Finite volume schemes for the biharmonic problem on general meshes

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 probl...

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Bibliographic Details
Published in:Mathematics of computation 2012-10, Vol.81 (280), p.2019-2048
Main Authors: EYMARD, R., GALLOUËT, T., HERBIN, R., LINKE, A.
Format: Article
Language:English
Subjects:
Approximation
Boundary conditions
Finite element method
Interpolation
Laplace transformation
Laplacians
Mathematical constants
Mathematical functions
Mathematics
Numerical Analysis
Perceptron convergence procedure
Stencils
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Summary: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 )-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.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-2012-02608-1