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Center for Nonlinear Studies |
Abstract: We consider a 2D diffusion problem in a domain with Dirichlet boundary conditions. The Maximum Principle asserts that the solution cannot have a maximum or a minimum within the interior of the underlying domain. So the continuous solution are between the values defined on the boundaries. The Discrete Maximum Principle (DMP) is the discrete form of the Maximum Principle. If the numerical solution violates DMP,i.e., some solutions are beyond the range defined by the boundary values, it will give wrong predictions due to the non-physical solutions or fluxes. Using Finite Element method, we discretize the domain into triangles. Then the problem becomes to solve a linear system Ax = b, with A, which is called the stiffness matrix, be symmetric and positive definite. If in addition, all off-diagonal entries of A are non-positive and each row sum of A is non-negative, then A is an M-matrix. If A is an M-matrix with diagonal dominance, then the numerical solution is guaranteed to satisfy DMP. It has been proven that for isotropic diffusion problems, if all the angles in triangulation of the mesh are not greater than \pi/2, then the stiffness matrix A is an M-matrix with diagonal dominance and the solution satisfies DMP. However, for anisotropic diffusion problems, the stiffness matrix obtained using general mesh will have some positive off-diagonal entries. Thus A is not an M-matrix, and the solution may violate DMP. In this study, we try to adapt mesh for 2D anisotropic diffusion problems so that the solution satisfies DMP. We first deduce sufficient conditions for a mesh such that it guarantees the corresponding stiffness matrix to be an M-matrix. Then we construct some meshes of this type for computation. Finally, we study the effect of different meshes on the numerical solution, and modify the meshes locally to improve the numerical solution. The goal is to develop mesh adaptation strategy such that the solution satisfies DMP with as little mesh modification as possible.
Host: Mentor: Daniil Svyatskiy