a new triangular and tetrahedral basis ... - brown university...spencer j. sherwin and george em...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 38, 3775-3802 (1 995)

A NEW TRIANGULAR AND TETRAHEDRAL BASIS FOR HIGH-ORDER (hp) FINITE ELEMENT METHODS

SPENCER J. SHERWIN AND GEORGE EM KARNIADAKIS*

Division of Applied Mathematics, 182 George Street, Box F, Brown University, Providence, R.I . 02912, U.S.A.

SUMMARY In this paper we describe the foundations of a new hierarchical modal basis suitable for high-order (hp) finite element discretizations on unstructured meshes. It is based on a generalized tensor product of mixed-weight Jacobi polynomials. The generalized tensor product property leads to a low operation count with the use of sum factorization techniques. Variable p-order expansions in each element are readily implemented which is a crucial property for efficient adaptive discretizations. Numerical examples demonstrate the exponential convergence for smooth solutions and the ability of this formulation to handle easily very complex two- and three-dmensional computational domains employing standard meshes.

KEY WORDS finite-elements; high-order; spectral expansions

1. INTRODUCTION

In the last two decades there have been some significant advances in the development of the hp-version of finite element methods. Szabo pioneered this method in the early 1970s by recognizing the relative strengths of h-type finite element methods and p-type Galerkin methods and implementing a combination of both. Since then several versions of this approach have been formulated for both solid mechanics and fluid dynamics in the founding papers References 1-4. More recently, non-conforming formulations were developed to allow greater flexibility in handling geometric complexities and local refinement requirements in the works of References 5-7 as well as curved boundaries.*

Non-conforming discretizations and variable p-order per element are key in establishing robust adaptive discretization strategies by exploiting the dual paths of convergence associated with this method. Non-conforming discretization allows h-refinement where a larger number of elements are introduced at a fixed (low) p-order in each element. Variable p-order per element allows flexible p-refinement where the skeleton mesh consisting of a smaller number of relatively large-sized elements remains unaltered whilst p is varied independently in each element. This has primarily been the research focus of References 5,9, 10 and other groups have developed similar adaptive (hp) discretizations.'

The basis or shape functions employed in the aforementioned formulations typically involve Legendre or Chebyshev one-dimensional polynomials. Multi-dimensional expansions are then constructed using tensor products for quadrilateral or hexahedral elements.' '* ' A basis directly associated with boundary and interior elemental nodes usually lacks hierarchy and furthermore

* Address for correspondence

CCC 0029-5981/95/223775-28 0 1995 by John Wiley & Sons, Ltd.

Received I September I994 Revised 12 January 1995

3776 S. J. SHERWIN AND G. E. KARNlADAKlS

a variable p-order per element is not readily a~hievable .~ However, a modal basis can be constructed consisting of vertex modes, edge modes, and interior modes which are hierarchical and allow the order along the boundaries as well as within each element to be variable.I2 In addition to the easier handling of non-uniform resolution requirements, hierarchical bases lead to better conditioning of the stiffness matrix' and thus fast iterative algorithms can be effectively employed in the solution algorithms.

The need for constructing p-type hierarchical bases in triangular domains became evident early on. Peano14 constructed a hierarchical triangular basis using area (barycentric) co-ordinates. This construction enforced conditions on the tangential derivatives at the midpoint along each side. Then appropriate continuity constraints were imposed for C o and C' elements. A sub- sequent variation of this construction introduced Legendre polynomials to avoid round-off error for high-order p-expansions." However, both approaches require special integration rules which become quite complicated at high-order p.'

In this paper we present a new basis for triangular and tetrahedral domains that is based on Cartesian co-ordinates and preserves a tensor product property. This property leads to sum factorization and low operation account, a crucial factor for p-type finite elements and spectral methods. Sum factorization refers to the property which allows us to evaluate an O ( N Z d ) summations in O ( N ( d + ')), For example, let us consider the inner product of an arbitrary function u(a ,b ) with an expansion basis &i(a ,b ) . This operation can be discretely computed using numerical Gaussian quadrature as the summation

where w(ai), w ( b j ) are the quadrature weights. Each summation has to be evaluated for all expansion modes over I,m. Therefore, if there are O ( N 2 ) expansion modes then the total operation count would be 0(N4) since we need to evaluate an O ( N 2 ) summation over the indices i , j for all O ( N 2 ) expansion modes over I , m. However, if the two-dimensional expansion basis is a product of two one-dimensional functions, i.e.,

we can rewrite the inner products by factoring out terms involving the summation over j :

j \ i /

Now this can be evaluated in O ( N 3 ) operations since the term in brackets can be computed to form S,(bj)

N

f ; (b j ) = C u(ai, bj)w(ai)i l(ai) i

which is an 0 ( N 3 ) operation because it is an O ( N ) summation over i at the O ( N ) points bj for the O ( N ) modes denoted by the I index. Having evaluated f , (b j ) , the inner product summation becomes

HIGH-ORDER ( h p ) FINITE ELEMENT METHODS 3777

which again is an O ( N 3 ) operation involving the O ( N ) summation overj for each of the O ( N 2 ) modes, I , m.

The proposed basis employs Jacobi polynomials of mixed weight to automatically accommod- ate exact numerical intergration using standard Gauss-Jacobi one-dimensional quadrature rules. In particular, exploiting the tensor product property of the basis multi-dimensional integrals can be evaluated efficiently in O ( N d + ” ) operations using the sum factorization technique as illustrated above. Similarly, the cost of evaluating derivatives or squares of a function is maintained at operation count O ( N ( d + l ) ) in Rd (where N is the number of modes) as in the quadrilateral or hexahedral spectral elements.’ The two-dimensional basis is based on Dubiner’s original ideas for spectral bases on triangle^,'^ for which a complete analysis and a formulation for Navier- Stokes equations is given in Reference 18.

This construction uses standard triangular/tetrahedral finite volume meshes”*20 since all that is required is a conforming triangular/tetrahedral discretization. Our experiments suggest that exponential convergence is obtained even for very distorted meshes. While standard Delaunay triangulization may provide an automatic procedure in generating unstructured meshes preven- ting sudden mesh distortions,2’ unlike other unstructured methods such a quality has not been observed to be necessary.

The paper is organized as follows, In Section 2 we introduce the new set of Cartesian co-ordinates for triangles and tetrahedra. In Section 3 we review Dubiner’s basis for a single domain in both two- and three dimensions, and in Sections 4 and 5 we develop the multi-domain two- and three-dimensional bases, respectively. In Section 6, we present numerical examples that demonstrate great flexibility in handling geometrically complex domains and exhibit exponential convergence for smooth solutions.

2. CO-ORDINATE MAPPINGS

We introduce here a set of mappings that are useful in defining the triangular and tetrahedral bases in terms of Cartesian co-ordinates attached to the transformed domain. The proposed high-order basis is not associated with any specific set of edge or interior nodes other than the vertices of the triangular or tetrahedral elements. Thus standard procedures of deriving this basis from a corresponding basis on a rectangular or hexahedral element are inappropriate.22 Never- theless, the following transformations motivate the use of a more convenient set of co-ordinates from the computational viewpoint.

We define the standard triangular and rectangular domains as shown in Figure 1, which are mathematically expressed as

T2={(+ ,$) l - 1 < # J , $ ; d ) + $ < O } R2 = {(@,“)I - 16 @,Y < 1)

The rectangular domain R 2 can be mapped into the triangular domain T 2 by the following transformat ions:

*=Y

+ = - 1 (1 + w 1 - ‘y)

2

and similarly the triangular domain T 2 can be mapped into the rectangular domain R 2 by the inverse transformation:

y r = *

3778 S. J. SHERWIN AND G. E. KARNlADAKlS

Figure 1. Rectangle to triangle transformation

As indicated in Figure 1, within T 2 the coordinate Q, has a value of - 1 along the line 4 = - 1 and a value of 1 along the line 4 + 9 = 0 except at the point ( 4 = - 1, = l), where @ is multi-valued. We know that CD is bounded in R 2 and therefore we expect the same to be true in T 2 . To show that this is the case at the degenerate point (4 = - 1, t+b = lj , we consider the substitution 4 = - 1 + E sin 6, 9 = 1 - E cos 6. Here 0 is defined in a counter clockwise manner from the vertical, as indicated in Figure 1, and c is a small perturbation such that when E = 0 we have (4 = - 1, II/ = 1). Substituting these values into the definition of 0 given by equation (1) we can find the limiting behaviour of the singularity:

(1 - 1 + Esin6) C D - 1 . 1 = 2 - 1 = 2 t a n O - l

(1 - I + ECOSU)

Since 0 6 8 Q 7r/4 we know that 0 < tan 8 Q 1 and so - 1 < Q,- 1, 6 1. It might appear strange to use a co-ordinate system which has a singular point but it should be noted that the singularity in the co-ordinates does not imply that the expansion is singular. We recall that both cylindrical and spherical co-ordinate systems have multivalued co-ordinates at the origin. The introduction of the co-ordinate Q, in the T 2 space is useful in defining the two-dimensional triangular basis since we can now describe the T 2 domain with the two co-ordinates a, b, where a = Q,, h = 'P = +. The advantage of these co-ordinates is that they are bounded by constants, ie. - 1 < a, b d 1, which proves efficient when calculating the integrals required in a Galerkin formulation.

The mapping shown in Figure 1 is the foundation for constructing a co-ordinate system in the tetrahedral domain T 3 starting from a co-ordinate system in the hexahedral domain R 3 . To ahieve this, we repeatedly apply the two-dimensional transformation (a, Y ) f-) (4, ~) in three steps. Schematically, this transformation is shown in Figure 2: In the first step we map R 3 into a triangular prism with the transformation:

c = t

b = h


Figure 2. Transformation from hexahedral co-ordinate system to tetrahedral co-ordinate system

(1 + a)(l - t ) r' = - 1 2

where (a, b, c) are the Cartesian co-ordinates defining the domain R3 = {(a, b, c ) I - 1 < a, b, c ,< 1). In the second step, we map the prism into a square-based pyramid applying the mapping:

t = t

b = b

- 1 (1 + r')(l - b)

2 r =

Finally, we map the pyramid into the tetrahedron T 3 via the transformation:

t = t

- 1 (1 + b)(l - t )

2 S =

r = r

where ( r , s, t) define the local co-ordinate system associated with T 3 = { - 1 < r, s, t; r + s + t < - 1). Correspondingly, the three-dimensional basis can be expressed via the initial set of

co-ordinates (a, b, c) as we demonstrate in Section 5. To summarize, we the write the hexahedral co-ordinates (a, b, c) in terms of the tetrahedral co-ordinates (r , s, t ) by repeatedly applying the

3780 S. J. SHERWIN AND G . E. KARNIADAKIS

constant ‘a’ planes constant ‘b’ planes constant ‘c’ planes

Figure 3. Constant planes of the co-ordinates (I, h and c on the standard tetrahedron

inverse transformation to arrive at:

For c = - 1 we recover the two-dimensional mapping. The constant planes represented by u, b, c in the T 3 space can be seen in Figure 3 . We note the

degeneracy of the co-ordinate system in the T 3 space. Planes of constant ‘a’ remain planes as this co-ordinate varies from a = - 1 to u = 1 and are dependent on all basic co-ordinates r, s and t . However, planes of constant ‘b’ degenerate to a line as this co-ordinate varies from b = - 1 to b = 1 although these planes only depend on the basis co-ordinates s and t . Finally, planes of constant ‘c’ degenerate to a point as this co-ordinate varies from c = - 1 to c = 1 and these planes only depend on the basis co-ordinate t .

3. DUBINER’S ORTHOGONAL BASIS

3.1. Tbco-dimensional basis

We wish to define a polynomial basis, denoted by lg:m(r,s), so that we can approximate the function f ( r , s) in the domain T 2 , i.e.

1 In

Here A,,, is the expansion coefficient for the expansion polynomial lg:m and (r, s) are the local co-ordinates within the triangle T Z . Dubiner” proposed a polynomial expansion basis for triangular domains, which is orthogonal in the Legendre inner product. The principal idea is to express the expansion bas’is in terms of a function which is a polynomial in both the standard co- ordinates ( r , s) as well as the transformed co-ordinates (a, b). We review this basis below.

Let us denote Pisp(x) as the dh-order Jacobi polynomial in the [ - 1,1] interval with the orthogonality relationship,

1

Pf*p((x)P“,.(x)(l - ~ ) “ ( l + X)’dx = 61, s- 1

HIGH-ORDER ( h p ) FINITE ELEMENT METHODS 378 1

15

10

a

5

0

-1 -0.5 0 0.5 1

Zeros

Figure 4. Zeros of the Jacobi Polynomial P;O(s) for m = 7. As alpha is increased we see that the zeros move towards the s = - I point.

where dl,, is the Kronecker-delta. Dubiner's orthogonal expansion basis is given by

We note that this is a polynomial in ( r , s ) since the (1 - s)' factor acting on the Pp30(2(1 + r)/(l - s) - 1 ) Jacobi polynomial produces an Ith-order polynomial in r, s. This factor also necessitates the 'unusual' Jacobi polynomial P:" '.'(s) in order to maintain orthogonality. In Figure 4, we see the zeros of the Jacobi polynomial P;OO(s) as a function of x for the case m = 7. The effect of increasing ct is to draw the zeros towards the s = - 1 point. This means that the zeros become declustered from the s = 1 point which is this degenerate vertex. It is possible that this property is significant in favourably scaling the derivative operator as seen in Reference 18.

The basis can also be expressed as the product of to polynomials in (a,b) space, i.e.

where 1 2

gr(a) = PY3'(a) and grm(h) = (1 - b)'P:f+'*o(h)

Dubiner refers to this property as a warped product to differentiate it from the standard tensor product associated with quadrilateral domains. A basis is said to have a warped product relative

3782 S . J. SHERWIN AND G . E. KARNIADAKIS

to an inner product ( , ) if and only if

1 2 The significance of this property is that the inner product between two polynomial basesff and di which both span a two-dimensiona12space can be expressed as the product of two one- dimensional inner products, (f , ;) and ( f , ; ) multiplied by a constant. This is important when evaluating integrals involving 'i:m(r,s) with ' j Z g ( r , s ) over T2, since it is possible to write the integral as the product of two line integrals as explained in Reference 18. Integrals involving the inner product of d$ = lg:m(r,s) with a functionf(r, s) can also be efficiently evaluated using the sum factorization technique. Finally, we note that the basis is complete in a polynomial space Py where YY is defined by

9 . = S~an{r's"}~lm)Ep

where

p = {(Im)10 < I , m; 1 < L, 1 + m < M } , L d M

We recall that l&m(r,s) is orthogonal in the Legendre internal product defined by

To demonstrate this result Since

J J 1 -

we can perform a co-ordinate transformation from (r, s) to (a, b).

I I

the inner product becomes

Now from the definition of the Jacobi polynomial, P%p((x), we see from relation (2) that the first integral is zero if 1 # p . When I = p the second integral is zero if m # q. This is because the polynomial (1 - b ) ' + p + l is the weight function for the second integral (see relation (2)) and so the orthogonality of the base 'g:,(r, s) is verified.

3.2 Three-Dimensional basis

the tetrahedron T 3 : The three-dimensional basis is defined similarly using the following triple warped product on

HIGH-ORDER ( h p ) FINITE ELEMENT METHODS

This can be expressed in terms of a, b, c as

d(a,b,c) c?( r , s , f )

J = - -

To show that this is orthogonal we first need to calculate the Jacobian J:

aa aa aa dr ds dt _ _ _

2b c'b db dr 8s at

_ _ _ _

ar dc ac ar as Z - -

(1 + (1 + r )

(1 + 4 2- (1 - f ) *

2 2

( - s - t ) ( - s - t ) Z ( - s - t ) Z

2 0

(1 - t )

3783

2 2 =

( - s - t ) (1 - t )

0 0 1

which if we express in terms of b,c noting that

we obtain

2

J = (L)(L) 1 - b 1 - c

Having determined the Jacobian we are now in a position to demonstrate the orthogonality of the basis by considering the inner product:

Using the orthogonality of Jacobi polynomials we know that

1 1

glgpda = PP*O(a)P;*O(a)da = 61,

when 1 = p we find that the second integral of Ilrnnpqr becomes

3784 S. J. SHERWlN A N D G . E. KARNlADAKlS

Finally, when 1 = p and rn = q the third integral of Ilmnpqr becomes

Therefore, we see that Ilmnpq, can be written as

which is clearly orthogonal. The orthogonal bases proposed by Dubiner would be ideal in both two- and three-dimensions

except that we want to discretize a complex-geometry solution domain into many triangular or tetrahedral subdomains, or elements. We are interested in discretizing 2nd-order problems and for this we need to enforce a C o continuity between each subdomain. C o continuity is not easily enforced but might be achieved by a constraint on all modes. A constraint of this sort would introduce a non-diagonal contribution to the matrices thereby destroying the most desirable property of the expansion. To overcome this problem, multi-domain triangular and tetrahedral expansions can be constructed using a more standard finite element approach. In this approach we decompose the expansion into vertex, edge and interior modes as we demonstrate in the next section.

4. MULTI-DOMAIN TRIANGULAR BASIS

We define a polynomial basis, denoted by ' i ; ; ( r , s), so that we can approximate the function , f ( x , y ) by a Co continuous expansion over 'K' triangles in the form:

HereT:,, is the expansion coefficient corresponding to the expansion polynomial '&A in the k t h triangular element (1 6 k 6 K ) ; ( x , y ) are the global spatial co-ordinates of the function and (r, s) are the local co-ordinates within any given triangle.

The basis is now split into interior and boundary modes where all interior modes are zero on the triangle boundary similar to the bubble modes used in p-type finite element^.^^.^^ The boundary modes can further be described in terms of vertex and edge modes. The vertex modes vary linearly from a unit value at one vertex to zero at the others. The edge modes only have magnitude along one edge and are zero at all other vertices and edges. Using the notation given in Figure 5 and recalling that PZqp(z) refers to the Jacobi polynomial we can write the multi-domain triangular basis as follows: Vertex modes:

1 _ 2 * E " A = (?)(?) 9

HIGH-ORDER (hp) FINITE ELEMENT METHODS 3785

Figure 5. Definition of standard triangle for multi-domain basis

Edge modes (2 < I ; 1 < rn 1 1 < L; I + rn < M )

'g?z* = (i)(i)(T) l + a 1 - b l + b P$I,(b)

'iTr3 = (T)(T)(T) 1 - b l + h PL',(b)

Here the indices Irn refer to the principal polynomial in a and b. Note that for the principal polynomial in b we exclude the factor ((1 - b)/2)' since this is related to the Ith-order polynomial in a. We have used L and M ( L < M ) to denote the total number of modes. This basis is once again complete in the polynomial space Yy as defined in the previous section.

The edge modes have exactly the same shape for in = 1 - 1. This is easily seen by considering the fact that along side 1, h = - 1; along side 2, a = 1 and along side 3, a = - 1 . Co continuity can therefore be enforced by matching the expansion coefficients of similar shaped modes along an edge. This means that there must be the same number of modes along any edge which can be satisfied by setting L = M . For this case the number of modes used in a basis is therefore dictated by the value of L. One interpretation of L is that the maximum order of the polynomial along any edge is ( L - 1). For example when L = 2 we only have 3 vertex modes which gives us a linear finite elements basis that has a maximum polynomial order of L - 1 = 1 along any edge. The minimum order expansion starts at L = 2 as L = 1 would indicates an expansion which is a constant.

3786 S. J. SHERWIN A N D G . E. KARNIADAKIS

The interior modes are similar to the orthogonal basis of Section 3.1 except for a ( (1 + z)/2)((1 - z)/2) factor in both the r- and s-dependent polynomials. It is this factor which ensures that the interior modes are zero on the boundaries. To help maintain orthogonality between the modes, the Jacobi polynomials in r now have weights of a = 1, @ = 1 which is beneficial as this makes (1 + r)(l - r ) the weight function in the inner product. The polynomial ( 1 - z)(l + z)P'*'(z) is related to the integral of the Legendre polynomial P0*O(z) by a constant factor," i.e.

2n P;*O(s)ds = - (1 - z)(l + z)P::'1(z) (3)

It is interesting to note that the integral of the Legendre polynomial is used to construct a hierarchical p-type finite element basis on quadrilateral^.'^-^^

Since the vertex modes are the same as the linear finite element basis by matching the vertex and edge modes we can join multiple triangular domains together in a contiguous fashion. The only complication is that some edge modes may be the negative of the adjacent matching mode. This point is illustrated in Plate 1, showing all the modes for L = 5. At this order there are three interior modes and three edge modes on each side. Along the edges one has a quadratic form, the next one has a cubic form and the last one has a quartic form. If we consider the case where side 1 of one triangular expansion meets side 2 of another triangular expansion then the cubic mode would be the negative of each other whilst all of the quadratic and quartic modes have the same sign. So the connectivity condition is enforced by initially stating that all triangular elements must be defined in an counter clockwise sense: if at any edge side 1 meets side 2 or like sides meet (e.g. side 3 with side 3), the odd modes of one of the triangular elements will be negated. This condition is not difficult to implement, since only the information that a mode must be negated need be stored. All local operations can be performed without knowledge of how the modes are connec- ted.

Another point that can be appreciated from Plate 1 is that the expansion order can be varied within a singular element. Setting L = M implies that the polynomial order along each edge is the same. However, to satisfy the C o continuity we only require that neighbouring elements have the same number of modes along an elemental edge. Therefore, we can allow the expansion order to be different along each edge and even within the interior. For example, on side 1 we could locally have an expansion order of L = 3 which would include the two vertex modes and the quadratic mode. Then on side 2 we could have an expansion order of L = 5 including all edges modes shown in Plate 1. Then for C o continuity we require that the expansion order of the adjacent element which meets side of 1 to be L = 3. Allowing the expansion to vary within each element means that the polynomial space which the expansion spans will be modified. Therefore, we can construct a conforming variable-order expansion over multiple elements.

As mentioned earlier not all the modes are orthogonal in this basis. To see what type of orthogonality does exist we consider the inner product of two sets of the interior modes over the T' space:

HIGH-ORDER (hp) FINITE ELEMENT METHODS 3787

where P,?.'(Z) is orthogonal in the inner product: 1

(P!*l(z),u(z))l,l = (1 + z)(l - z)P;*'(z)U(z)dz s- 1

The value of the integral will be zero if u(z) is a polynomial of order ( I - 1) or less, since in this case we can express u ( z ) as u(z) = Cf:AOjP!9'(z). We can write the first integral of (4) as

r l

J (1 + a)(l - a)P:i:(a)f,(a)da, wheref,(a) = (1 + a)(l - a)P:.',(a) - 1

This integral will be zero if 1 - 2 > p . Similarly, it will also be zero if p - 2 > 1 and both expressions can be summarized as 11 - p 1 > 2. The practical significance of this orthogonality is evident when evaluating the mass matrix. Considering the interior mode component of the mass matrix, ( glm , g p q ), the above orthogonality result says that the matrix will have an upper bandwidth of ( L - 3 ) + ( L - 4) + ( L - 5 ) - 1 x 3L (recall that in the interior the maximum value of 1 is L - 3). However, if we now consider the second integral of equation (4) for the case when 11 - pI = 2 it can be shown that the upper bandwidth can be reduced to ( L - 3) + ( L - 4) + 1 = 2L. Considering the case where 1 - p = 2, so p = 1 - 2, the second integral of (4) can be written as

-2'"'c""r -2'"""o'

f l I

J (1 + b)(l - b) 21-1 P,Zl* 21 1 1 f,(b)db, wheref,(b) = P%!-15*1db - 1

This integral is zero if m - 1 > q and similarly it is also zero for the case where p - 1 = 2 and q - 1 > m. So if 11 - pI = 2, then the off-diagonal of the mass matrix are zero if Im - q1 > 1 which means the mass matrix has an upper bandwidth of ( L - 3 ) + ( L - 4) + 1. Finally, it should also be noted that there is similar orthogonality between the interior and edge modes and this introduces further sparsity into the mass matrix structure. This structure is evident in Figure 6 which shows the form of the mass matrix, on a single triangle, for L = 15.

4.1. Triangular multi-domain basis in area co-ordinates

Area co-ordinates, otherwise known as triangular or barycentric co-ordinates, are commonly used in the finite element literature. This co-ordinate system is illustrated in Figure 7 for the standard triangle. Any point in the triangle is described by three co-ordinates L1, L2 and L3. These co-ordinates can be interpreted as the ratio of the areas A l , A2 and A 3 over the total area A = A , + A 2 + A 3 , i.e.

A3 L3 = 2 L p A ' A1 L 2 = A ' A2

Therefore L 1 , L2 and L3 have a unit value at the vertices marked 1, 2 and 3 in Figure 7, respectively. By definition these co-ordinates have the property:

L1 + Lz + L3 = 1

Ll = f ( 1 - r) - f ( 1 + s)

L2 = f (1 + r) L3 = +( l + s)

and they can be expressed in terms of r, s as

3788

3(

(L3!

d r -T

c *

I

.n

.... ...... * . ....... ..... ....... ..... ....... ....... ....... ....... ....... ....... ....... ....... ..... ..... .... .... ...... ...... .... ..... ... ...... .... ....... ..... ....... ...... ....... ....... ....... ...... ....... ....... ...... ...... ..... ..... .... .... ...... .... ....... ..I.. .... ....... ...... .... ....... ....... .... ....... ... ....... ...... .... ....... ....... ....... ...... ...... ..... ..... .... .... .... ..,,.a. ...... ...... ....... ..... ....... ....... ....... ....... ....... ....... .... .... ...... ..... ..... ..,.* ..... .... ....... .... ....... .... ....... ... .... ....... ......

Boundary-Boundary Matrix ... ::.. !!I. [I:.

' i i

.:: '!

- 3 ;

.:: .. ... :. - A ... ...

S. J . SHERWIN A N D G. E. KARNlADAKlS

- . * . -... ... " ... ... ." ". ." .".. :.. ............. . . ......... ............... ....

~i!!:. 2:. :: ::: Interior-Boundary *!:::i:. Matrix ':i& yg;

Figure 6. The structure of the mass matrix on the standard triangle for L = 15. The matrix is symmetric. The boundary modes are listed first followed by the interior modes which have a banded structure. The matrix rank is L ( L + 1)/2

Figure 7. Notation for area co-ordinate system. A given point in the triangle is described by three co-ordinates L, , L2 and f., which can be interpreted as the ratio of areas A, , A, and A, over the total area

Since a = 2(1 + r)/(j - S) - 1 and b = s we can express the area co-ordinates in terms of a, b as

(1 - U ) (1 - b) L , =-- 2 2

(1 + U ) (1 - h) L 2 = - - 2 2


(1 + 6) L3 = - 2

Manipulating these relationships we can find a, b in terms of L 1 , L2 and L3 :

Finally, we can recast the expansion basis presented in the previous section in terms of the area co-ordinates as

Vertex modes

Edge modes ( 2 < I; 1 < mil< L; 1 + m < M )

1 - 2 w d c 2

g l m = L2L,P;_',(2L3 - 1)

1 - 2"dc 1

g l m = LlL3P;_L1(2L3 - 1 )

Interior modes ( 2 < I; 1 < m 11 < L; I + m < M )

1 -2'"'C"O' = L,L2L3P:r:(-)(I - L3)'-2P:l:l'.'(2L3 - 1) S l m 1 - L3 (7)

Not surprisingly the basis has maintained its product form now being (L2 - Ll) / ( l - L3) and L3 . To appreciate that all modes can be expressed polynomials in (L2 - Ll ) / ( l - L3) and L3 we note that the vertex modes can be recast as

a function of as products of

L2 - L1 I - - 1 - 2 " C " " I 1; L3 \

910 = ( 1 - L3)

3790 S. J. SHERWIN AND G. E. KARNIADAKIS

We can compare this basis with the hierarchic shape function for triangles given in Szabo and Babuska’s book.” Both bases are built up from linear finite element and so the vertex modes ( 5 ) are identical. The shape of the modes on side 1 were given as

(side 1 ) - Ni - L,L,Vi(L2 - L l ) , i = 2 , . . . , p

Here cpi(z) is proportional to the P/L\(Z) Jacobi polynomial. The other edge modes given in Szabo and Babuska’s book are said to be defined analogously which implies that they have the form

N(side 2) - i - LZL3(PdL3 - L2)

- L l L 3 ( P i ( L 1 - L 3 ) p i d e 3 ) -

i

These modes are symmetric in the sense that they decay in a similar fashion from any face to the opposite vertex. However, unlike the modes given in (6) they do not maintain a consistent product form since these modes are function of the three variables L2 - L 1 , L3 - L2 and L1 - L 3 . It is also not clear that an integration scheme can be readily constructed using these variables. The general form of the modes on sides 2 and 3 are similar to those shown in (6b) and (6c).

Finally, Szabo and Babuska’s expansion have the same first internal modes

N\O’ = L1L2L3

whereas the other interior modes are generated by multiplying N\” by Legendre polynomials and products of Legendre polynomials. For example,

N\O’ = L1L2L3Pygo(L2 - L , )

N y = L 1 L 2 L 3 P y ( 2 L 3 - 1)

If we were to cast these modes in terms of the interior modes given in equation (7) they would have the form

(0) - L L L PO 0 N,, - 1 2 3 ILZ(L2 - Ll)p;-o1(2L3 - 1) Clearly, these modes are distinct from those given in equation (7) since they are functions of the Jacobi polynomials P;:\((L2 - Ll)/(l - L 3 ) ) and Pi11:”(2L3 - 1) rather than Pf:”,L, - L , ) and P$!1(2L3 - 1). This is important because the form used in equation (7) was designed to maintaining orthogonality in the mass and Helmholtz matrices. The use of these rather unusual polynomials is also salient in maintaining an O ( L 2 ) scaling of the weak advection operator.” To our knowledge, it had not been shown that Szabo and Babuska’s basis will demonstrate this property. Once again the interior modes shown in equation (7) can be constructed from a product of polynomials in (L2 - Ll)/(l - L3) and L3. However, Szabo and Babuska’s basis are constructed from polynomials in L2 - L1 and 2L3 - 1. This is different than the product form of the edge modes which destroys the ability to use the sum factorization technique.

5. MULTI-DOMAIN TETRAHEDRAL BASIS

We define a polynomial basis, denoted by ‘gLn (r, s), so that we can approximate the function f(x, y , z ) by a Co continuous expansion over ‘K‘ sub-domains of the form

HIGH-ORDER ( h p ) FINITE E L E M E N T METHODS 3791

Vertices Edges Faces

Figure 8. Tetrahedron notation

Heref:,,, is the expansion coefficient corresponding to the expansion polynomial '42" in the kth sub-domain; (x, y, z) are the global spatial co-ordinates of the function and (r, s, t) are the local co-ordinates within any given tetrahedron.

Having defined the co-ordinates (a, b, c) in Section 2 we can now describe the expansion basis. Using the notation given in Figure 8 we have

Vertex modes

1-3"'"" (1 ; U ) ( l ; b ) ( 1 ;C) g100 = - - ~

1-3'ens 1 + u 1 - b 1 - C gloo = (i)(T)(T)

go10 = (T)(T) goo1 = (F)

1-3'e''C 1 + b 1 - c

1p3.enn

Edge modes (2 < I; 1 < m,n - I < L;f + m < M ; I + m + n < N )

1 - 3cd%c I (1 ; u ) ( l ; u ) P: : : (U) (1 ~ ;b)l(l ~ ;c)' groo = - __

1-3cdgr2 (l;U)(l;b)(l;b) Pk',(b) (1;')"" - g l m o = - - -

3192 S. J. SHERWIN A N D G. E. KARNIADAKIS

1-3'dsr5 ( l ; u ) ( l ~ b ) ( l ; c ) ( l ; c ) P,' "1 (c) glen = - - ~ ~

1 - 3 ~ ~ ' ~ ~ (1 ; b ) (1 ; c) (1 c ) P,' '_I1 (c) g o l n = - - -

Face modes (2 < I ; 1 < m,n - 1 < L ; I + m c M ; I + m + n < N )

1-3'""' (1 ~ -

a)( 1 a ) (1 b ) ( I - - b))' P;)':/-'(b) (1 - ; c)~'~ P:::(a) g l m o =

1-3 9 xlmn f a c e 3 = (' :")(' ')(' 2 b ) P ~ ~ l ( b ) ( ~ ) ( ~ ) m ' l P ~ m ~ l , l ~ c )

1 - a l + h 1 - b l + c l - c m + l i;:i 4= (--> (7) (i) P;! (b) (I) ( i) P:!!' q (c)

Interior modes (2 < I ; 1 d m,n - 1 < L; I + m < M ; 1 + m + n < N )

where the indices lmn in glmn refer to the order of the principal polynomial in a, b and c respectively, and L, M, N define the total number of ,modes.

As can be seen, the basis is split into four types of modes: interior,,fuces, edges and vertices. Iso-contours of representative modes for an expansion order of L = M = N = 5 are shown in Plate 2. The interior modes are zero at all boundaries whilst the face modes have non-zero contributions in the interior and on one face only. The edge modes have magnitude along one edge and are zero at all vertices and other edges. The vertex modes vary from a unit value at one vertex to zero at all other vertices. These modes are identical to linear finite element modes. Every mode is a polynomial in (a, b, c) space as well as ( r , s, t ) space. Once again to enforce Co continuity we require that the number of modes along any edge or any face are the same between two connecting elements. This condition can be satisfied by setting L = M = N ; for an L order expansion there are L ( L + 1) ( L + 2)/6 modes in three-dimensions. The expansion space is complete in a polynomial space YY defined by:

q9 = Span{rfsmtn},fmn,,,

where

p = { ( I , m, n) 10 < I , m, n; I i- m + n < L }

The order of polynomials in ( r , s, t ) that can be resolved for a given expansion order L when L = M = N can be seen in Figure 9. Here we have added the L = 1 order expansion which is the constant plane. On each plane of this diagram the order of each term is constant and equal to ( L - 1) .


Figure 9. Pascal’s diagram for h-p tetrahedra: Span of the polynomial space for expansions of order L = 1 to L = 6.

Similar to the two-dimensional expansion the order of the expansion can be varied within an element. This is possible since we only require that the number of modes at connecting faces and edges are the same. This requirement does not restrict us from allowing the expansion to change at different faces and edges or even in the interior of the expansion. So it is possible to have a quadratic-type expansion on one face and a quartic expansion on another. Therefore, it is possible to construct a conforming variable-order expansion in a very natural fashion.

In Figure 10 we see the structure of the mass matrix at an expansion order of L = M = N = 15. The ordering is such that the vertices are listed first, followed by the edges, then by the faces and finally the interior. For the interior modes the ‘n’ index is allowed to run fastest followed by the ‘m’ and then the ‘ I ’ index. Once again a definite structure is evident. The banded structure of the interior-interior matrix can be explained in exactly the same way as that of the interior-interior matrix for the two-dimensional basis as shown in Section 4. However, in this case the bandwidth is O(L2). It is worth noting that the stiffness matrix has a similar bandwidth as the mass matrix since the partial orthogonality of the basis is preserved for higher derivatives.

5.1. Multi-domain connectivity

In a standard finite element discretization there are 12 permutations in which a tetrahedron can connect to another tetrahedron as there are four faces and three ways that a face can be orientated. For our multi-domain expansion there are only four permutations. This would initially appear to be restrictive as it is not immediately evident that we can construct a matching

3794 S. J. SHERWIN AND G . E. KARNIADAKIS

- Vertices

Edges face 1

face 2

Faces face 3

face 4

Interior

Figure 10. The structure of the mass matrix on the standard tetrahedron for L = M = N = 15. The matrix is symmetric and has a rank of L ( L + 1)(L + 2)/6

connectivity with these reduced permutations given any conforming discretization using tetrahedra. However, in this section we shall show that such connectivity requirements can be satisfied implying that the connectivity constraint is non-restrictive.

To understand why only four out of the 12 potential tetrahedral connectivities are permissible we must consider the surface co-ordinate lines. The salient surface co-ordinate lines are shown in Figure Il(a). Here we see the lines that the co-ordinates ‘a’ and ‘ b make on the surface of the standard tetrahedron. Clearly, there is a definite clustering of these lines at two vertices. One vertex lies in the base of the tetrahedron and will be referred to as the base degenerate vertex (this vertex is the same as vertex C in Figure 8). The second vertex will be referred to as the most degenerate vertex (this vertex is the same as vertex D in Figure 8). Now the connectivity restriction requires that these co-ordinate lines are orientated in the same direction which means that a degenerate vertex must meet a degenerate vertex. This criterion is derived from the requirement that the surface modes must have a conforming shape when two tetrahedral elements meet. To help illustrate this criterion we shall introduce a diagrammatic representation of the degenerate vertices as shown in Figure l l(b). A triangle in the plane of the base of the tetrahedron points to the base degenerate vertex whilst a circle through the top vertex represents the most degenerate vertex. Finally, Figure 1 1 (c) illustrates one permissible connectivity where the base degenerate vertex meets the most degenerate vertex. There are, therefore, three types of connections: A most degenerate vertex meets another most degenerate vertex; a base degenerate vertex meets a most degenerate vertex; or a base degenerate vertex meets a base degenerate vertex. Finally, it should be noted that in any connectivity pattern the tetrahedra can always be rotated about the most degenerate vertex thereby changing the position of the base degenerate vertex.

If we assume that we have a conforming discretization we can generate the local orientation of the tetrahedra using the following algorithm. We assume that we have a list of vertices and we know a list of elements which touch each vertex, This list of elements will be called a vertex group and all elements are assumed to have a tag of zero.


Mastdegenerate /vertex

a) Surface co-ordinates lines

Most dqperate ..-/vertex

Brue degenerate

b) Diagrammatic representstion c) Face connectivity where base degenerate vertex meets most degenerate vertex of surface co-ordinates

Figure 1 I . Tetrahedron connectivity is dictated by the connectivity of the surface co-ordinates as indicated in (a). The surface co-ordinates can be represented diagrammatically as shown in (b). For the connectivity condition to be satisfied

the degenerate vertices must meet in a manner similar to that shown in (c)

For every vertex in the list:

(1) Orientate all elements with a tag of one in this vertex group so that their base degenerate

(2) Orientate all elements with a tag of zero in this vertex group so that their most degenerate vertex points at this vertex. Then set their tags to two.

vertex points at this vertex. Then set their tags to one.

This algorithm visits all vertices in the mesh and if this is the first time the elements in the vertex group have been visited the most degenerate vertex is orientated at this vertex. If this is the second time the elements in the vertex group have been visited then set the base degenerate vertex to this vertex. To see how this works we can consider the example shown in Figure 12.

Here we assume that we are given a discretization of a box using six tetrahedra as shown in Figure 12a. Starting our algorithm we begin with vertex A. Since all elements have a tag of zero at this point we go straight to the second part of the algorithm and orientate all elements that touch this vertex so that their most degenerate vertices point to A. Therefore tetrahedra HBDA and BHEA are orientated as shown in Figure 12(b) and now have a tag set to one. Continuing to the next vertex B we see that all elements belong to this vertex group. The first part of the algorithm is to orientate the elements with a tag of one to have their base degenerate vertex pointing at B. So the tetrahedra HBDA and BHEA are rotated as shown in Figure 12(c) and their tags are set to two. The second part of the algorithm then orientates all the other tetrahedra to have their most degenerate vertex pointing at B. The connectivity is actually satisfied at this point since the orientation the faces have on the boundaries is irrelevant. However, if we continue the algorithm

3796 S. J. SHERWIN AND G. E. KARNIADAKIS

A D

a) Initial d w e h t i o n of box b) First vertex group

d) Find element orkntatlon at end of algorithm

Figure 12. Setting up the required connectivity for the discretization of a box as shown in (a). Vertex A is given as the first most degenerate vertex as shown in (b). In (c) vertex B is then given as the second most degenerate vertex and the base degenerate vertices from group one are aligned to satisfy connectivity. The final element orientation is shown in

figure (d)

looping through the vertices consecutively we end up with the tetrahedra orientated as shown in Figure 12(d).

Clearly, the connectivity is not unique since any elements that have their most degenerate vertex pointing at E can be rotated about E. However, we have demonstrated that it is possible to satisfy the connectivity requirements imposed by the co-ordinate system and thereby imply that the requirement is non-restrictive. This implies that standard unstructured mesh algorithms used in CFD’9.2’ can be used here to generate the mesh. Work is currently underway to construct an automatic proceedure to perform this task at very low computational cost.=

5.2. Tetrahedral multi-domain basis in volume co-ordinates

Similar to the use of area or triangular co-ordinates in two dimensions there are volume or tetrahedral co-ordinates in three dimensions. The volume co-ordinates denoted by L 1 , L2, L3 and L4 are defined to have a unit value at the vertices marked 1,2,3 and 4 in Figure 13. Each co-ordinate decays linearly to zero at all other vertices. For the standard tetrahedral space that we have adopted (ie. T 3 = { - 1 ,< r , s , t ; I + s + t < - I}) the


(- 1 .- 1 ,-1)

Figure 13. In the standard tetrahedral space the tetrahedral co-ordinates L 1 , L 2 . L 3 and L4 are defined to have a unit value at the vertices marked 1, 2, 3 and 4, respectively. Each co-ordinate decays from this unit value to zero at all other

vertices

co-ordinates are defined by

- ( l + r + s + t ) (1 + r ) , L 2 = -

2 2 L 1 =

Once again these co-ordinates have the property that

Ll + L2 + L3 + L4 = 1

These co-ordinates can be expressed in terms of (a, b, c) as

(1 + a) (1 - b) (1 - c) , L 2 = - - - (1 - a) (1 - 6) (1 - c) L 1 = - - - 2 2 2 2 2 2

(1 + c) , L 4 = - (1 + b) (1 - C) L 3 = - - 2 2 2

If we re-arrange these expressions we can find a, b,c in terms of L 1 , L 2 , L3 and L4, i.e.

L2 - L1 (1 - L3 - L4)

a = 2L2 - 1 = (1 - L3 - L4)

c = 2L4 - 1

Therefore, we can write the tetrahedral expansion given in Section 5 in terms of the volume co-ordinates as

3798

Vertex modes

S . J . SHERWIN A N D G. E. KARNIADAKIS

1-3'c."'

go10 = L3

Edge modes (2 d I; 1 < m, n - 1 < L; 1 + m < M ; 1 + rn + n < N )

1 - 3 c d ~ e 4

g l o n = L,L4P:.',(2L4 - 1)

1 - 3 W c b

goln = L3L4P:>l1(2L4 - 1)

Face modes (2 < I ; 1 < m, n - 1 < L; I + m c M ; 1 + m + n < N )

L2 - L1 1 - L3 - L4

L2 - Ll - 3r4cc 2

glen = L1L2L,P::;( 1 - L 3 - J(1 - L3 - L4)1-2P,2!-1',1(2L4 - 1)

-3rAcc 3

g l m n = L2L3L4P;I1

- 3f"' 4

g l m n = L1L3L4P;11 ( __- ?L4 1 ) ( 1 - L4)m-l P,2m::'3'(2L4 - 1)

3799 HIGH-ORDER (kpj FINITE ELEMENT METHODS

The expansion is now a product of polynomials in (L2 - Ll)/(l - L3 - L4), 2L3/(1 - L4) and L4.

The Szabo and Babuska basis for the tetrahedron also uses the linear finite element as the vertex modes. Their edge modes are similar to the ones described in Section 4.1 and the edge between vertex 1 and 2 (i.e. side 1 ) has the form

Nilq2’ = LIL2(pi(L2 - Ll) , i = 2,. . . , p

As mentioned in Section 4.1, cp,(z) is proportional to the P,!L\(z) Jacobi polynomial. The other edge modes are defined analogously. Similar to the triangular construction these modes have a rotational symmetry although they do not maintain any consistent product form. The face modes of Szabo and Babuska’s basis are also analogous to the triangular definition having the form on the face defined by vertices 2, 3 and 4 (i.e. face 3):

A y j 3 . 4 ’ = L2L3L4Pp.o(L3 - L2)P9’0(2L4 - 1).

The interior modes are defined as

N!o;,, = L,L2L3L4PpS0(L2 - L1)P9’O(2L3 - l)P;-O(2L4 - 1 )

Although the general construction of the face and interior modes is similar to the basis we have presented above there are some notable distinctions. This basis only uses the Legendre polynomial Pp*’(z) instead of combinations of the general Jacobi polynomials. This is important since it is the introduction of the general Jacobi polynomial that helps maintain orthogonality. Also Szabo and Babuska’s basis does not use consistent polynomial variables in all the modes which means that we cannot apply the sum factorization technique.

6. NUMERICAL RESULTS

In this section we shall give some illustrative examples in both two- and three-dimensional domains which demonstrate the exponential convergence property of the expansion basis. The analysis of the two-dimensional basis (c = - 1) as well as its application to the incompressible Navier-Stokes equations has been described in Reference 18.

The first example is the two-dimensional domain shown in Plate 3. This domain uses a variety of triangular elements of different aspect ratios and orientations in an almost random triangular- ization. Within this domain, we have solved an elliptic Helmholtz problem of the form

V 2 U ( X , Y ) - U ( X , Y ) =fb, Y )

The exact solution that we considered was

d ( x - 15)2 + ( y - 8)2))

Also shown in Plate 3 are the L,, H I errors plotted with respect to expansion order. Exponential convergence is observed as indicated by the asymptotic linear behaviour of the curves on this lin-log plot. We note that the solution domain does not include the region immediately around x = 15, y = 8 since within this region it is not possible to bound all the derivatives of U ( X , Y).

3 800 S. J. SHERWIN AND G . E. KARNlADAKlS

Figure 14. Projection of a rotating solution in a helical domain. Here we see an iso-contour of the projection in two orientations

The first three-dimensional example is a projection of the function

To project this function we have performed a Galerkin type approximation of the solution using a collocation evaluation of the solution at the quadrature points.'* Essentially, this projection involves inverting a mass matrix system which is an equivalent operation, in our formulation, to inverting a Helmholtz matrix system. As shown in Figure 14 the function is projected in a helical domain with a similar rotation as the solution. This figure shows an iso-contour of the projection in two orientations.

Figure 15 demonstrates the exponential convergence possible with p-type refinement. Here we see the L, and L2 errors on a lin-log plot. Whilst a projection is not a solution to a partial differential equation does indicate how accurately a solution can be resolved by the expansion basis. Once again the asymptotically straight line implies exponential convergence with expansion order.

The final result shown in Plate 4 demonstrates the three-dimensional curvilinear capabilities of the algorithm. We see an engineering-type-geometry discretized using K = 144 tetrahedral elements. The curved surface was approximated using an iso-parametric representation. Within this domain we consider the solution to a Poisson problem and a solution of the form

u(x, y, z) = sin(48(x, z))cos(ny)exp( -r(x, 2))

O(x, z) = atan(x, z)

r(x, 2) = Jix2 + z2)

As can be seen the solution is easily represented in cylindrical co-ordinates although the calculation was performed in a Cartesian co-ordinate system. All Dirichlet boundary conditions


lo-. _..; ........

10-3 -..i ........ L 0 L

W 10.6 _..i ........

10-m -..i .......

10- :..; ..... t. I i

10-7 -L- 3 4 5 6 7 8 O 1 0 1 1 1 2 1 3

Expansion Order

Figure 15. Exponential convergence of the projection in Figure 14 in the L , and L2 norms

. 4 ' ", \

Figure 16. Converge in the L , and H , normal as s function of expansion order for the solution to the Poisson problem shown in Plate 4.

3802 S. J . SHERWIN AND G . E. KARNIADAKIS

were imposed. Figure 16 shows the error in both the L , and H I norms as a function of expansion. Exponential convergence rate is once again evident from the form of the log-lin plot.

ACKNOWLEDGEMENTS

We would like to acknowledge the assistance of T. Warburton in the construction of three- dimensional tetrahedral meshes and for his contribution in the tetrahedral connectivity issue. This work was supported by the Office of Naval Research, Air Force Office of Scientific Research and the National Science Foundation.

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REFERENCES

1. Babuska, B. A. Szabo and I . N. Katz, ‘The p-version of the finite element method’, SIAM J . Numer. Anal., 18, 515 (1981). 1. Babuska and M. R. Dorr, ‘Error estimates for the combined h- and p-version of the finite element method’, Numer. Math., 37, 257 (1981). A. T. Patera, ‘A spectral method for fluid dynamics: Laminar Row in a channel expansion’, J . Cornput. Phys., 54,468 (1984). G. E. Karniadakis, E. T. Bullister and A. T. Patera. ‘A spectral clement method for solution of two- and three- dimensional time dependent Navier-Stokes equations’. in Finite Element Methods lo r Nonlinear Problems, Springer, Berlin, 1985, p. 803. L. Demkowicz, J. T. Oden, W. Rachowicz and 0. Hardy, ‘Toward a universal h-p adaptive finite element strategy’, Part I . Constrained approximation and data structure. Comput. Methods Appl. Mech. Eng., 77, 79 (1989). C. Bernardi, Y. Maday and A. T. Patera, ‘A new conforming approach to domain decomposition: the mortar element method’, in Nonlinear Partial Differential Equations and Their Applications, Pitman and Wiley, London, 1992. R. D. Henderson and G. E. Karniadakis, ‘Unstructured spectral element methods for simulation of turbulent flows’, J . Cornput. Phys., 1995, in press. 1. Babuska and 9. Q. Guo, ‘The h-p version of the finite element method for domains with curved boundaries’, SIAM J . Numer. Anal., 25, 837 (1988). J. T. Oden, L. Demkowicz, W. Rachowicz and T. A. Westermann, ‘Toward a universal h-p adaptive finite element strategy. part 2. A posteriori error estimation’, Comput. Methods Appl. Mech. Eny., 77, 113 (1989). W. Rachowich, T. J. Oden and L. Demkowicz, ‘Toward a universal h-p adaptive finite element strategy part 3. Design of h-p meshes’, Comput. Methods Appl. Mech. Eng., 77, 181 (1989). 0. C. Zienkiewicz, J. G. Zhu and N. G. Gong, Effective and practical h-p adaptive analysis strategies for the finite element method’, Inr. j . numer. methods my., 28, 879 (1989). B. Szabo and 1. Babuska, Finite Element Analysis, Wiley, New York, 1991. 0. C. Zienkiewicz and R. L. Taylor, The Finite Element Method, Vol. 1, McGraw-Hill, New York, 1989. A. Peano, ‘Hierarchies of comforming finite elements for plane elasticity and plate bending’, Comput. Math. Appl., 2, 221 (1976). D. A. Dunavant, ‘High degree efficient symmetrical Gaussian quadrature rules for the triangle’, Int. j . numer. methods eng., 21, 1985. C . Canuto, M. Y. Husanini, A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988. M. Dubiner, ‘Spectral methods on triangles and other domains’, J. Sci. Cornput., 6 , 345 (1991). S. J. Sherwin and G. E. Karniadakis, ‘A triangular spectral element method; applications to the incompressible Navier-Stokes equations’, Comput. Meth. Appl. Mech. Eng., 1995, in press. T. J. Baker, ‘Developments and trends in three-dimensional mesh generation’, Appl. Numer. Math., 5, 275 (1989). K. Morgan, J. Peraire and J. Peiro, ‘Unstructured grid methods for compressible flows’, in AGARD Report R-787 Unstructured Grid Methods for Aduection Dominated Flows, AGARD, Paris, 1992. pp. 5.1-5.39. N. P. Weatherill and 0. Hassan, ‘Efficient three-dimensional Delaurnay triangularisation with automatic point creation and imposed boundary constaints’. Int. j . numer. methods eng., 37, 2005 (1994). T. J. R. Hughes, The Finite Element Method Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1987. 1. Babuska and M. Suri, ‘The p and h-p versions of the finite element method: Basic principles and properties’, Technical Report, Institute for Physical Science and Technology, University of Maryland, 1992. J. T. Oden, ‘Optimal hp-finite element methods’, Technical Report TICOM Report 92-09, University of Texas at Austin. 1992. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1970. T. C. Warburton, S . J. Sherwin and G. E. Karniadakis, ‘Extrusion of 3d meshes from 2d templates for computational mechanics problems’, in preparation.

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