AN UNCONSTRAINED MIXED METHOD FOR THE BIHARMONICPROBLEM∗

CESARE DAVINI† AND IGINO PITACCO†

SIAM J. NUMER. ANAL. c© 2000 Society for Industrial and Applied MathematicsVol. 38, No. 3, pp. 820–836

Abstract. In this work we present a finite element method for the biharmonic problem basedon the primal mixed formulation of Ciarlet and Raviart [A mixed finite element method for thebiharmonic equation, in Symposium on Mathematical Aspects of Finite Elements in Partial Differ-ential Equations, C. de Boor, ed., Academic Press, New York, 1974, pp. 125–143]. We introduce adual mesh and a suitable approximation of the constraint that enables us to eliminate the auxiliaryvariable with no computational effort. Thus, the discrete problem turns to be governed by a systemof linear equations with symmetric and positive definite coefficients and can be solved by classicalalgorithms. The construction of the stiffness matrix is obtained by using Courant triangles and canbe done with great efficiency.

Key words. numerical method, mixed finite elements, Kirchhoff plates

AMS subject classifications. 65N12, 65N15, 65N30

PII. S0036142999359773

1. Introduction. In the early 70s Glowinski [8] proposed a nonconforming finiteelement method for the resolution of the Dirichlet problem for the biharmonic operator

∆2u = f on Ω ⊂ 2,u = 0 on ∂Ω,

u,n= 0 on ∂Ω(1.1)

based on the introduction of a discrete operator ∆h defined in C0(Ω) finite elementspaces. Two different approximations by piecewise polynomials of degree one andtwo, respectively, were considered and a general solution procedure based on Uzawa’salgorithm was given for either approximation schemes. While plain convergence wasproved in the paper of Glowinski, error estimates were given only later. For themethod using polynomials of degree k ≥ 2 the rate of convergence was determined byCiarlet and Glowinski [6], who studied the problem in the framework of the mixedmethods according to the scheme of Ciarlet and Raviart [3]; convergence results forthe approximation by linear finite elements were established by Scholz [10]. In [8]Glowinski noted that, by a slight modification, the discrete problem associated withthe use of polynomials of degree one could be reformulated as an unconstrained mini-mum problem and solved by ordinary algorithms. Rather surprisingly, in the followingyears this direct method did not receive attention, but it was practically overwhelmedby the rapid growth of the mixed methods.

In [7] we independently worked out a similar discretization method for elasticplates of Kirchhoff theory and called it lumped strain method (LSM). Our approachwas inspired by a geometrical point of view. We introduced a generalization of thenotions of mean and Gaussian curvature that carry over to polyhedra, and we intro-duced a sequence of functionals defined in discrete spaces whose minimizers convergeto the solution of the original problem in a sense weaker than what is usually required.

∗Received by the editors August 4, 1999; accepted for publication (in revised form) March 16,2000; published electronically August 17, 2000.

http://www.siam.org/journals/sinum/38-3/35977.html†Dipartimento di Ingegneria Civile, via delle Scienze 208, 33100 Udine, Italy (cesare.davini@

dic.uniud.it, igino.pitacco@dic.uniud.it).

820

AN UNCONSTRAINED MIXED METHOD 821

Thus, we arrived at a nonconforming direct method whose distinguishing feature isthat it takes account of discontinuity in the first derivatives by a relaxation of theenergy functional, rather than by the introduction of constraints as aside conditionsto the problem as is done in the mixed methods. While an implementation for vari-ous plate shapes and boundary conditions has been done [11] and has shown excellentperformances of the method, in [7] we gave the proof of convergence only for convexpolygonal plates simply supported at the boundary. In that case the Gaussian cur-vature gives rise to a null Lagrangian and the equilibrium of the plate turns to begoverned by a biharmonic problem. In particular, the LSM reduces to Glowinski’sdirect method.

In this paper we turn attention to the biharmonic problem and complete ourprevious analysis in various ways. We extend it to cover the case of smooth boundariesand both simple support and clamping boundary conditions, and give the respectiveorders of convergence. Thus, Glowinski’s direct method is complemented with ad hocerror estimates.

In order to accomplish the scope, we embed the problem in the framework ofthe mixed methods following the primal formulation of Ciarlet and Raviart [3], seealso [4], and show that the Glowinski-LSM method emerges quite naturally from theintroduction of a dual mesh and an approximation of the constraint that makes itexplicit by reducing the minimization problem to an unconstrained one. Then, thestandard architecture of the mixed methods is adapted in order to get error estimatesfor the examined boundary value problems. Extending the estimates of [6] and [10]to Glowinski’s method of degree one, we find in particular that the convergence rateof Ciarlet and Glowinski applies to the simply supported case, while Scholz’s holdstrue for the clamped case. Incidentally, the analysis emphasizes that the explicitconstruction of the dual mesh is not really needed for actual computations; althoughfor the clamped case a restriction on the area of the dual elements is requested.This frees us from the need of respecting a geometrical condition, described as H3hypothesis in [8], when choosing the dual mesh.

After recalling the basic ideas of the primal mixed method in the next section,the LSM for the simply supported and clamped cases is illustrated in sections 3 and 4;convergence and estimates are studied in section 5 . In the final part of the paperwe give numerical applications that emphasize some computational features and thequality of convergence.

2. The primal mixed formulation. In order to introduce ideas and notationlet us consider the class of biharmonic problems that take the variational form

J(z) = infv∈H2

E(Ω)

1

2

∫Ω

(∆v)2dx− 〈f, v〉 ,(2.1)

where f ∈ H−1 (Ω), Ω is a domain in 2 with smooth boundary and H2E(Ω) stands

either for H2(Ω) ∩ H10 (Ω) for the simply supported case, or H

20 (Ω) for the clamped

case. Following the theory of the mixed methods and in particular [4], instead ofproblem (2.1), one can minimize the functional

F (v, vψ) :=1

2

∫Ω

(vψ)2dx− 〈f, v〉(2.2)

over the pairs (v, vψ) ∈ H2E(Ω)×L2(Ω) whose components are required to satisfy the

equality ∆v = vψ. Accordingly, by a pure change of notation, (2.1) is written as a

822 CESARE DAVINI AND IGINO PITACCO

constrained minimization problem

F (u) = infv∈Gb(v,q)=0∀q∈L2(Ω)

F (v),(2.3)

where v = (v, vψ), G = H2E(Ω)× L2(Ω),

F (v) :=1

2a(v,v)− l(v)(2.4)

with a(v,w) :=∫Ω

vψwψdx, l(v) := 〈f, v〉, and

b(v, q) :=

∫Ω

(vψ −∆v) qdx, q ∈ L2(Ω).

Problem (2.3) can be weakened in the following sense. First, by restricting the spaceof the test functions q to Q = H1

0 (Ω) or Q = H1 (Ω) for the simply supported andthe clamped cases, respectively, and writing b(v, q) in the form

b(v, q) =

∫Ω

∇v · ∇q + vψq dx, q ∈ Q.(2.5)

Second, by setting the minimization problem

F (u) = infv∈Vb(v,q)=0∀q∈Q

F (v)(2.6)

in the larger space V = H10 (Ω) × L2(Ω) endowed with the natural product norm

‖v‖2V = |v|21 + ‖vψ‖2

0. Here and in what follows we write |·|m and ‖·‖m instead of|·|Hm(Ω) and ‖·‖Hm(Ω), respectively. Sometimes we use the notation |·|m,ω instead of

the complete one |·|Hm(ω) and Hm for Hm (Ω).

Using standard arguments, problem (2.6) can be restated as the saddle pointproblem

F (u) = infv∈V

supq∈Q

(F (v) + b(v, q))(2.7)

whose optimality conditions are

a(u,v) + b(v, p) = l(v) ∀v ∈ V,b(u, q) = 0 ∀q ∈ Q.

(2.8)

In order to discretize the problem associated with the mixed formulation, let therebe given two finite element spaces X0h ⊂ H1

0 and Yh ⊂ L2, where the index h refersto a mesh size. Defined Vh = X0h × Yh and Qh = X0h, we have to look for a pair(uh, ph) ∈ Vh ×Qh which is solution of

a(uh,vh) + b(vh, ph) = l(vh) ∀vh ∈ Vh,b(uh, qh) = 0 ∀qh ∈ Qh.

(2.9)

It is easy to show that, if (u, p) is the solution of the mixed problem (2.8), thenu belongs to H3 ∩ H1

0 , with uψ = −p = ∆u, and coincides with the solution z ofproblem (2.1). Existence and uniqueness of the solution of problems of type (2.8) are

AN UNCONSTRAINED MIXED METHOD 823

Fig. 1. Discretization of the domain.

guaranteed by the theorem of Brezzi and Fortin, see [2, Proposition 1.1 and 1.3], underthe condition that a(·, ·) be coercive on Ker(B), where B is the operator defined by

B : V → Q′, 〈Bv, ·〉Q′×Q = b (v, ·),

and that Im(B) be closed. It can be shown that the two conditions hold for the simplysupported case. For the clamped case, on the contrary, it turns out that Im(B) is notclosed and the existence of the saddle point is to be proved by alternative ways; seeCiarlet [4]. With regards to the discrete problem (2.9), existence and uniqueness ofsolution is also ensured by Brezzi’s theorem because Im(Bh) is always closed, since weare dealing with finite-dimensional spaces, and it remains to prove the convergence ofthe discrete solutions to the continuous one. A sufficient condition for convergence isthat the Ladyzhenskaya–Babuska–Brezzi (L.B.B.) condition be satisfied uniformly [2,Theorem 2.1]. Again, the L.B.B. condition does not hold for the clamped case andconvergence must be proved specifically; see [4].

The method presented in the next section is based on a slight modification of(2.9) that consists in finding a couple (uh, ph) ∈ Vh ×Qh solution of

a(uh,vh) + bh(vh, ph) = l(vh) ∀vh ∈ Vh,bh(uh, qh) = 0 ∀qh ∈ Qh,

where bh(·, ·) is an approximation of b(·, ·) in a sense to be made precise.3. The simply supported case. Consider the approximation of the biharmonic

problem of the previous section with H2E = H2 ∩H1

0 . Let Th = Ti be a sequenceof regular triangulations in the sense of Ciarlet [4], i.e., such that the ratio betweenh = supi diam(Ti) and ρh = infi supS diam(S);S a disk contained inTi is boundedby a constant independent of h. Here the Nh nodes of the mesh sit at the vertices ofthe triangles Ti, as illustrated in Figure 1 . Denoting

Ωh =

⋃Ti∈Th

Ti

we require that Ωh ⊂ Ω and invades Ω from inside, with supy∈∂Ωhdist(y, ∂Ω) = O(h2).

The boundary nodes of Ωh need not sit on ∂Ω.

824 CESARE DAVINI AND IGINO PITACCO

We also introduce a dual mesh Th = Ti whose Nh polygonal elements arecentered at the nodes xi of Th and have edges crossing the sides of the trianglesconcurrent in xi, as in Figure 1, where the dual elements are drawn with dashed lines.We assume that the dual mesh is also regular in the above sense.

Let Xh ⊂ H1(Ωh) be the space of functions that are linear on the triangles Ti andcontinuous in Ωh. When needed we regard Xh as a subspace of H

1(Ω) imagining toextend the functions outside Ωh in any of the standard ways for curved boundaries.In particular, we have∫

Ωh

∇vh · ∇qh dx =∑

i,j=1,...,Nh

Cijvh(xj)qh(xi) ∀vh, qh ∈ Xh,(3.1)

where Cij :=∫Ωh

∇ϕi · ∇ϕjdx, the ϕi being the base functions in Xh such that

ϕi (xj) = δij .We denote by X0h the space of functions Xh∩H1

0 (Ωh) and by Yh the space of thefunctions which are constant on the Ti. In what follows we will regard X0h and Yhas subspaces of H1

0 (Ω) and L2(Ω), respectively, by extending their functions to zerooutside Ωh. Note that with this choice the way Xh is extended to H

1(Ω) is irrelevantin the following analysis. Let Vh = X0h × Yh and Qh = X0h. With this choice, for(vh, qh) ∈ Vh ×Qh the bilinear form b(·, ·) becomes

b(vh, qh) =∑

i,j=1,...,Nh

Cijvh(xj)qh(xi) +∑

i=1,...,Nh

vψh(xi)

∫Ti

qh dx.(3.2)

Expanding qh|Ti around point xi, qh (x) = qh (xi)+∇qh (x) (x− xi) for x ∈ Ti, we get

b(vh, qh) =∑

i=1,...,Nh

∣∣∣Ti

∣∣∣ vψh(xi) + ∑j=1,...,Nh

Cijvh(xj)

qh(xi) +O(vψh , qh),(3.3)

where

|O(vψh , qh)| ≤ ‖vψh‖L2 |qh|H1 h.(3.4)

Accordingly, we approximate the constraint functional b (·, ·) by

bh(vh, qh) =∑

i=1,...,Nh

∣∣∣Ti

∣∣∣ vψh(xi) + ∑j=1,...,Nh

Cijvh(xj)

qh(xi)(3.5)

and consider the discrete problem in the form

a(uh,vh) + bh(vh, ph) = l(vh) ∀vh ∈ Vh,bh(uh, qh) = 0 ∀qh ∈ Qh,

(3.6)

which is equivalent to

F (uh) = infvh∈Vhbh(vh,qh)=0∀qh∈Q

1

2a(vh,vh)− 〈f,vh〉 .(3.7)

Now,

a(vh,vh) =∑

i=1,...,Nh

∣∣∣Ti∣∣∣ vψh(xi)2.(3.8)

AN UNCONSTRAINED MIXED METHOD 825

Moreover, as vh (xi) = qh (xi) = 0 at the boundary nodes, vh satisfies the constraintbh(vh, qh) = 0 ∀qh ∈ Qh if and only if

vψh(xi) = −∣∣∣Ti

∣∣∣−1 ∑j∈I

Cijvh(xj) ∀i ∈ I,(3.9)

where I is the set of the values taken by the index of the internal nodes. We denote byB the set of values of the boundary nodes. Then, problem (3.7) reduces to calculating

infvh∈X0h

vψh(xi), i∈B

1

2

∑i,j,k∈I

∣∣∣Ti∣∣∣−1

CijCikvh(xj)vh(xk)(3.10)

+1

2

∑i∈B

vψh(xi)2∣∣∣Ti

∣∣∣− 〈f, vh〉 .

It follows that optimality requires vψh(xi) = 0 i ∈ B, which is the discrete counter-part of the natural boundary condition for the simply supported case. The problemcan then be reformulated as the unconstrained minimization problem, as follows.

Set

Jh(vh) :=1

2

∑i,j,k∈I

∣∣∣Ti∣∣∣−1

CijCikvh(xj)vh(xk)− 〈f, vh〉(3.11)

and find uh ∈ X0h such that

Jh(uh) = infvh∈X0h

Jh(vh).(3.12)

Remark. With a suitable choice of the dual mesh, it is possible to give a geomet-rical interpretation of the linear form

∑j=1,...,Nh

Cijvh(xj) i = 1, . . . , Nh, obtaining

a more transparent expression for the energy functional. For a function v ∈ H2(Ω),the average of the mean curvature evaluated over Ti is K

im(v) =

1|Ti|

∫Ti∆v dx or,

equivalently,

Kim(v) =

1∣∣∣Ti∣∣∣∫∂Ti

v,n ds,(3.13)

where n is the outward normal to the boundary ∂ Ti. In [7] we extended the notion ofmean curvature to the functions of Xh by defining it according to (3.13) and provingthat, if the dual elements have boundaries crossing the sides of the primal elementsat the middle points, the equality∫

Ωh

∇vh · ∇qh dx = −∑i

∣∣∣Ti∣∣∣Ki

m(vh)qh(xi) ∀vh, qh ∈ Xh(3.14)

holds true. Thus, comparing with (3.1), it follows that

Kim(vh) = −

∣∣∣Ti∣∣∣−1 ∑

j=1,...,Nh

Cijvh(xj) ∀vh ∈ Xh.(3.15)

Accordingly, the constraint conditions (3.9) become

vψh(xi) = Kim(vh) ∀i ∈ I,(3.16)

826 CESARE DAVINI AND IGINO PITACCO

and Jh(vh) reads

Jh(vh) =1

2

∑i∈I

Kim(vh)

2∣∣∣Ti

∣∣∣− 〈f, vh〉 ,(3.17)

which is the discrete functional considered in [7]. The term Kim(vh)

2|Ti| is the discretecounterpart of the strain energy of the plate element Ti, and Jh(·) is a relaxation ofJ(·) from H2 ∩H1

0 to X0h.

4. The clamped case. When dealing with boundary conditions correspondingto the clamped case, the above discretization method requires a modification. Sup-pose that displacement and normal derivative be prescribed to vanish on the wholeboundary and consider the problem

J(z) = infv∈H2

0

1

2

∫Ω

(∆v)2dx− 〈f, v〉 .(4.1)

To put it in the mixed form we just need to choose V = H10 × L2 and Q = H1 in

the above framework and write the saddle point condition. With this choice of Qthe constraint condition v ∈ Ker (B) implies that vψ = ∆v, with v ∈ H2 ∩H1

0 , andv,n= 0 on ∂Ω; see Ciarlet [4].

The mixed problem is discretized by choosing

Vh = X0h × Yh

and Qh = Xh. Then, the constraint equation bh(vh, qh) = 0 ∀qh ∈ Qh:

∑i=1,...,Nh

∑j=1,...,Nh

Cijvh (xj) +∣∣∣Ti

∣∣∣ vψh(xi) qh(xi) = 0 ∀qh ∈ Qh(4.2)

becomes

vψh(xi) = −∣∣∣Ti

∣∣∣−1 ∑j∈I

Cijvh (xj) ∀i ∈ I ∪B.(4.3)

Thus, if we take account of (4.3) in the discrete problem and impose optimality withrespect to the values of vψh(xi) for i ∈ I∪B, we get at the unconstrained minimizationproblem

Jh(uh) = infvh∈X0h

1

2

∑i∈I∪Bj,k∈I

∣∣∣Ti∣∣∣−1

CijCikvh (xj) vh (xk)− 〈f, vh〉 .(4.4)

Note that the space X0h where the functional Jh is minimized takes account of theboundary condition on the displacement only, whereas the presence of the conditionon the normal derivative modifies the objective functional by requiring that it includethe curvatures vψh(xi) of the boundary elements Ti, i ∈ B.

5. Convergence and error estimates. The equivalence of the minimum for-mulation and the saddle point formulation has been established for both types ofboundary conditions provided that the solution of the biharmonic problem belongs toH3(Ω), see [4], for the clamped case. Thus, in order to justify the method outlined

AN UNCONSTRAINED MIXED METHOD 827

above, it remains to discuss the convergence of the discrete solutions (uh, ph) to thesaddle point (u, p). In the following we give the error estimates for the two cases,separately.

Proposition 1 (coercivity of a(·, ·) in Ker (Bh); Ciarlet [4]). Let Vh = X0h×Yhand X0h ⊂ Qh. Then, there exists a constant α0 > 0 such that

a(vh,vh) ≥ α0 ‖vh‖2V ∀vh ∈ Ker(Bh).(5.1)

Proof. Recalling that bh(vh, qh) = b(vh, qh) +O(vh, qh), we have

b(vh, qh) +O(vh, qh) = 0, ∀vh ∈ Ker(Bh) ∀qh ∈ Qh.(5.2)

Choosing qh = vh in (5.2) we get

|vh|21 +∫

Ω

vhvψhdx+O(vh, vh) = 0 ∀vh ∈ Ker(Bh)(5.3)

and, by the Cauchy–Schwarz inequality,

|vh|21 ≤ ‖vh‖0 ‖vψh‖0 + |O(vh, vh)| ∀vh ∈ Ker(Bh).(5.4)

Recalling (3.4) and applying Poincare inequality it follows that

|vh|1 ≤ (h+ c) ‖vψh‖0 ∀vh ∈ Ker(Bh),(5.5)

where c is the Poincare constant. Thus, by the definition a(vh,vh) = ‖vψh‖20, for

h ≤ 1, it follows that

a(vh,vh) ≥ 1

(1 + c)2 |vh|21 ∀vh ∈ Ker(Bh),(5.6)

and then, by summing a(vh,vh) on the two sides,

a(vh,vh) ≥ 1

2 (1 + c)2 ‖vh‖2

V ∀vh ∈ Ker(Bh),(5.7)

which is (5.1) with α0 =1

2(1+c)2.

We premise two statements that characterize our approximation of the constraintfunctional b (·, ·) by bh (·, ·).

Proposition 2. For q ∈ Q let Phq be the projection of its restriction q|Ωh on Qh

with respect to the inner product in H1. Then, for every vh ∈ Vh

b (vh, q) = bh (vh, Phq) +O(vh, q),(5.8)

with

|O(vh, q)| ≤ c ‖vh‖V ‖q‖H1 h.(5.9)

Proof. Put

b (vh, q) = b (vh, q − Phq) + b (vh, Phq) .(5.10)

Then, by arguments (3.3)–(3.4) it turns out that

b(vh, Phqh) = bh(vh, Phqh) +O(vh, Phqh),(5.11)

828 CESARE DAVINI AND IGINO PITACCO

where

|O(vh, Phq)| ≤ c ‖vψh‖0 ‖q‖1 h.(5.12)

Furthermore,

b (vh, q − Phq) =

∫Ωh

∇vh · ∇ (q − Phq) dx+

∫Ωh

vψh (q − Phq) dx.

Recalling that ∫Ωh

∇vh · ∇ (q − Phq) dx = −∫

Ωh

vh (q − Phq) dx,

because vh ∈ Qh, it follows that

b (vh, q − Phq) = −∫

Ω

vh (q − Phq) dx+

∫Ω

vψh (q − Phq) dx,

where we have again taken into account that vh = vψh = 0 outside Ωh. Thus,

|b (vh, q − Phq)| ≤ ‖vh‖V ‖q − Phq‖0(5.13)

and, by the approximation properties of the projection,

|b (vh, q − Phq)| ≤ c ‖vh‖V ‖q‖1 h.(5.14)

Taking (5.10)–(5.14) into account proves the proposition.Proposition 3. If u ∈ Ker(Bh) and u ∈ Ker(B), then

|u− u|1 ≤ c ‖uψ − uψ‖0 +(|u|2 + ‖uψ‖0

)h.(5.15)

Proof. The condition u ∈ Ker(Bh) implies that b(u, qh) = O(u, qh) ∀qh ∈ Qh

(see (3.3)), while b(u, qh) = 0 ∀qh ∈ Qh. Subtracting these two equalities yieldsb(u− u, qh) = O(u, qh) ∀qh ∈ Qh; i.e.,∫

Ω

∇ (u− u) · ∇qhdx =

∫Ω

(uψ − uψ) qhdx+O(u, qh).(5.16)

Let P0h(u) be the projection of u in X0h defined by∫Ω∇ (P0h(u)− u) · ∇qhdx =

0 ∀qh ∈ X0h. Thus, (5.16) can be rewritten as∫Ω

∇ (u− P0h(u)) · ∇qhdx =

∫Ω

(uψ − uψ) qhdx+O(u, qh) ∀qh ∈ X0h.(5.17)

Taking qh = u − P0h(u) in (5.17), using the Cauchy–Schwarz inequality, (3.4), andthe Poincare inequality successively, one has

|u− P0h(u)|1 ≤ c ‖uψ − uψ‖0 + ‖uψ‖0 h.(5.18)

Note that u ∈ Ker (B) implies that u ∈ H2. Then, from basic error estimates [5],one gets

|u− P0h(u)|1 ≤ |u|2 h.(5.19)

Hence (5.15) follows from the triangle inequality.

AN UNCONSTRAINED MIXED METHOD 829

Thus, two functions in Ker (Bh) and Ker (B) are close to each other if their partsuψ and uψ are close.

The following proposition is a slight modification of a classical result.Proposition 4 (see Brezzi and Fortin [2, Proposition 2.4]). Let (u, p) ∈ V ×Q

be the solution of the continuous problem and (uh, ph) ∈ Vh ×Qh the solution of therelated discrete problem. Then

‖u− uh‖V ≤ c1 infwh∈Ker(Bh)

‖u−wh‖V + c2 ‖p‖1 h,(5.20)

where c1 and c2 are constants independent of h.Proof. Let wh ∈ Ker(Bh), so that wh−uh ∈ Ker(Bh). By the coercivity of a(·, ·)

in Ker(Bh),

α0 ‖wh − uh‖V ≤ supvh∈Ker(Bh)

a(wh − uh,vh)

‖vh‖V.(5.21)

Therefore, observing that a(wh − uh,vh) = a(wh − u,vh) + a(u− uh,vh), it followsthat

α0 ‖wh − uh‖V ≤ supvh∈Ker(Bh)

a(wh − u,vh)

‖vh‖V(5.22)

+ supvh∈Ker(Bh)

a(u− uh,vh)

‖vh‖V.

Furthermore, by subtracting the first line of (2.8) from the first line of (3.6) and takingaccount that vh ∈ Ker(Bh), we get

a(u− uh,vh) = −b(vh, p).(5.23)

Hence, Proposition 2 yields

a(u− uh,vh) = O(vh, p),(5.24)

where O(vh, p) satisfies inequality (5.9). Using (5.23) in (5.22), by the continuity ofa(·, ·) we obtain

α0 ‖wh − uh‖V ≤ ‖a‖ ‖wh − u‖V + c ‖p‖1 h.(5.25)

Inequality (5.20) follows then from (5.25) by the triangle inequality ‖u− uh‖V ≤‖u−wh‖V + ‖wh − uh‖V , with appropriate constants c1 and c2.

Propositions 3 and 4 reduce the proof of convergence for the LSM to the researchof an element u ∈ Ker(Bh) such that uψ is arbitrarily close to uψ when h → 0. Thenext theorem gives the convergence rate for the simply supported case.

Theorem 5. Let (u, p) ∈ V ×Q, with V = H10 ×L2 and Q = H1

0 , be the solutionof the continuous problem for the simply supported case and (uh, ph) ∈ Vh × Qh thesolution of the related discrete problem. Then

‖u− uh‖V ≤ c ‖u‖3 h,(5.26)

where c is a constant independent of h.

830 CESARE DAVINI AND IGINO PITACCO

Proof. Let uh = (uh, uψh) ∈ Vh be defined as follows:

uψh (xi) =∣∣∣Ti

∣∣∣−1∫Ti

uψdx ∀i ∈ I ∪B,(5.27)

and uh ∈ X0h defined by

∣∣∣Ti∣∣∣−1 ∑

j∈ICij uh (xj) = −uψh(xi) ∀i ∈ I(5.28)

so that bh (uh, qh) = 0 ∀qh ∈ X0h; see (3.9). Such a uh exists because the matrix Cij

i, j ∈ I is positive definite, as follows from (3.1) by taking qh = vh

∫Ω

∇v2hdx =

∑i,j∈I

Cijvh (xi) vh (xj) ∀vh ∈ X0h.(5.29)

Note that the values uψh(xi) for i ∈ B do not enter the constraint condition, so thereis no hindrance to choosing them according to (5.27). It follows that uh is completelydefined by (5.27) together with the condition uh ∈ Ker(Bh). Recall that u ∈ H3 underour assumptions, because of the regularity theory, and uψ = ∆u. By the Poincare

inequality, from (5.27) we get ‖uψh − uψ‖20,Ti

≤ ch2 |uψ|21,Ti , or, summing up over theindices i ∈ I ∪B,

‖uψh − uψ‖20 ≤ ch2 |uψ|21 .(5.30)

Taking into account that ‖uψh‖0 ≤ ‖uψ‖0, from Proposition 3 and inequality (5.30)we get

|uh − u|1 ≤ c ‖u‖3 h.(5.31)

Again inequalities (5.30) and (5.31) yield that ‖uh − u‖V ≤ ch ‖u‖3. The assertion ofthe theorem follows by choosing wh = u in (5.20) and observing that p = ∆u.

Remark. The convergence of the LSM for the simply supported case was given in[7] by direct arguments and in a totally different context. In particular, no mentionwas made of the multiplier p which appears rather redundant in a problem that iseventually unconstrained. An observation at the end of section 3 implies that uψhconverges to uψ in L

2, so the paper establishes convergence both for u and ∆u. Whilethe convergence rate was not estimated in that paper, here Theorem 5 provides itsorder.

For the clamped case unqualified convergence of LSM was first proved by Glowin-ski [8]. As it stands, the proof is written for a datum f in L2, i.e., for a solutionu ∈ H4. In fact, it is possible to modify the proof and show that one has convergencealso under the minimum regularity requirement that u ∈ H2. To get the rate of con-vergence here we assume that the elements Ti of the dual mesh have an area equal toone-third of the area of suppϕi and that the solution has bounded second derivatives.The estimate we find coincides with that given by Scholz [10], who follows a differentmixed formulation requiring that Yh ⊂ Xh. In both cases the result is based on a L

∞

error estimate for the Ritz projection due to Nitsche [9].

AN UNCONSTRAINED MIXED METHOD 831

Let us premise the following lemmas.Proposition 6. Let us define

Rh(i)v :=

∫Ωh

vϕidx∫Ωh

ϕidx,

Rhv : =∑

i∈I∪BRh(i)vχTi

,

(5.32)

where χTiis the characteristic function of Ti. Then, the following approximation

result holds true:

‖Rhv − v‖0,Ωh ≤ C|v|1,Ωh ∀v ∈ H1(Ω),(5.33)

with C independent of h.Proof. Let Si = suppϕi. Observing that Rh(i), regarded as a linear operator

from L2(Si) to the space of constant functions on Si, leaves invariant the constantpolynomials, by standard results the following inequality holds true:

‖Rh(i)v − v‖0,Si ≤ c|v|1,Sihi(5.34)

with c independent of h; see, for instance, [5, Theorem 15.3]. The parameter hi in(5.34) is the diameter of the smaller circle containing Si and is bounded by 2h forevery polygon Si.

Now

‖Rh(i)v − v‖0,Ti≤ ‖Rh(i)v − v‖0,Si(5.35)

as Ti ⊂ Si, and, from (5.34),

‖Rh(i)v − v‖20,Ti

≤ 4c2|v|21,Sih2.(5.36)

Hence, by summing inequalities (5.36) over the index i we get

‖Rhv − v‖20,Ωh

≤ 4c2h2∑i

|v|21,Si .(5.37)

Breaking |v|21,Si into the sum∑

α |v|21,T iα over the triangles of the primal mesh con-tained in Si and observing that each triangle is contained in at most three differentSi, it follows that ∑

i∈I∪B|v|21,Si ≤ 3

∑Ti∈Th

|v|21,Ti ≤ 3|v|21,Ω.(5.38)

Therefore using this inequality in (5.37) we get the thesis.Proposition 7. If B and Bh are the constraint operators for the clamped case,

and (u, uψ) ∈ Ker(B) is such that u ∈ W 2,∞ ∩ H3, then there exists(wh, wψh

) ∈Ker(Bh) with the property

‖wψh − uψ‖0,Ω ≤ c1‖u‖2,∞h12 log h+ c2 ‖uψ‖1,Ω h(5.39)

and

‖wh − u‖1,Ω ≤ c‖u‖2,∞h,(5.40)

where uψ = ∆u.

832 CESARE DAVINI AND IGINO PITACCO

Proof. Let wh = P0h(u) be the Ritz projection of u into X0h defined in Proposi-tion 3. Then, choose wψh such that wh = (wh, wψh) ∈ Ker(Bh), i.e.,∫

Ω

∇wh · ∇qhdx+∑

i∈I∪Bwψh(xi)|Ti|qh(xi) = 0 ∀qh ∈ Xh.(5.41)

Recalling that u ∈ Ker(B) we have∫

Ω

∇u · ∇qh + uψqhdx = 0 ∀qh ∈ Xh.(5.42)

Choosing qh ∈ X0h and subtracting (5.42) from (5.41) we get

∑i∈I

wψh(xi)|Ti|qh(xi)−∑i∈I

qh(xi)

∫Ω

uψϕidx = 0 ∀qh ∈ X0h;(5.43)

i.e.,

wψh(xi) = |Ti|−1

∫Ω

uψϕidx ∀i ∈ I.(5.44)

Now, subtracting (5.42) from (5.41), with qh ∈ Xh, and taking (5.43) into accountwe get

∑i∈B

qh(xi)

∫Ω

(∇wh −∇u) · ∇ϕidx

+∑i∈B

(wψh(xi)|Ti| −

∫Ω

uψϕi dx

)qh(xi) = 0;

(5.45)

i.e.,

wψh(xi) = |Ti|−1

∫Ωh

uψϕidx+ εi ∀i ∈ B,(5.46)

where εi = ε′i + ε′′i and

ε′i = |Ti|−1

∫Ω\Ωh

uψϕi dx,

ε′′i = |Ti|−1

∫Ω

∇(u− wh) · ∇ϕi dx.

(5.47)

Now it is easily seen that

ε′i ≤ c‖u‖2,∞h

and, by Nitsche’s estimate ‖u − wh‖1,∞ ≤ c1‖u‖2,∞h log h, together with |∇ϕi| ≤ch−1, that

ε′′i ≤ c‖u‖2,∞ log h.

Then it follows that

|εi| ≤ c‖u‖2,∞ log h ∀i ∈ B.(5.48)

AN UNCONSTRAINED MIXED METHOD 833

Thus, while in the internal polygons Ti the function wψh coincides with the pro-jection Rhuψ, on the boundary polygons they differ from each other by εi. So, we have

‖wψh −Rhuψ‖20,Ω =

∑i∈B

‖εi‖20,Ti

,(5.49)

and, taking into account that

∑i∈B

‖εi‖20,Ti

≤ c2‖u‖22,∞(log h)

2∑i∈B

|Ti| = c2‖u‖22,∞(log h)

2|ΩB |(5.50)

by (5.48), we get

‖wψh −Rhuψ‖0,Ω ≤ c|∂Ω| 12 ‖u‖2,∞(log h)h12 .(5.51)

By the triangle inequality ‖wψh − uψ‖0,Ω ≤ ‖wψh −Rhuψ‖0,Ω + ‖Rhuψ − uψ‖0,Ωh +‖uψ‖0,Ω\Ωh , where the last term is bounded by c‖u‖2,∞h. Thus, from (5.51) and(5.33) we get the proof of (5.39). Inequality (5.40) is a straightforward consequenceof L2-estimates for the Ritz projection of functions in H2.

Using the previous result in Proposition 4 yields the error estimateTheorem 8. Let (u, p) ∈ V ×Q, with V = H1

0 ×L2 and Q = H1, be the solutionof the continuous problem for the clamped case and let u ∈ W 2,∞ ∩ H3. Then, if(uh, ph) ∈ Vh ×Qh is the solution of the related discrete problem, we have

‖u− uh‖V ≤ c1 ‖u‖2,∞ h12 lnh+ c2 ‖∆u‖1 h.(5.52)

Likely, when u ∈ H4, by the embedding H4 ⊂ W 2,∞ this bound can be giventhe form

‖u− uh‖V ≤ c ‖u‖4 h12 lnh.(5.53)

6. Properties of the discrete problem and applications. Let us considerthe general discrete problem: find uh ∈ X0h such that

Jh(uh) = infvh∈X0h

1

2

∑i∈I∪Bj,k∈I

∣∣∣Ti∣∣∣−1

CijCikvh (xj) vh (xk)− 〈f, vh〉 ,(6.1)

where B = j : xj ∈ ∂ Ω for the clamped case and B = ∅ for the simply supportedcase. For convenience we denote with iα, α = 1, . . . ,m, the elements of I ∪ B andwith jα, α = 1, . . . , n, the elements of I. Accordingly, v ∈ n will be the vectorwith components vα = vh (xjα), α = 1, . . . , n. Furthermore, let b ∈ n be the vectorwhose components are defined by bα = 〈f, ϕjα〉 , α = 1, . . . , n, and K ∈ n ×n thesymmetric matrix defined by

Kαγ =∑

β=1,...,m

∣∣∣Ωiβ

∣∣∣−1

CiβjαCiβjγ , α, γ = 1, . . . , n.(6.2)

With the introduced notation, problem (6.1) can be rewritten as

infv∈n

1

2Kv · v − b · v,(6.3)

834 CESARE DAVINI AND IGINO PITACCO

and the minimizer u ∈ n solves the linear system

Ku = b.(6.4)

To build the stiffness matrix K one has first to compute the matrix C ∈ N ×N

Cij =

∫Ω

∇ϕi · ∇ϕjdx, i, j = 1, . . . , N,(6.5)

where N is the total number of nodes in the triangulation Th. C is the matrix arisingfrom the assemblage of linear (Courant) triangles, and its evaluation is a standardtask in the framework of C0 elements. Since ∇ϕj are piecewise constant, C is veryeasy and time inexpensive to compute because it does not require the use of numericalintegration. Furthermore C has a highly sparse structure that implies an adequatelyhigh level of sparsity of the matrix K according to (6.2).1

It is worth noting that for the simply supported case there is no need to computeK. In fact we have B = ∅ and K = CD−1C, where C ∈ n × n is a symmetricpositive definite matrix defined by Cαβ =

∫Ω∇ϕjα · ∇ϕjβdx, jα, jβ ∈ I, and D is

the diagonal matrix defined by Dαα = |Ωjα |, jα ∈ I. Accordingly, system (6.4)reduces to

CD−1Cu = f(6.6)

and can be efficiently treated by introducing a new unknown s = D−1Cu and solvingsequentially the two linear systems

Cs = f ,Cu = Ds.

(6.7)

This operation requires only one factorization of the matrix C and two backward-substitutions. Clearly this simplification reflects the fact that for the simply supportedcase the biharmonic problem can be split into two second order problems to be solvedin sequence.

It is also worth saying that the use of a dual mesh is not a burden because allthat we need in order to calculate D is the area of the dual elements. There is norestriction on it in the simply supported case, whereas for the clamped case, accordingto Proposition 6 one has to suppose that the area of the dual element Tp be equalto one-third of the area of the corresponding mp triangles of Th concurring in thenode xp.

Elements⋃mp

i=1 ωip in Figure 2, where ωi

p is the quadrilateral region defined by

vertices xp, yip, z

ip, y

i+1p with zip the center of gravity of the triangle T i

p and yip the

midpoints of the sides of T ip, conform to the requirement. We call such dual elements

barycentric. Although it does not satisfy the requirement, in general, by the sakeof comparison here we consider also the possibility of taking the circumcenters ofthe triangles T i

p in lieu of the zip when defining the ω

ip. Note that the latter choice

is possible if the primal triangulation satisfy the Delunay condition and the dualelements turns to be the associated Voronoı polygons.

The performances of the method depend upon the choice of the dual elements asthey appear in numerical experiments below. We numerically analyzed a uniformly

1A comparative evaluation of the computational cost of the calculation and assemblage of K isgiven in [11] for various boundary value problems of the plate theory.

AN UNCONSTRAINED MIXED METHOD 835

Fig. 2. A patch of elements.

Fig. 3. Error‖uψh−uψ∞‖

L2

‖uψ∞‖L2

vs. mesh parameter h.

loaded square plate with two different boundary conditions using both types of dualelements. Calling (uh, uψh) the solution obtained for a triangulation characterized

by a mesh amplitude h, in Figure 3 we plotted the relative error‖uψh−uψ∞‖

L2

‖uψ∞‖L2

vs.

the parameter h of a sequence of refined triangulations of the domain. The referencesolution (u∞, uψ∞) is obtained by a very fine discretization (3969 degrees of freedom).The four plotted curves are labeled with S-V, S-B, C-V, C-B. The first letter refers tothe boundary condition: S for the simply supported case, and C for the clamped case.The second letter refers to the choice of the dual elements: V for the Voronoı elements,and B for the barycentric ones. Observe that the rate of convergence for the simplysupported case is the same for both choices of the dual mesh. The situation is differentand rather surprising for the clamped case, where the use of the Voronoı elementsgreatly improves the rate of convergence. To give an idea of the rate of convergence,in Table 1 we report the relative error in the evaluation of the displacement andthe Laplacian of the displacement in the center of the plate for both the simply

836 CESARE DAVINI AND IGINO PITACCO

Table 1Percent errors vs. degrees of freedom.

DOF εu % S. εu % C. ε∆u % S. ε∆u % C.

1 28.2 173 13.1 57.5

9 0.09 43.0 -1.04 14.2

49 -0.11 12.4 -0.286 4.13

225 -0.04 3.10 -0.075 1.01

961 0 0.57 -0.012 0.208

supported (S) and the clamped (C) cases for various triangulations. Reported in thefirst column are the numbers of the degrees of freedom for the triangulations. Theresults are obtained with Voronoı dual elements, and the errors are evaluated withrespect to the values of Fourier series solutions found in [1].

REFERENCES

[1] R. Bares, Berechnungstafeln fur Platten und Wandenscheiben, Bauverlag GmbH, Wiesbaden,Germany, 1979.

[2] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag,New York, 1991.

[3] P. G. Ciarlet and P.A. Raviart, A mixed finite element method for the biharmonic equation,in Mathematical Aspects of Finite Elements in Partial Differential Equations, Proceedingsof a Symposium Conducted by the Mathematic Research Center, University of Wisconsin,Madison, 1974, C. de Boor ed., Academic Press, New York, 1974, pp. 125–143.

[4] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam,1978.

[5] P. G. Ciarlet, Basic error estimates for elliptic problems, in Handbook of Numerical Analysis,Vol. 2, P. G. Ciarlet and J.L. Lions, eds., North Holland, Amsterdam, 1991, pp. 17–351.

[6] P. G. Ciarlet and R. Glowinski, Dual iterative techniques for solving a finite element ap-proximation of the biharmonic equation, Comput. Methods Appl. Mech. Engrg., 5 (1975),pp. 277–295.

[7] C. Davini and I. Pitacco, Relaxed notions of curvature and a lumped strain method for elasticplates, SIAM J. Numer. Anal., 35 (1998), pp. 677–691.

[8] R. Glowinski, Approximations externes, par Elements Finis de Lagrange d’Ordre Un et Deux,du Probleme de Dirichlet pour l’Operateur Biharmonique. Methode Iterative de Resolutiondes Problemes Approches, in Topics in Numerical Analysis, J.J.H. Miller, ed., AcademicPress, London, 1973, pp. 123–171.

[9] J.A. Nitsche, An L∞-convergence of finite element approximations, in Mathematical Aspectsof Finite Element Methods, Lecture Notes in Math. 606, Springer, Berlin, 1977, pp. 261–274.

[10] R. Scholz, A mixed method for 4th order problems using linear finite elements, RAIRO Model.Math. Anal. Numer., 12 (1978), pp. 85–90.

[11] S. Suraci, Un nuovo metodo di approssimazione in analisi strutturale: applicazioni e confronticon il metodo degli elementi finiti, thesis, Universita degli studi di Udine, Udine, Italy, 1997.