Raphaèle Herbin

Robert Eymard

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1283–1286Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comPartial Differential Equations/Numerical AnalysisApproximation of the biharmonic problem using piecewise linear finiteelementsApproximation d’un problème biharmonique par élément fini P1Robert Eymard a, Raphaèle Herbin b,1a Laboratoire d’analyse et de mathématiques appliquées, UMR CNRS 8050, Université Paris-Est, 5, boulevard Descartes, Champs-sur-Marne,77454 Marne-la-Vallée cedex 2, Franceb Laboratoire d’analyse, topologie et probabilités, UMR CNRS 6632, Université de Provence, 39, rue Joliot-Curie, 13453 Marseille, Francea r t i c l e i n f o a b s t r a c tArticle history:Received 8 September 2010Accepted 3 November 2010Available online 17 November 2010Presented by Philippe G. CiarletWe propose an approximation of the solution of the biharmonic problem in H20(Ω)which relies on the discretization of the Laplace operator using nonconforming continuouspiecewise linear finite elements.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous proposons une approximation de la solution du problème bi-harmonique dans H20(Ω)basée sur la discrétisation du Laplacien par éléments finis P1 continus mais non conformes.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionA large number of discretization methods for the biharmonic operator has been proposed in the recent past. The mostclassical is probably the conforming finite element method. For fourth order problems, the conforming finite element spacemust be a finite dimensional subspace of the Sobolev space H2(Ω). Hence elementary basis functions are sought such thatthe reconstructed global basis functions on Ω belong to C1(Ω). On Cartesian meshes, such basis functions are found bygeneralizing the one-dimensional P 3 Hermite finite element to the multi-dimensional framework. This task becomes muchmore difficult on more general meshes and involves rather sophisticated finite elements such as the Argyris finite elementon triangles in 2D, which unfortunately requires 21 degrees of freedom [7]. Hence nonconforming FEMs have also beenwidely studied: see e.g. [7, Section 49], [8], and references therein, and [3,4] for more recent works. Discontinuous Galerkinmethods have also been recently developed and analysed [10–13]; error estimates have been derived for polynomials ofdegree greater or equal to two or three. Other methods which have been developed for fourth order problems includemixed methods [5] (see also references therein), [14], and compact finite difference methods [6,2,1]. All the above methodsare high order methods, and therefore, rather computationally expensive and may not be so easy to implement. Recently,a cheaper low order method based on the discretization of the Laplace operator by a cell centred finite volume schemewas proposed [9]. The idea in the present paper is to show that the discretization of the Laplace operator by the piecewiselinear finite element method is also possible in order to obtain a convergent low order scheme for the biharmonic operator.E-mail addresses: robert.eymard@univ-paris-est.fr (R. Eymard), raphaele.herbin@cmi.univ-mrs.fr (R. Herbin).1 Work supported by Groupement de recherche MOMAS, PACEN/CNRS.1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.11.0021284 R. Eymard, R. Herbin / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1283–1286Fig. 1. Definition of the domain.Throughout this paper, d ∈ N \ {0} denotes the space dimension, Ω is an open polygonal bounded and connected subsetof Rd , with Lipschitz-continuous boundary ∂Ω , and f ∈ L2(Ω),  ∈ L2(Ω) and g ∈ (L2(Ω))d . Under these assumptions, weconsider the following problem: findu ∈ H20(Ω); ∀v ∈ H20(Ω),∫Ωu(x)v(x)dx =∫Ω( f (x)v(x) + g(x) · ∇v(x) + (x)v(x))dx, (1)where H20(Ω) is the closure in H2(Ω) of the set C∞c (Ω) of infinitely continuously differentiable functions with compactsupport in Ω .2. Definition of the schemeLet Ω be a polyhedral domain in Rd , with d ∈ N . We consider a classical simplicial mesh T of Ω [7, Section 6]; wedenote by hT the supremum of the diameters of all S ∈ T , and by θT the infimum of the angles of all triplets of verticesof the simplex S , for any simplex S ∈ T .Let V be the finite set of the vertices of the mesh, and V S the set of the vertices of a given simplex S ∈ T . Let Vext bethe set of all vertices of the mesh lying on the boundary and Vint = V \ Vext the set of the interior vertices. For all S ∈ Tand all z ∈ V S , we denote by K S,z the subset of S of all points whose barycentric coordinate related to z is greater thanthat related to any z′ ∈ V S with z′ = z (see Fig. 1). We then denote for all z ∈ V by Kz the union of all K S,z , for all S ∈ Tsuch that z ∈ V S , which leads to ⋃s∈V Kz = Ω . We denote by Vz the set of all y ∈ V such that there exists S ∈ T with{y, z} ⊂ V S . For any z ∈ V , let ϕz be the piecewise affine basis function of the P1 finite element, such that ϕz(z) = 1 andϕz(z′) = 0 for all z′ ∈ V \ {z}. We then denote by V T the vector space spanned by all functions ϕz , z ∈ V (then any u ∈ V Tis such that u = ∑z∈V u(z)ϕz).Next, the linear form z : V T → R for z ∈ V is defined by∀u, v ∈ V T ,∫Ω∇u(x) · ∇v(x)dx = −∑z∈V|Kz|v(z)zu,and the discrete Laplace operator T : V T → L2(Ω) is defined by∀u ∈ V T , T u(x) =∑z∈Vzu1Kz (x), for a.e. x ∈ Ω.Note that, if we denoteT z y = −∫Ω∇ϕz(x) · ∇ϕy(x)dx,we get, using the property∑y∈V T z y = 0 since∑y∈V ϕy(x) = 1 for all x ∈ Ω ,zu = 1|Kz|∑T z y(u(y) − u(z)), ∀z ∈ V.y∈VzR. Eymard, R. Herbin / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1283–1286 1285We then have∀u ∈ V T , ∀v ∈ V T ,∫Ω∇u(x) · ∇v(x)dx = −∫ΩP T v(x)T u(x)dx, (2)where P T v denotes the piecewise constant function defined by the values v z in Kz , for all z ∈ V . The discrete approximationspace is then defined byV T ,0 ={u ∈ V T , u(z) = 0 for all z ∈ Vext},and the approximation scheme consists in findingu ∈ V T ,0; ∀v ∈ V T ,0,∫ΩT u(x)T v(x)dx =∫Ω(f (x)v(x) + g(x) · ∇v(x) + (x)T v(x))dx. (3)3. Convergence analysisThe convergence analysis relies on the following sequence of results, the proof of which is given in a forthcoming paper:Lemma 3.1 (Existence, uniqueness and estimate on the solution of (3)). There exists C > 0, only depending on Ω and θ , such that forany u ∈ V T ,0 satisfying (3), then‖u‖L2(Ω)  C(‖ f ‖L2(Ω) + ‖g‖L2(Ω)d + ‖‖L2(Ω)),‖∇u‖L2(Ω)d  C(‖ f ‖L2(Ω) + ‖g‖L2(Ω)d + ‖‖L2(Ω)),‖T u‖L2(Ω)  C(‖ f ‖L2(Ω) + ‖g‖L2(Ω)d + ‖‖L2(Ω)).As a consequence, there exists one and only one u ∈ V T ,0 such that (3) holds.The key point for the proof of the preceding result is the relation (2) with v = u, together with the inequality‖u − P T u‖L2(Ω)  hT ‖∇u‖L2(Ω)d .Lemma 3.2 (Compactness of a sequence of approximate solutions). Let (Tm)m∈N be a sequence of classical simplicial discretizationsof Ω [7, Section 6], such that hTm tends to 0 as m → ∞. Assume that there exists θ > 0 with θ < θTm for all m ∈ N. Let (um)m∈Nbe a sequence of functions such that um ∈ V Tm,0 for all m ∈ N. Assume that the sequence (Tm um)m∈N is bounded in L2(Ω); thenthere exists a subsequence of (Tm)m∈N , again denoted (Tm)m∈N , and u ∈ H20(Ω), such that the corresponding subsequence (um)m∈Nsatisfies:(i) um → u in L2(Ω),(ii) ∇um → ∇u in L2(Ω)d,(iii) Tm um → u weakly in L2(Ω),as m → ∞.The difficult point in the proof of the above lemma is the H20(Ω) regularity of u. This point is proved using that,prolonging u by 0 in Rd \ Ω , then u ∈ H1(Rd) together with u ∈ L2(Rd).Lemma 3.3 (Interpolation of regular functions with compact support). Let θ > 0 with θ < θT . Let ϕ ∈ C2c (Ω) and let a =d(support(ϕ), ∂Ω). Then there exists ĨT ϕ ∈ V T ,0 and C > 0 only depending on θ such that(i) ‖ĨT ϕ − ϕ‖L2(Ω)  ChT |ϕ|2a2 ,(ii) ‖∇ĨT ϕ − ∇ϕ‖L(Ω)d  ChT |ϕ|2a2 ,(iii) ‖T ĨT ϕ − T ϕ‖L2(Ω)  ChT |ϕ|2a2 , where |ϕ|2 = maxi, j=1,d ‖∂2i jϕ‖L∞(Ω) , and T ϕ is the piecewise constant function equalin Kz to1|Kz |∫Kzϕ(x)dx, for all z ∈ V .In the preceding lemma, the function ĨT ϕ is obtained by multiplying the approximate solution in V T ,0 of the Dirichletproblem with right-hand side −ϕ by a regularization function whose second order derivatives are bounded by C/a2.1286 R. Eymard, R. Herbin / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1283–1286Theorem 3.4 (Convergence of the scheme). Let u ∈ H20(Ω) be the solution of Problem (1); let T be a conformal simplicial mesh of Ωand uT ∈ V T ,0 be the solution of (3). Then, as hT tends to 0 with 0 < θ  θT :(i) uT converges in L2(Ω) to u,(ii) ∇uT converges in L2(Ω)d to ∇u,(iii) T uT converges in L2(Ω) to u.The proof of the convergence theorem is obtained by first extracting a sequence of approximate solutions thanks toLemmas 3.1 and 3.2, then setting in (3) v = ĨT ϕ given by Lemma 3.3 and letting hT tend to 0.Remark 3.5 (Order of convergence). An error estimate of order 1 may be proven if the exact solution has a compact supportin Ω . Using some triangular meshes, numerical results show an order 2 error on u, an order 1 error on ∇u and an order1/2 on u.References[1] M. Ben-Artzi, I. Chorev, J.-P. Croisille, D. Fishelov, A compact difference scheme for the biharmonic equation in planar irregular domains, SIAM J. Numer.Anal. 47 (4) (2009) 3087–3108.[2] M. Ben-Artzi, J.-P. Croisille, D. Fishelov, A fast direct solver for the biharmonic problem in a rectangular grid, SIAM J. Sci. Comput. 31 (1) (2008)303–333.[3] C. Bi, L. 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Approximation of the biharmonic problem using piecewise linear finite elements

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Approximation of the biharmonic problem using piecewise linear finite elements

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