknowns, method K

would appear twice as efficient as method U (see [111). With a more

realistic work count, method K would asymptotically be the least

efficient method considered.

II

106

and 4 are the same as the slopes for the same methods in Figure 7-1.

Local mesh refinement, in effect, transforms variables so the problem

looks like it has a smooth solution on a uniform mesh. Methods + and X

are fairly efficient in all cases, refining the mesh near the

singularity, and refining the order where the solution is smooth (see

Figure 7-6).

Figures 7-7 and 7-8 present the errors due to applying the

"approximate error" and "residuals and Jumps" procedures to Prot-lem 2.

The errors are similar to the ones shown in Figure 7-5.

Figure 7-9 presents the errors due to applying the "true error"

procedure to Problem 3. The slope of the line for method U corresponds

to an O(h) convergence rate and an O(h -2 ) work count. The line for

method I appears to level out, reflecting an O(k 4 ) work count and slow

convergence in k. The slope of the line for method 2 reflects an

improving (( N-112 ) convergence rate and an O(N) work count, where N is

the number of elements. At the modest accuracies required, the mesh is

refined not only along the line singularities at the edges of the

boundary layer, but in the boundary layer as well (see Figure 7-10). We

expect the slope of the line for method 2 to approach -1 asymptotically,

reflecting an O(NO) convergence rate. The slopes of the lines for

methods 3, 4, +, and I reflect convergence rates tending toward O(N-).

Figure 7-11 depicts a sample grid for method +. Note that the shape of

the boundary layer cannot be discerned from the grid.

. . .. ... . . u m m lm m m wm m m omm •

r1

107

Figures 7-12 and 7-13 present the errors due to applying the

"approximate error" and "residuals and jumps" procedures to Problem 3.

The errors are similar to the ones shown in Figure 7-9.

In this test, methods + and X appear to be the most robust, since

they are fairly efficient for all three problems, regardless of whether

the errors are large or small. Each of the other methods does poorly

for some problems in some cases. Table 7-3 s=-arizes the observed

efficiency a, where

Illeli (the number of nonzeroe8)-a .

Method J Problem 1 II Problem 2 JJ Problem3 JIII II-J It

U J 1/2 JJ 1/4 JJ < 1/2 JJ

,I K (k-1)/2 II 1/4 II : 1/2 IIII I II II2 II 1/2 II 1/2 II > 1/2 IIIIII-- - II II II

II 3 II 1 II 1 II 1 IIIIII II II -,II

4 II 3/2 II 3/2 It : 1 II

+ II (k-1)/2 II (k-1)/2 II - 1 IIIIII - II II IIX II (k-X)/2 II (k-1)/2 I 1 IIIIII II II It

Table 7-3: Efficiency susmary

7.4 Error estimator behavior

Figures 7-14 - 7-19 depict the relative errors p and & in the error

estimators for the tests run using the "approximate error" and

* -

108

"residuals and jumps" procedures respectively.

In the "approximate error" procedure, the error is usually

estimated within a factor of two (-.5 j p j 1), and always

-.7 1 - 1.6

p appears to converge to zero in methods U and 2 for Problem 1, and in

methods 2 and 3 for Problem 2 (see Section 7.5). For Problem 1. p

varies somewhat erratically in methods + and X, especially when k, is

large. This may be partly due to ill-conditioning.

In the "residuals and jumps" procedure, the error is always

estimated within a factor of two. In particular, in method 3,

-.23 Sa .20,

and g appears to converge to zero. In methods + and X, 2 varies more

smoothly than p. Thus, in this test, the "residuals and Jumps"

procedure appears to be slightly more robust than the "approximate

error" procedure.

7.5 Error estimator convergence

As further evidence of apparent error estimator convergence, we now

present the behavior of the four error estimators 111,111. 111111,

1lll1l, and a. with relative errors p. p, p. and & respectively, for a

sample variable-coefficient problem with a smooth solution. k and

h - 1/n are uniform. S contains piecewise polynomials one order higher

-- i it " II -I. ... .. . " -1- '-

109

than S. We have observed similar error estimator behavior for other

problems with smooth solutions.

Table 7-4 depicts the relative errors in the error estimators for

the Dirichlet problem

- D (exyD u) - D (exyD u) + u/(l+x+y) - f(xy)x x y y

in the interior of 0,

u(x,y) = 0 on 80,

where f(x,y) is chosen so that u(x,y) = exy sin(nx) sin(xy).

In Table 7-4, the error estimators are always within a factor of

two of Illelll. None of the error estimators converge to Illelil with k

increasing and h fixed, but some appear to converge with h decreasing

and k fixed. 111;111 and 111.111 appear to converge when k j 4.

Iligill appears to converge when k ( 4. but does not converge when

k - 4. The convergence rates are not clear. When k 2 or 3. i behaves

like a constant times Illelil.

110

if n 1f1116.111 if-- l-I-l III I

k=2 2 .181(+1) .053 .026 .026 II .264 .877 .028 .024 .022 II .378 .436 .012 .011 .010 II .40

16 .218 .0039 .0037 .00331I .4032 .109 .0011 .0010 .0009 II .40

1- - I 1llell pI I-In I11.1 II II

k-3 2 I .515 -.46 -.45 -.45 II -.154 I .857(-1) II -. 12 -.11 -. 12 II -.00888 i .194(-1) II -.022 -.022 -.022 II .018

16 I .477(-2) II -. 0041 -.0038 -.0043 II .013

In pI1111 I

k=4 2 I .443 .33 II .29 .29 -.44 II4 I .426(-1) .12 II .10 .039 -.39 II8 I .446(-2) .043 II .038 -.11 -.35 II

16 I .508(-3) .019 II .018 -.25 -.32 II

II n II lllelll II P II £ II P II 2 II

k-5 II 2 II .671(-l) II .35 II .34 II .34 II -.36 IIII 4 II .426(-2) II .23 II .22 II .18 I-.42 IIII 8 II .256(-3) II .20 II .19 II -.026 II -.48 II

II n II lllelll II II II P II 2II----II I-- I I--**~-I I II I I---I I

k-6 II 2 II .118(-1) II .31 II .29 II .29 II -.25 IIII 4 II .508(-3) II .23 II .22 II .099 II -.29 IIII 8 II .156(-4) II .32 II .32 II .077 II -.26 II

II n II1lllell II p II 2 II P II 2 III II--I I - I ----- II II II

k-7 II 2 II .213(-2) II .36 II .35 .35 3 II -.14 IIII 4 II .380(-4) II .36 II .35 II .31 II-.19 II

Table 7-4t Error estimator behavior

1* -- *.iA0l2

111

Figure 7-1: Problem 1 - true error

SMOOTH - TRUE ERROR

1. DE-O2 x-+

I.OE-03

I.-01. OE-0

1.ED . .... ' .... ..... : I ....

1.OE01 !.OE02 1. OE*O3 1. OE04 .OE+05NONZEROES

U H UNIFBORM K=2K H = .5 . K RDPTIVE2 HRRDPTIVE , K = 23 H A PTIVE .K : 34 HRDPTIYE K = 44 H RDRPTIVE , K RDAPTIVE. DISCRETEX H ADAPTIVE , K FDRPTIVE. SYMMETRIC

112

Figure 7-2: Sample grid for problem 1 (method +,residuals and Jumps)

VT 34 -313

4

4 4

113

Figure 7-3: Problem 1 - approximate error

SMOOTH - RPPROXIMRTE ERROR1. OE+OO

1. OE-O

1.OE-02

S1. OE-03LI

1. OE-04

,x xi 1. OE-05 ~X+)4+

1.OE+O1 1.DE+02 1.OE+03 1.OE+04 1.OE+0S

NONZEROES

U H UNIFORM K = 2K H = .5 K ADAPTIVE2 H ADAPTIVE . K = 23 H ADAPTIVE . K = 34 H ADAPTIVE . K = 4+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE , K ADAPTIVE. SYMMETRIC

I

114

Figure 7-4: Problem 1 - residuals and jups

SMOOTH - RESIDURLS RNO JUMPS1. OE-O00'

1. OE-02

1.OE-03La-

1. OE-04 +X+X

1. OE-05 X

xx

1.OE-06 . . ...... ' . . . . . . .. -I , . . .....1.OE+O 1. OE+02 1. OE+03 1.OE+04 1. OE+05

NONZERBES

U H UNIFORM . K = 2K H = .5 . K ADAPTIVE2 H ADAPTIVE . K = 2S H ADAPTIVE . K = 34 H RDAPTIVE .K = 4+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

115

Filgre 7-5: Problem 2 - true error

POINT SINGULRRITY - TRUE ERROR1. DE+DO

1. CE-02

I.OE-02 -

. 4 x

I.DE-D2

1.OEO1 1. OE.02 1. OE 03 1.OE+04 1. OE 05

NONZEROES

U H UNIFORM K = 2K H = .5 , K ADAPTIVE2 H ADAPTIVE, K = 23 H ADAPTIVE K = 3I H ADAPTIVE K= 4+ H ADAPTIVE , K ADAPTIVE. DISCRETEX H ADAPTIVE , K ADAPTIVE. SYMMETRIC

116

Figtr 7-6: Sample grid for problem 2 (method +, residuals and Jumps)

3 2

22

2 2

222 2

117

Figure 7-7: Problem 2 - approximate error

POINT SINGULARITY - RPPROXIMATE ERROR1.OE+O0

1. oE-oI

1. OE-01LUJ

1.OE-O3

1.OE-04 I .

1.OE+O1 1. OE+02 1. OE+03 1. OE+04 1.OE05

NONZERSES

U H UNIFORM . K = 2K H = .5 . K RRPTIVE2 H ADAPTIVE . K = 23 H RDPTIVE , K = 34 H RDRPTIVE . K = 4+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE . K RDAPTIVE. SYMMETRIC

118

Figure 7-3; Problem 2 - residuals and jumps

POINT SINGULARITY - RESIDUALS AND JUMPS1. OE O0 I. . .pl . .. .. .. . . .. .

I. OE-01

1. OE-O2

1.OE-03

1.OE-041.OE+O1 1.OE+02 1.OE+03 1.0E,04 1.OE+OS

NONZEROES

U H UNIFORM . K = 2K H = .5 . K ADAPTIVE2 H ADAPTIVE . K = 23 H ADAPTIVE , K = 3I H ADAPTIVE . K = 4+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

119

Figure 7-0: Problem 3 -true error

LINE SINGULARITY -TRUE ERROR1. OE+01

1. OE.OO

1.OE-02

1.OE.O1 1.0E+02 1.OE+03 I.0E+04 1. OE+05NONZEROES

U H UNIFORM K =2K H = .5 ,K ADAPT IVE2 H ADAPTIVE . K =23 H ADAPTIVE . K =34 H ADAPTIVE . K =4+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

120

Figusre 7-10: Sample grid for problem 3 (method 2, residuals ant jumps)

OWN"

121

Figure 7-11: Sample grid for problem 3 (method +o residuals and jumps)

3 3 44

3 3

3 3 4

3 3 3 3 3 33 3

3 3 3 3 3 3

33 3 3 3 3 3

3 3 33 3 3 3 3 3

2 3 3 3 32 2 2

_ _ _ _3 3 3 32~2

3 2 3 32 22 2 2 2

S 22

2 2

2 2 2 2 3 322 22 2 2 3 3 3

2 2 2 2

3 3 3 3 3 3

3 3 3 3 3 3

1

122

Figure 7-12: Problem 3 -approximate error

1.O+O1 LINE SINGULARITY A PPROXIMATE ERROR

1.0E+00

1. OE-01

i.OE.1 1.~eO2 NONZEROES

KH = .5 .K ADAPTIVE2H ADAPTIVE. K =2

3H ADAPTIVE . K =

+H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

123

Figure 7-13: Problem 3 - residuals and Jumps

LINE SINGULARITY - RESIDUALS AND JUMPS1.OE+O1. ......... ,

1. OE.OO

oU I.OE-O01 +

I.OE-O02

1.OE+O1 I.OE+02 1.OE+03 1.OE+04 1.OE+05

NONZEROES

U H UNIFORM K : 2K H = .5 .K RDAPTIVE2 H ADAPTIVE K = 23 H RDRPTIVE K = 34 H ADAPTIVE, K = 4+ H ADAPTIVE , K ADAPTIVE, DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

I.

124

FiSure 7-14: Problem 1 - approximate error

SMOOTH - RPPROXIMRTE ERROR2.000. ......................

X

0.000

+x

1.000

0.000 ' .- J . +

-1.000. . .. . . .. . . . . ., , , . . .1.OE+O1 1.OE+02 1. OE+03 1.0E+04 1. OE05

NONZEROES

U H UNIFOR , K = 2K H = .5 . K ADAPTIVE2 H ADAPTIVE . K = 23 H ADAPTIVE , K = 34H ADAPTIVE K =4

+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

125

Fiure 7-15: Problem 1 - residuals and Jumps

SMOOTH - RESIDUALS RND JUMPS2.000

1.000

0.000

X + X x *x

-1.000l.OE+01 1. E+02 1.OE+03 1. OE+04 1. OE+05

NONZEROES

U H UNIFOR ,K : 2K H = .5 K RDRPTIVE2 H RPTIVE, K = 23 H RPTIVE K = 34 H ADAPTIVE K = 4+ H ADAPTIVE , K ADAPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMIETRIC

126

Figure 7-16: Problem 2 -approximate error

POINT SINGULRRITY -RPPROXIMRTE ERROR

2.00

1.000

0.000

-1.00......I1.OE+01 1. OE+a2 1. 0E403 1. OE404 1.ODE+05

NONZEROES

U H UNIFORM . K =2K H =.5S SKADAPTIVE

2 HRADAPTIVE.K =23 H ADPTIVE K =34 M ADAPTIVE. K =4+ H ADAPTIVE . K ADAPTIVE. DISCRETEX H ADAPTIVE , K ADAPTIVE. SYMMETRIC

127

Figure 7-17: Problem 2 - residuals and Jumps

POINT SINGULARITY - RESIDUALS RND JUMPS2.000. ........ ,. ....

1.000

-0.000

1.OE+01 1.OE+02 1.OE+03 1. OE+04 1.OE.05NONZEROES

U H UNIFORM . K : 2K H : .5 . K OAPTIVE2 H RDTIVE .K = 23 H DAPTIVE , K a 34I HADRPTIVE . K : 4+ H ADRPTIVE . K ADRPTIVE. DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

128

Figure 7-18: Problem 3 - approximate error

LINE SINGULARITY - RPPROXIMRTE ERROR2.000

1.000

-I.000

-1.000 ... . . .......... . ..... .I . . . . .I . . . .

1. OE+01 1. OE+02 1. OE+O3 1. OE+04 1. OE+05

NONZEROES

U H UNIFIRM o K : 2K H : .5 , K ADAPTIVE2 H ADAPTIVE .K = 23 H ADAPTIVE . K = 34 H AOPTIVE , K = 4+ H AOPTIVE . K ADAPTIVE, DISCRETEX H ADAPTIVE . K ADAPTIVE. SYMMETRIC

.. .. , .I;,

129

Figuze 7-19: Problem 3 -residuals and jumpa

LINE SINGULARITY -RESIDUALS AND JUMPS2. ID[

~w -1.D14 .OE.01 I.0E402 1.OE.03 1.OE+04 1.OE+0S

NONZEROES

U H UN!FU -K .(2K H =.5 K ADAPTIVE2 HRADAPTIVE.K =23 H ADAPTVE ,K z34 M ADAPTIVE ,K z

XH ADAPTIVE .~~ K DPIE YMTI

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PATE

IILMEI