Unit - Centered Finite Volume Scheme For Diffusion Equation On Arbitrary Polygon Meshes

In this thesis, based on linearity-preserving criterion, we propose some cell-centered finite volume schemes for2D anisotropic diffusion problems on arbitrary polygonal meshes. Here, linearity-preserving criterion has been used as an revelatory method in the derivation of some cell-centered schemes on distorted grids, which requires that the derivation of a scheme is exact whenever the solution is a linear function and the diffusion coefficient is a constant tensor on each mesh cell. The observation from other peoples’works indicates that the violation of linearity-preserving property might result in a loss of accuracy on distorted meshes, and thus it is the fundamental criterion from our point of view.Firstly, we introduce a general framework for the construction of a certain kind of cell-centered finite volume schemes on arbitrary polygonal meshes. And then we re-derive the traditional nine-point scheme and a certain multi-point flux approximation (MPFA) scheme by this general frame. Secondly, based on this general frame, we construct two linearity-preserving and cell-centered finite volume schemes. The main feature of the first one lies in the introduction of two auxiliary unknowns on each cell edge, and then the scheme has both cell-centered primary unknowns and cell-edge based auxiliary unknowns. The auxiliary unknowns are interpolated by the multi-point flux approximation (MPFA) technique, which reduces the scheme to a completely cell-centered one. The second one is based on the so-called harmonic averaging points located at the interface of heterogeneity. The new scheme is locally conservative and has a compact stencil, which reduces to a nine-point one on structured quadrilateral meshes. Under some general and easily verified assumptions, we obtained the coercivity and stability results in Hl norm through a discrete functional approach. Finally, based on this general frame, and together with a certain reconstruction algorithm, we obtain a family of linearity-preserving schemes. Moreover, we investigate the relations between the general scheme and some existing schemes, which include the aforementioned two schemes suggested in this thesis.In each chapter, we present some numerical experiments to show the accuracy and efficiency of the new schemes. Many dedicate experiments using grids with unstruc-tured, severely distorted and arbitrary (continuous or discontinuous, homogeneous or heterogeneous) anisotropic diffusion tensors show the good performance of the proposed scheme (quadratic convergence rate for the approximate solution).