Finite volume schemes for the biharmonic problem on general meshes

Eymard, R.; Gallouët, T.; Herbin, R.; Linke, A.

doi:10.1090/S0025-5718-2012-02608-1

Finite volume schemes for the biharmonic problem on general meshes
HTML articles powered by AMS MathViewer

by R. Eymard, T. Gallouët, R. Herbin and A. Linke PDF

Math. Comp. 81 (2012), 2019-2048 Request permission

Abstract:

During the development of a convergence theory for Nicolaides’ extension of the classical MAC scheme for the incompressible Navier-Stokes equations to unstructured triangle meshes, it became clear that a convergence theory for a new kind of finite volume discretizations for the biharmonic problem would be a very useful tool in the convergence analysis of the generalized MAC scheme. Therefore, we present and analyze new finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions on grids which satisfy an orthogonality condition. We prove that a piecewise constant approximate solution of the biharmonic problem converges in $L^2(\Omega )$ to the exact solution. Similar results are shown for the discrete approximate of the gradient and the discrete approximate of the Laplacian of the exact solution. Error estimates are also derived. This part of the paper is a first, significant step towards a convergence theory of Nicolaides’ extension of the classical MAC scheme. Further, we show that finite volume discretizations for the biharmonic problem can also be defined on very general, nonconforming meshes, such that the same convergence results hold. The possibility to construct a converging lowest order finite volume method for the $H^2$-regular biharmonic problem on general meshes seems to be an interesting result for itself and clarifies the necessary ingredients for converging discretizations of the biharmonic problem. All these results are confirmed by numerical results.

References

L. Agelas, D. A. Di Pietro, and R. Masson. A symmetric and coercive finite volume scheme for multiphase porous media flow problems with applications in the oil industry. In Finite volumes for complex applications V, pages 35–51. ISTE, London, 2008.
M. Ben-Artzi, I. Chorev, J.-P. Croisille, and D. Fishelov, A compact difference scheme for the biharmonic equation in planar irregular domains, SIAM J. Numer. Anal. 47 (2009), no. 4, 3087–3108. MR 2551159, DOI 10.1137/080718784
Matania Ben-Artzi, Jean-Pierre Croisille, and Dalia Fishelov, A fast direct solver for the biharmonic problem in a rectangular grid, SIAM J. Sci. Comput. 31 (2008), no. 1, 303–333. MR 2460780, DOI 10.1137/070694168
Chun-jia Bi and Li-kang Li, Mortar finite volume method with Adini element for biharmonic problem, J. Comput. Math. 22 (2004), no. 3, 475–488. MR 2056302
Susanne C. Brenner and Li-Yeng Sung, $C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains, J. Sci. Comput. 22/23 (2005), 83–118. MR 2142191, DOI 10.1007/s10915-004-4135-7
Franco Brezzi and Michel Fortin, Mixed and hybrid finite element methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205, DOI 10.1007/978-1-4612-3172-1
Guo Chen, Zhilin Li, and Ping Lin, A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow, Adv. Comput. Math. 29 (2008), no. 2, 113–133. MR 2420868, DOI 10.1007/s10444-007-9043-6
P. G. Ciarlet and J.-L. Lions (eds.), Handbook of numerical analysis. Vol. II, Handbook of Numerical Analysis, II, North-Holland, Amsterdam, 1991. Finite element methods. Part 1. MR 1115235
Philippe Destuynder and Michel Salaun, Mathematical analysis of thin plate models, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 24, Springer-Verlag, Berlin, 1996 (English, with French summary). MR 1422248, DOI 10.1007/978-3-642-51761-7
R. Eymard, J. Fuhrmann, and A. Linke. MAC schemes on triangular meshes. WIAS in: Finite Volumes for Complex Applications VI, Problems and Perspectives, Springer Proccedings in Mathematics, 2011, pp. 399–407.
Robert Eymard and Thierry Gallouët, $H$-convergence and numerical schemes for elliptic problems, SIAM J. Numer. Anal. 41 (2003), no. 2, 539–562. MR 2004187, DOI 10.1137/S0036142901397083
Robert Eymard, Thierry Gallouët, and Raphaèle Herbin, Finite volume methods, Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 713–1020. MR 1804748, DOI 10.1086/phos.67.4.188705
R. Eymard, T. Gallouët, and R. Herbin, A cell-centered finite-volume approximation for anisotropic diffusion operators on unstructured meshes in any space dimension, IMA J. Numer. Anal. 26 (2006), no. 2, 326–353. MR 2218636, DOI 10.1093/imanum/dri036
R. Eymard, T. Gallouët, and R. Herbin, Cell centred discretisation of non linear elliptic problems on general multidimensional polyhedral grids, J. Numer. Math. 17 (2009), no. 3, 173–193. MR 2573566, DOI 10.1515/JNUM.2009.010
R. Eymard, T. Gallouët, and R. Herbin, Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes SUSHI: a scheme using stabilization and hybrid interfaces, IMA J. Numer. Anal. 30 (2010), no. 4, 1009–1043. MR 2727814, DOI 10.1093/imanum/drn084
R. Eymard, T. Gallouët, R. Herbin, and J.-C. Latché, Analysis tools for finite volume schemes, Acta Math. Univ. Comenian. (N.S.) 76 (2007), no. 1, 111–136. MR 2331058
Thierry Gallouët, Raphaèle Herbin, and Marie Hélène Vignal, Error estimates on the approximate finite volume solution of convection diffusion equations with general boundary conditions, SIAM J. Numer. Anal. 37 (2000), no. 6, 1935–1972. MR 1766855, DOI 10.1137/S0036142999351388
M. Gander and G. Wanner. From Euler, Ritz and Galerkin to modern computing. to appear in SIAM Review, 2011.
Emmanuil H. Georgoulis and Paul Houston, Discontinuous Galerkin methods for the biharmonic problem, IMA J. Numer. Anal. 29 (2009), no. 3, 573–594. MR 2520159, DOI 10.1093/imanum/drn015
Thirupathi Gudi, Neela Nataraj, and Amiya K. Pani, Mixed discontinuous Galerkin finite element method for the biharmonic equation, J. Sci. Comput. 37 (2008), no. 2, 139–161. MR 2453216, DOI 10.1007/s10915-008-9200-1
C. A. Hall, J. C. Cavendish, and W. H. Frey, The dual variable method for solving fluid flow difference equations on Delaunay triangulations, Comput. & Fluids 20 (1991), no. 2, 145–164. MR 1123813, DOI 10.1016/0045-7930(91)90017-C
F. H. Harlow and J. E. Welch. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of fluids, 8(12):2182–2189, 1965.
Igor Mozolevski, Endre Süli, and Paulo R. Bösing, $hp$-version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation, J. Sci. Comput. 30 (2007), no. 3, 465–491. MR 2295480, DOI 10.1007/s10915-006-9100-1
J. Nicolaides, T. A. Porsching, and C. A. Hall. Covolume methods in computational fluid dynamics. In M. Hafez and K. Oshma, editors, Computation Fluid Dynamics Review, pages 279–299. John Wiley and Sons, New York, 1995.
R. A. Nicolaides, Analysis and convergence of the MAC scheme. I. The linear problem, SIAM J. Numer. Anal. 29 (1992), no. 6, 1579–1591. MR 1191137, DOI 10.1137/0729091
S. V. Patankar. Numerical heat transfer and fluid flow. Series in Computational Methods in Mechanics and Thermal Sciences, Minkowycz and Sparrow Eds., 1980.
W. Ritz. Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik. Journal für die reine und angewandte Mathematik:111-114, 1857, Analysis and Applications, 135:1–61, 1908.
Endre Süli and Igor Mozolevski, $hp$-version interior penalty DGFEMs for the biharmonic equation, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 13-16, 1851–1863. MR 2298696, DOI 10.1016/j.cma.2006.06.014
Tongke Wang, A mixed finite volume element method based on rectangular mesh for biharmonic equations, J. Comput. Appl. Math. 172 (2004), no. 1, 117–130. MR 2091134, DOI 10.1016/j.cam.2004.02.002