A POSTERIORI ESTIMATES FOR CONFORMING KIRCHHOFFPLATE ELEMENTS∗

TOM GUSTAFSSON† , ROLF STENBERG‡ , AND JUHA VIDEMAN§

Abstract. We derive a residual a posteriori estimator for the Kirchhoff plate bending prob-lem. We consider the problem with a combination of clamped, simply supported and free boundaryconditions subject to both distributed and concentrated (point and line) loads. Extensive numericalcomputations are presented to verify the functionality of the estimators.

Key words. Kirchhoff plate, C1 elements, a posteriori estimates

AMS subject classifications. 65N30

1. Introduction. The purpose of this paper is to perform an a posteriori erroranalysis of conforming finite element methods for the classical Kirchhoff plate bendingmodel. So far this has not been done in full generality as it comes to the boundaryconditions. Most papers deal only with clamped or simply supported boundaries, see[29] for conforming C1 elements, [9, 17, 29] for the mixed Ciarlet–Raviart method([11]), and [8, 7, 19, 14, 21, 30, 22] for discontinuous Galerkin (dG) methods. Thefew papers that do address more general boundary conditions, in particular free, are[5, 20] in which the nonconforming Morley element is analysed, [3, 4] where a newmixed method is introduced and analysed, and [19] where a continuous/discontinuousGalerkin method is considered. One should also note that the Ciarlet–Raviart methodcannot even be defined for general boundary conditions. Free boundary conditionscould be treated using dG methods following an analysis similar to the one presentedhere.

In this study, we will derive a posteriori estimates using conforming methods andallowing for a combination of clamped, simply supported and free boundaries. Inaddition, we will investigate the effect of concentrated point and line loads, which arenot only admissible in our H2-conforming setting but of great engineering interest,on our a posteriori bounds in numerical experiments. We note that finite elementapproximation of elliptic problems with loads acting on lower-dimensional manifoldshas been considered by optimal control theory, see [16] and all the references therein.

The outline of the paper is the following. In Section 2, we recall the Kirchhoff-Loveplate model by presenting its variational formulation and the corresponding boundaryvalue problem. We perform this in detail for the following reasons. First, as notedabove, general boundary conditions are rarely considered in the numerical analysisliterature. Second, the free boundary conditions consist of a vanishing normal momentand a vanishing Kirchhoff shear force. These arise from the variational formulationvia successive integrations by parts. It turns out that the same steps are neededin the a posteriori analysis in order to obtain a sharp estimate, i.e. both reliable

∗Funding from Tekes – the Finnish Funding Agency for Innovation (Decision number3305/31/2015) and the Finnish Cultural Foundation is gratefully acknowledged, as as well as thefinancial support from FCT/Portugal through UID/MAT/04459/2013.†Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100,

00076 Aalto, Finland e-mail: (tom.gustafsson@aalto.fi).‡Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100,

00076 Aalto, Finland e-mail: (rolf.stenberg@aalto.fi).§CAMGSD/Departamento de Matematica, Instituto Superior Tecnico, Universidade de Lisboa,

Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal (jvideman@math.tecnico.ulisboa.pt).

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and efficient. In the following two sections, we present the classical conforming finiteelement methods and derive new a posteriori error estimates. In the last section, wepresent the results of our numerical experiments computed with the triangular Argyriselement. We consider the point, line and square load cases with simply supportedboundary conditions in a square domain as well as solve the problem in an L-shapeddomain with uniform loading using different combinations of boundary conditions.

2. The Kirchhoff plate model. The dual kinematic and force variables inthe model are the curvature and the moment tensors. Given the deflection u of themidsurface of the plate, the curvature is defined through

(2.1) K(u) = −ε(∇u),

with the infinitesimal strain operator defined by

(2.2) ε(v) =1

2

(∇v +∇vT

),

where (∇v)ij = ∂vi∂xj

. The dual force variable, the moment tensor M , is related to K

through the constitutive relation

(2.3) M(u) =d3

12CK(u),

where d denotes the plate thickness and where we have assumed an isotropic linearlyelastic material, i.e.

(2.4) CA =E

1 + ν

(A +

ν

1− ν(trA)I

), ∀A ∈ R2×2.

Here E and ν are the Young’s modulus and Poisson ratio, respectively. The shearforce is denoted by Q = Q(u). The moment equilibrium equation reads as

(2.5) div M(u) = Q(u),

where div is the vector-valued divergence operator applied to tensors. The transverseshear equilibrium equation is

(2.6) − divQ(u) = l,

with l denoting the transverse loading. Using the constitutive relationship (2.4), astraightforward elimination yields the well-known Kirchhoff–Love plate equation:

(2.7) A(u) := D∆2u = l,

where the so-called bending stiffness D is defined as

(2.8) D =Ed3

12(1− ν2).

Let Ω ⊂ R2 be a polygonal domain that describes the midsurface of the plate.The plate is considered to be clamped on Γc ⊂ ∂Ω, simply supported on Γs ⊂ ∂Ωand free on Γf ⊂ ∂Ω as depicted in Fig. 1. The loading is assumed to consist of adistributed load f ∈ L2(Ω), a load g ∈ L2(S) along the line S ⊂ Ω, and a point loadF at an interior point x0 ∈ Ω.

2

Next, we will turn to the boundary conditions, which are best understood fromthe variational formulation. (Historically, this was also how they were first discoveredby Kirchhoff, cf. [27].) The elastic energy of the plate as a function of the deflectionv is 1

2a(v, v), with the bilinear form a defined by

(2.9) a(w, v) =

∫Ω

M(w) : K(v) dx =

∫Ω

d3

12C ε(∇w) : ε(∇v) dx,

and the potential energy due to the loading is

(2.10) l(v) =

∫Ω

fv dx+

∫S

gv ds+ Fv(x0).

Defining the space of kinematically admissible deflections

(2.11) V = v ∈ H2(Ω) : v|Γc∪Γs = 0, ∇v · n|Γc = 0,

minimization of the total energy

(2.12) u = argminv∈V

1

2a(v, v)− l(v)

leads to the following problem formulation.

Problem 1 (Variational formulation). Find u ∈ V such that

(2.13) a(u, v) = l(v) ∀v ∈ V.

To derive the corresponding boundary value problem, we recall the followingintegration by parts formula, valid in any domain R ⊂ Ω

(2.14)

∫R

M(w) : K(v) dx

=

∫R

div M(w) · ∇v dx−∫∂R

M(w)n · ∇v ds

=

∫R

A(w) v dx+

∫∂R

Q(w) · n v ds−∫∂R

M(w)n · ∇v ds,

At the boundary ∂R, the correct physical quantities are the components in the normaln and tangential s directions. Therefore, we write

(2.15) ∇v =∂v

∂nn +

∂v

∂ss

and define the normal shear force and the normal and twisting moments as

(2.16)Qn(w) = Q(w) · n, Mnn(w) = n ·M(w)n,

Mns(w) = Msn(w) = s ·M(w)n.

With this notation, we can write

(2.17)

∫∂R

Q(w) · n v ds−∫∂R

M(w)n · ∇v ds

=

∫∂R

Qn(w)v ds−∫∂R

(Mnn(w)

∂v

∂n+Mns(w)

∂v

∂s

)ds,

3

and thus rewrite the integration by parts formula (2.14) as

(2.18)

∫R

M(w) : K(v) dx

=

∫R

A(w) v dx+

∫∂R

Qn(w)v ds

−∫∂R

(Mnn(w)

∂v

∂n+Mns(w)

∂v

∂s

)ds.

The key observation for deriving the correct boundary conditions is that, at anyboundary point, a value of v specifies also ∂v

∂s . Defining the Kirchhoff shear force(cf. [12, 24, 13])

(2.19) Vn(w) = Qn(w) +∂Mns(w)

∂s

an integration by parts on a smooth part S of ∂R yields

(2.20)

∫SQn(w)v ds−

∫SMns(w)

∂v

∂sds =

∫SVn(w) ds−

∣∣∣baMns(w)v,

where a and b are the endpoints of S.We are now in the position to state the boundary value problem for the Kirchhoff

plate model. Assuming a smooth solution u in (2.13), we have

(2.21)

a(u, v) =

∫Ω

A(u)v dx+

∫∂Ω

Qn(u)v ds

−∫∂Ω

(Mnn(u)

∂v

∂n+Mns(u)

∂v

∂s

)ds.

With the combination of clamped, simply supported and free boundary conditions at∂Ω = Γc ∪ Γs ∪ Γf , we have for any v ∈ V ,

(2.22)

∫∂Ω

Qn(u)v ds−∫∂Ω

(Mnn(u)

∂v

∂n+Mns(u)

∂v

∂s

)ds

=

∫Γf

Qn(u)v ds−∫

Γf

Mns(u)∂v

∂sds−

∫Γs∪Γf

Mnn(u)∂v

∂nds.

In the final step, we integrate by parts at the free part of the boundary. To this end,let Γf = ∪m+1

i=1 Γif , with Γif smooth. Integrating by parts over Γif yields

(2.23)

∫Γif

Qn(u)v ds−∫

Γif

Mns(u)∂v

∂sds =

∫Γif

Vn(u)v ds−∣∣∣cici−1

Mns(u)v

where c0 and cm+1 are the end points of Γf and ci, i = 1, . . . ,m, its successive interiorcorners. Combining equations (2.21)–(2.23), and noting that v(c0) = v(cm+1) = 0,gives finally

(2.24)

a(u, v) =

∫Ω

A(u)v dx−∫

Γs∪Γf

Mnn(u)∂v

∂nds

+

m+1∑i=1

∫Γif

Vn(u)v ds−m∑i=1

(Mns(u)|ci+ −Mns(u)|ci−

v(ci),

4

where Mns(u)|ci± = limε→0+Mns(u)|ci+ε(ci±1−ci)Choosing v ∈ V in such a way that three of the four terms in (2.24) vanish and

the test function in the fourth term remains arbitrary and repeating this for eachterm, we arrive at the following boundary value problem:

• In the domain we have the distributional differential equation

(2.25) A(u) = l in Ω,

where l is the distribution defined by (2.10).• On the clamped part we have the conditions

(2.26) u = 0 and∂u

∂n= 0 on Γc.

• On the simply supported part it holds

(2.27) u = 0 and Mnn(u) = 0 on Γs.

• On the free part it holds

(2.28) Mnn(u) = 0 and Vn(u) = 0 on Γif , i = 1, . . . ,m.

• At the interior corners on the free part, we have the matching condition onthe twisting moments

(2.29) Mns(u)|ci+ = Mns(u)|ci− for all corners ci, i = 1, . . . ,m.

f

Fg

ci

Γc

Γs

Γf

Fig. 1. Definition sketch of a Kirchhoff plate with the different loadings and boundary conditions.

3. The finite element method and the a posteriori error analysis. Thefinite element method is defined on a mesh Ch consisting of shape regular triangles.We assume that the point load is applied on a node of the mesh. Further, we assumethat the triangulation is such that the applied line load is on element edges. We denotethe edges in the mesh by Eh and divide them into the following parts: the edges inthe interior E ih, the edges on the curve of the line load ESh ⊂ E ih, and the edges on the

free and simply supported boundary, Efh and Esh, respectively. The conforming finiteelement space is denoted by Vh. Different choices for Vh are presented in Section 4.Note that we often write a . b (or a & b ) when a ≤ Cb (or a ≥ Cb) for some positiveconstant C independent of the finite element mesh.

5

Problem 2 (The finite element method). Find uh ∈ Vh such that

(3.1) a(uh, vh) = l(vh) ∀vh ∈ Vh.

Let K and K ′ be two adjoining triangles with normals n and n′, respectively,and with the common edge E = K ∩K ′. On E we define the following jumps

(3.2) JMnn(v)K|E = Mnn(v)−Mn′n′(v)

and

(3.3) JVn(v)K|E = Vn(v) + Vn′(v).

In the analysis, we will need the Girault–Scott [15] interpolation operator Πh :V → Vh for which the following estimate holds

(3.4)

∑K∈Ch

h−4K ‖w −Πhw‖20,K +

∑E∈Eh

h−1E ‖∇(w −Πhw)‖20,E

+∑E∈Eh

h−3E ‖w −Πhw‖20,E . ‖w‖22 and ‖Πhw‖2 . ‖w‖2.

Note that the Girault–Scott interpolant uses point values at the vertices of the mesh.We use this property in the proof of Theorem 1 to derive a proper upper bound forthe error in terms of the edge residuals.

Next, we formulate an a posteriori estimate for Problem 2. The local error indi-cators are the following:

• The residual on each element

h2K‖A(uh)− f‖0,K , K ∈ Ch.

• The residual of the normal moment jump along interior edges

h1/2E ‖JMnn(uh)K

∥∥0,E

, E ∈ E ih.

• The residual of the jump in the effective shear force along interior edges

h3/2E ‖JVn(uh)K− g‖0,E , E ∈ ESh ,

h3/2E ‖JVn(uh)K‖0,E , E ∈ E ih \ ESh .

• The normal moment on edges at the free and simply supported boundaries

h1/2E ‖Mnn(uh)

∥∥0,E

, E ∈ Efh ∪ Esh.

• The effective shear force along edges at the free boundary

h3/2E ‖Vn(uh)‖0,E , E ∈ Efh .

The global error estimator is then defined through

(3.5)

η2 =∑K∈Ch

h4K‖A(uh)− f‖20,K +

∑E∈ESh

h3E‖JVn(uh)K− g‖20,E

+∑

E∈Eih\ESh

h3E‖JVn(uh)K‖20,E +

∑E∈Eih

hE‖JMnn(uh)K∥∥2

0,E

+∑E∈Efh

h3E‖Vn(uh)‖20,E +

∑E∈Efh∪E

sh

hE‖Mnn(uh)∥∥2

0,E.

6

Theorem 1 (A posteriori estimate). The following estimate holds

(3.6) ‖u− uh‖2 . η.

Proof. Let w = u − uh and w := Πhw ∈ Vh be its interpolant. In view of thewell-known coercivity of the bilinear form a and Galerkin orthogonality, we have

(3.7)‖u− uh‖22 . a(u− uh, w) = a(u− uh, w − w)

= l(w − w)− a(uh, w − w).

Since x0 is a mesh node and the interpolant uses nodal values, we have

(3.8) F(w(x0)− w(x0)

)= 0,

and hence

(3.9) l(w − w) = (f, w − w) + 〈g, w − w〉S .

From integration by parts over the element edges, using the fact that the interpolantuses values at the nodes, it then follows that

(3.10)

‖u− uh‖22. (f, w − w) + 〈g, w − w〉S − a(uh, w − w)

= (f, w − w) + 〈g, w − w〉S−∑K∈Ch

(A(uh), w − w)K + 〈Qn(uh), w − w〉∂K

− 〈Mns(uh), ∂∂s (w − w)〉∂K − 〈Mnn(uh), ∂∂n (w − w)〉∂K

= (f, w − w) + 〈g, w − w〉S−∑K∈Ch

(A(uh), w − w)K + 〈Vn(uh), w − w〉∂K

− 〈Mnn(uh), ∂∂n (w − w)〉∂K.

Regrouping and recalling definitions (3.2) and (3.3), yields

(3.11)

‖u− uh‖22.∑K∈Ch

(f −A(uh), w − w)K

−∑E∈ESh

〈JVn(uh)K− g, w − w〉E −∑

E∈Eih\ESh

〈JVn(uh)K, w − w〉E

−∑E∈Eih

〈JMnn(uh)K, ∂∂nE

(w − w)〉E

−∑E∈Efh

〈Vn(uh), w − w〉E −∑

E∈Efh∪Esh

〈Mnn(uh), ∂∂nE

(w − w)〉E .

The asserted a posteriori estimate now follows by applying the Cauchy–Schwarz in-equality and the interpolation estimate (3.4).

7

Instead of the jump terms in the estimator η, we could consider the normal andtwisting moment jumps

h1/2E ‖JMnn(uh)K

∥∥0,E

, h1/2E ‖JMns(uh)K

∥∥0,E

,

and the normal shear force jumps

h3/2E ‖JQn(uh)K‖0,E , h

3/2E ‖JQn(uh)K− g‖0,E .

In this case we cannot, however, prove the efficiency, i.e. the lower bounds.Next, we will consider the question of efficiency. Let fh ∈ Vh be the interpolant

of f and define

(3.12) oscK(f) = h2K‖f − fh‖0,K .

Similarly, for a polynomial approximation gh of g on E ⊂ S we define

(3.13) oscE(g) = h3/2E ‖g − gh‖0,E .

In the following theorem, ωE stands for the union of elements sharing an edge E. Inits proof, we will adopt some of the techniques used in [18].

Theorem 2 (Lower bounds). For all vh ∈ Vh it holds

h2K‖A(vh)− f‖0,K . ‖u− vh‖2,K + oscK(f), K ∈ Ch,(3.14)

h1/2E ‖JMnn(vh)K‖0,E . ‖u− vh‖2,ωE

+∑K⊂ωE

oscK(f), E ∈ E ih,(3.15)

h3/2E ‖JVn(vh)K‖0,E . ‖u− vh‖2,ωE

+∑K⊂ωE

oscK(f), E ∈ E ih \ ESh ,(3.16)

h3/2E ‖JVn(vh)K− g‖0,E . ‖u− vh‖2,ωE

+∑K⊂ωE

oscK(f) + oscE(g), E ∈ ESh ,(3.17)

h1/2E ‖Mnn(vh)‖0,E . ‖u− vh‖2,ωE

+∑K⊂ωE

oscK(f), E ∈ Efh ∪ Esh,(3.18)

h3/2E ‖Vn(vh)‖0,E . ‖u− vh‖2,ωE

+∑K⊂ωE

oscK(f), E ∈ Efh .(3.19)

Proof. Denote by bK ∈ P6(K) the sixth order bubble that, together with its first-order derivatives, vanishes on ∂K, i.e. let bK = (λ1,Kλ2,Kλ3,K)2, where λj,K are thebarycentric coordinates for K. Then we define

(3.20) γK = bKh4K(A(vh)− fh) in K and γK = 0 in Ω \K,

for vh ∈ Vh. The problem statement gives

(3.21) aK(u, γK) = (f, γK)K ,

where aK(u, γK) =∫KM(u) : K(γK) dx. We have

(3.22)

h4K‖A(vh)− fh‖20,K . h4

K‖√bK(A(vh)− fh)‖20,K

= (A(vh)− fh, γK)K

= (A(vh), γK)K − (f, γK)K + (f − fh, γK)K

= aK(vh − u, γK) + (f − fh, γK)K .

8

The local bound (3.14) now follows from applying the continuity of a, the Cauchy–Schwarz inequality and inverse estimates.

Next, consider inequality (3.15). Suppose E = K1 ∩K2 for the triangles K1 andK2; thus ωE = K1 ∪K2. Let λE ∈ P1(ωE) be the linear polynomial satisfying

(3.23) λE |E = 0, and∂λE∂nE

= 1,

and let p1 be the polynomial that satisfies p1|E = JMnn(vh)K|E and ∂p1∂nE|E = 0.

Moreover, let p2 ∈ P8(ωE) be the eight-order bubble that takes value one at themidpoint of the edge E and, together with its first-order derivatives, vanishes on∂ωE . Define w = λEp1p2. Since

(3.24)∂w

∂nE

∣∣∣E

=∂λE∂nE

JMnn(vh)Kp2 = JMnn(vh)Kp2,

scaling yields the equivalence

(3.25)‖JMnn(vh)K‖20,E ≈

∥∥∥ ∂w∂nE

∥∥∥2

0,E≈ ‖√p2 JMnn(vh)K‖20,E

=⟨JMnn(vh)K, ∂w∂nE

⟩E.

Furthermore, since

(3.26)∂w

∂s

∣∣∣E

= 0, w|E∪∂ωE= 0 and ∇w|∂ωE

= 0,

the integration by parts formula (2.18) yields

(3.27)⟨JMnn(vh)K, ∂w∂nE

⟩E

= −∫ωE

M(vh) : K(w) dx+ (A(vh), w)ωE.

Extending w by zero to Ω \ ωE , we obtain from the problem statement (2.13)

(3.28)

∫ωE

M(u) : K(w) dx− (f, w)ωE= 0.

Hence, using the Cauchy–Schwarz inequality, we get from (3.27)

(3.29)

⟨JMnn(vh)K, ∂w∂nE

⟩E

=

∫ωE

M(u− vh) : K(w) dx+ (A(vh)− f, w)ωE

. ‖u− vh‖2,ωE|w|2,ωE

+ ‖A(vh)− f‖0,ωE‖w‖0,ωE

.

By scaling, one easily shows that

(3.30) |w|2,ωE. h

−1/2E

∥∥∥ ∂w∂nE

∥∥∥0,E

and ‖w‖0,ωE. h

3/2E

∥∥∥ ∂w∂nE

∥∥∥0,E

.

The estimate (3.15) then follows from (3.24), (3.25), (3.29), (3.30), and the alreadyproved bound (3.14).

Since (3.16) follows from (3.17) with g = 0, we prove the latter. Due to theregularity condition imposed on the mesh there exists for each edge E a symmetric

9

pair of smaller triangles (K ′1,K′2) that satisfy ω′E = K ′1 ∪K ′2 ⊂ ωE , see Fig. 2. Let

w′ = p′2(JVn(vh)K− gh) where p′2 is the eight-order bubble that takes value one at themidpoint of E and, together its first-order derivatives, vanishes on ∂ω′E . By the normequivalence, we first have

(3.31) ‖JVn(vh)K− gh‖20,E ≈ ‖w′‖20,E . 〈JVn(vh)K− gh, w′〉E .

Next, we write

(3.32) 〈JVn(vh)K− gh, w′〉E = 〈JVn(vh)K− g, w′〉E + 〈g − gh, w′〉E .

Due to symmetry, ∂w′

∂n |E = 0, and hence (2.18) and (2.20) give

(3.33)

〈JVn(vh)K− g, w′〉E = 〈JVn(vh)K, w′〉E − 〈g, w′〉E

=

∫ω′E

M(vh) : K(w′) dx− (A(vh), w′)ω′E − 〈g, w′〉E .

Extending w′ by zero to Ω \ ω′E , the variational form (2.13) implies that

(3.34)

∫ω′E

M(u) : K(w′) dx− (f, w′)ω′E − 〈g, w′〉E = 0 .

Hence,

(3.35) 〈JVn(vh)K− g, w′〉E =

∫ω′E

M(vh − u) : K(w′) dx+ (f −A(vh), w′)ω′E

and the Cauchy–Schwarz inequality, scaling estimates and (3.14) give

(3.36)

〈JVn(vh)K− g, w′〉E. h

−3/2E

(‖u− vh‖2,ω′E + h2

K‖A(vh)− f‖0,ω′E)‖w′‖0,E

. h−3/2E

(‖u− vh‖2,ωE

+∑K∈ωE

oscK(f))‖w′‖0,E .

The asserted estimate then follows from (3.31), (3.32) and (3.36).The estimates (3.18), (3.19), are proved similarly to the bounds (3.15) and (3.16),

respectively.

The above estimates provide the following global bound.

Theorem 3. It holds

(3.37) η . ‖u− uh‖2 + osc(f) + osc(g),

where

(3.38) osc(f) =

√∑K∈Ch

oscK(f)2 and osc(g) =

√∑E∈ESh

oscE(g)2.

10

E

K2

K ′2

K1

K ′1

Fig. 2. A depiction of the sets ωE (the entire polygon) and ω′E (the grey area). The trianglesK′1 and K′2 are symmetric with respect to the edge E.

4. The choice of Vh. Let us briefly discuss some possible choices of conformingfinite elements for the plate bending problem. Each choice consists of a polynomialspace P and of a set ofN degrees of freedom defined through a functional L : C∞ → R.We denote by xk, k ∈ 1, 2, 3, the vertices of the triangle and by ek, k ∈ 1, 2, 3,the midpoints of the edges, i.e.

(4.1) e1 =1

2(x1 + x2), e2 =

1

2(x2 + x3), e3 =

1

2(x1 + x3).

The simplest H2-conforming triangular finite element that is locally H4(K) ineach K is the Bell triangle.

Definition 1 (Bell triangle, N = 18).

P = p ∈ P5(K) : ∂p∂n ∈ P3(E) ∀E ⊂ K(4.2)

L(w) =

w(xk), for 1 ≤ k ≤ 3,

∂w∂xi

(xk), for 1 ≤ k ≤ 3 and 1 ≤ i ≤ 2,

∂2w∂xi∂xj

(xk), for 1 ≤ k ≤ 3 and 1 ≤ i, j ≤ 2.

(4.3)

Even though the polynomial space associated with the Bell triangle is not thewhole P5(K) it is still larger than P4(K). This can in some cases complicate theimplementation. Moreover, the asymptotic interpolation estimates for P5(K) are notobtained. This can be compensated by adding three degrees of freedom at the mid-points of the edges of the triangle and increasing accordingly the size of the polynomialspace.

11

Definition 2 (Argyris triangle, N = 21).

P = P5(K),(4.4)

L(w) =

w(xk), for 1 ≤ k ≤ 3,

∂w∂xi

(xk), for 1 ≤ k ≤ 3 and 1 ≤ i ≤ 2,

∂2w∂xi∂xj

(xk), for 1 ≤ k ≤ 3 and 1 ≤ i, j ≤ 2,

∂w∂n (ek), for 1 ≤ k ≤ 3.

(4.5)

The Argyris triangle can be further generalized to higher-order polynomial spaces,cf. P. Solın [26]. Triangular macroelements such as the Hsieh–Clough–Tocher triangleare not locally H4(K) and therefore additional jump terms are present inside theelements. Various conforming quadrilateral elements have been proposed in the liter-ature for the plate bending problem cf. Ciarlet [10]. The proofs of the lower boundthat we presented do not directly apply to quadrilateral elements, but the techniquescan be adapted to them as well.

5. Numerical results. In our examples, we will use the fifth degree Argyristriangle. On a uniform mesh for a solution u ∈ Hr(Ω), with r ≥ 2, we thus have theerror estimate [10]

(5.1) ‖u− uh‖2 . hs|u|r,

with s = minr − 2, 4. Since the mesh length is related to the number of degrees offreedom N by h ∼ N−1/2 on a uniform mesh, we can also write

(5.2) ‖u− uh‖2 . N−s/2|u|r.

If the solution is smooth, say r ≥ 6, we thus have the estimates

(5.3) ‖u− uh‖2 . h4 and ‖u− uh‖2 . N−2.

In fact, the rate N−2 is optimal also on a general mesh since, except for a polynomialsolution, it holds [2, 1]

(5.4) ‖u− uh‖2 & N−2.

In the adaptive computations we use the following strategy for marking the ele-ments that will be refined [29].

Algorithm 1. Given a partition Ch, error indicators ηK , K ∈ Ch and a thresholdθ ∈ (0, 1), mark K for refinement if ηK ≥ θmaxK′∈Ch ηK′ .

The parameter θ has an effect on the portion of elements that are marked, i.e. forθ = 0 all elements are marked and for θ = 1 only the element with the largest errorindicator value is marked. We simply take θ = 0.5 which has proven to be a feasiblechoice in most cases.

The set of marked elements are refined using Triangle [25], version 1.6, by requir-ing additional vertices at the edge midpoints of the marked elements and by allowingthe mesh generator to improve mesh quality through extra vertices. The defaultminimum interior angle constraint of 20 degrees is used.

The regularity of the solution depends on the regularity of the load and the cornersingularities, cf. [6]. Below we consider two sets of problems, one where the regularityis mainly restricted by the load, and another one where the load is uniform and thecorner singularities dominate.

12

5.1. Square plate, Navier solution. A classical series solution to the Kirch-hoff plate bending problem, the Navier solution [28], in the special case of a unitsquare with simply supported boundaries and the loading

(5.5) f(x) =

f0, if x ∈ [ 1

2 − c,12 + c]× [ 1

2 − d,12 + d],

0, otherwise,

reads

(5.6) u(x, y) =16f0

Dπ6

∞∑m=1

∞∑n=1

sin mπ2 sin nπ

2 sinmπc sinnπd

mn(m2 + n2)2sinmπx sinnπy.

In the limit c −→ 0 and 2cf0 −→ g0 we get the line load solution

(5.7) u(x, y) =8g0

Dπ5

∞∑m=1

∞∑n=1

sin mπ2 sin nπ

2 sinnπd

n(m2 + n2)2sinmπx sinnπy,

and in the limit c, d −→ 0 and 4cdf0 −→ F0 we obtain the point load solution

(5.8) u(x, y) =4F0

Dπ4

∞∑m=1

∞∑n=1

sin mπ2 sin nπ

2

(m2 + n2)2sinmπx sinnπy.

From the series we can infer that the solution is inH3−ε(Ω), H7/2−ε(Ω) andH9/2−ε(Ω),for any ε > 0, for the point load, line load and the square load, respectively. in thethree cases. On a uniform mesh, one should thus observe the convergence ratesN−0.5, N−0.75, and N−1.25.

An unfortunate property of the series solutions is that the partial sums convergevery slowly. This makes computing the difference between the finite element solutionand the series solution in H2(Ω) and L2(Ω)-norms a challenging task since the finiteelement solution quickly ends up being more accurate than any reasonable partial sum.In fact, the ”exact” series solution is practically useless, for example, for computingthe shear force which is an important design parameter.

The H2(Ω)-norm is equivalent to the energy norm,

(5.9) |||v||| =√a(v, v),

with which the error is straightforward to compute. In view of the Galerkin orthogo-nality and symmetry, one obtains

(5.10) |||u− uh|||2 = a(u− uh, u) = l(u− uh),

i.e. the error is given by

(5.11) |||u− uh||| =√l(u− uh).

This is especially useful for the point load for which

(5.12) |||u− uh||| =√F0

(u( 1

2 ,12 )− uh( 1

2 ,12 )).

13

Evaluating the series solution at the point of maximum deflection gives [28]

(5.13)

u( 12 ,

12 ) =

4F0

Dπ4

∞∑m=1

∞∑n=1

(sin mπ2 sin nπ

2 )2

(m2 + n2)2

=4F0

Dπ4

∞∑m=1

(sin

mπ

2

)2 ∞∑n=1

(sin nπ

2

)2(m2 + n2)2

=F0

2Dπ3

∞∑m=1

(sin mπ2 )2(sinhmπ −mπ)

m3(1 + coshmπ).

We first consider a point load with F0 = 1, d = 1, E = 1 and ν = 0.3, and comparethe true error with the estimator η. In this case, we have the approximate maximumdisplacement u( 1

2 ,12 ) ≈ 0.1266812, computed by evaluating and summing the first 10

million terms of the series (5.13). Starting with an initial mesh shown in Fig. 3, werepeatedly mark and refine the mesh to obtain a sequence of meshes, see Fig. 4 wherethe values of the elementwise error estimators are depicted for four consecutive meshes.Note that the estimator and the adapted marking strategy initially refine heavily inthe neighborhood of the point load as one might expect based on the regularity of thesolution in the vicinity of the point load.

In addition to the adaptive strategy, we solve the problem using a uniform meshfamily where we repeatedly split each triangle into four subtriangles starting from theinitial mesh of Fig. 3. The energy norm error and η versus the number of degrees offreedom N are plotted in Fig. 5. The results show that the adaptive meshing strategyimproves significantly the rate of convergence in the energy norm. In Fig. 5, we havealso plotted, for reference, the slopes corresponding to the expected convergence rateO(N−0.5) for uniform refinement and the optimal convergence rate for P5 elements,O(N−2).

In Fig. 5 is is further revealed that the energy norm error and the estimator ηfollow similar trends. This is exactly what one would expect given that the estimatoris an upper and lower bound for the true error modulo an unknown constant. Thisis better seen by drawing the normalized ratio η over |||u− uh|||, see Fig. 6. Since theestimator correctly follows the true error and an accurate computation of norms like‖u−uh‖2 is expensive, the rest of the experiments document only the values of η andN for the purpose of giving idea of the convergence rates.

We continue with the line load case taking g0 = 1 and d = 13 , and using the

same material parameter values as before. The initial and final meshes are shown inFig. 7. The estimator can be seen to primarly focus on the end points of the line load.The values of η and N are visualized in Fig. 9 together with the expected and theoptimal rates of convergence. Again the adaptive strategy improves the convergenceof the total error in comparison to the uniform refinement strategy. The local errorestimators and the adaptive process are presented in Fig. 8.

We finish this subsection by solving the square load case with f0 = 1, c = d = 13

and the same material parameters as before. The initial and the final meshes areshown in Fig. 10. The convergence rates are visualized in Fig. 12 and the local errorestimators in Fig. 11. An improvement in the convergence rate is again visible in theresults.

5.2. L-shaped domain. Next we solve the Kirchhoff plate problem in L-shapeddomain with uniform loading f = 1 and the following three sets of boundary condi-tions:

14

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Fig. 3. Initial and six times refined meshes in the point load case.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

0.36

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.005

0.010

0.015

0.020

0.025

0.030

0.035

0.040

0.045

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

0.0175

0.0200

0.0225

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.0015

0.0030

0.0045

0.0060

0.0075

0.0090

0.0105

0.0120

Fig. 4. Elementwise error estimators in the point load case.

15

102 103

10−2.5

10−2

10−1.5

N

|||u−uh|||

UniformAdaptive

102 103

10−1

100

O(N−0.5)

O(N−2)

N

η

UniformAdaptive

Fig. 5. The results of the point load case.

1. Simply supported on all boundaries.2. Clamped on all boundaries.3. Free on the edges sharing the re-entrant corner and simply supported along

the rest of the boundary.Due to the presence of a re-entrant corner, the solutions belong to H2.33(Ω), H2.54(Ω)and H2.64(Ω) in the cases 1, 2 and 3, respectively(see [23]). As before, we use fifth-order Argyris elements to demonstrate the effectiveness of the adaptive solution strat-egy. The initial and the final meshes are shown in Fig. 13. The resulting total errorestimators and unknown counts are visualized in Fig. 14.

Acknowledgements. The authors thank the two anonymous referees and Prof.A. Ern for comments that improved the final version of the paper.

REFERENCES

16

100 200 300 400 500 600

0.9

1

1.1

N

c−1

η|||u−uh|||

Fig. 6. The efficiency of the estimator in the point load case. The normalization parameter cis chosen as the mean value of the ratio η/|||u− uh|||.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Fig. 7. Initial and 6 times refined meshes for the line load case.

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17

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.0005

0.0010

0.0015

0.0020

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0.2

0.4

0.6

0.8

1.0ηK

0.00015

0.00030

0.00045

0.00060

0.00075

0.00090

0.00105

0.00120

0.00135

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0.2

0.4

0.6

0.8

1.0ηK

0.00004

0.00008

0.00012

0.00016

0.00020

0.00024

0.00028

0.00032

0.00036

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.000015

0.000030

0.000045

0.000060

0.000075

0.000090

0.000105

0.000120

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.000008

0.000016

0.000024

0.000032

0.000040

0.000048

0.000056

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.000003

0.000006

0.000009

0.000012

0.000015

0.000018

0.000021

0.000024

Fig. 8. Elementwise error estimators for the line load case.

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18

103 10410−4

10−3

10−2

10−1

O(N−0.75)

O(N−2)

N

η

UniformAdaptive

Fig. 9. Results of the line load case.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Fig. 10. Initial and 8 times refined meshes for the square load case.

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19

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.00015

0.00030

0.00045

0.00060

0.00075

0.00090

0.00105

0.00120

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035

0.00040

0.00045

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

0.00014

0.00016

0.00018

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.00001

0.00002

0.00003

0.00004

0.00005

0.00006

0.00007

0.00008

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.000008

0.000016

0.000024

0.000032

0.000040

0.000048

0.000056

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.000004

0.000008

0.000012

0.000016

0.000020

0.000024

0.000028

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.0000015

0.0000030

0.0000045

0.0000060

0.0000075

0.0000090

0.0000105

0.0000120

0.0000135

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0ηK

0.0000008

0.0000016

0.0000024

0.0000032

0.0000040

0.0000048

0.0000056

0.0000064

Fig. 11. Elementwise error estimators in the square load case.

20

103 104

10−4

10−3

10−2

O(N−1.25)

O(N−2)

N

η

UniformAdaptive

Fig. 12. Results of the square load case.

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21

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

1.0 0.5 0.0 0.5 1.01.0

0.5

0.0

0.5

1.0

Fig. 13. The initial (top right) and the final meshes with different boundary conditions. Theboundary conditions at the re-entrant corner are either simply supported (top right), clamped (bottomleft) or free (bottom right). Surprisingly enough, the meshes for the simply supported and clampedboundaries end up being exactly the same.

22

103 104

10−0.2

100

100.2

O(N−0.17)

O(N−2)

N

η

UniformAdaptive

103 104

10−1

100

O(N−0.27)

O(N−2)

N

η

UniformAdaptive

103 104

10−1.5

10−1

10−0.5

O(N−0.32)

O(N−2)

N

η

UniformAdaptive

Fig. 14. L-shaped domain results. Simply supported (top), clamped (middle) and free (bottom)boundary conditions on the re-entrant corner.

23