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Center for Nonlinear Studies |
The overview of the Mimetic Finite Difference (MFD) method with the focus on its capabilities to satisfy DMP will be given in this talk. The Maximum Principle is one of the most important properties of solutions of partial differential equations. Its numerical analog, the Discrete Maximum Principle (DMP), is one of the properties that are very difficult to incorporate into numerical methods, especially when the computational mesh contains distorted or degenerate cells or the problem coefficients are highly heterogeneous and anisotropic. A violation of DMP may lead to numerical instabilities such as oscillations and to non-physical solutions, for example, when heat flows from a cold material to a hot one. Some physical quantities, like concentration and temperature, are non-negative by their nature and their approximations should be non-negative as well. The family of the Mimetic Finite Difference (MFD) methods provides flexibility in the choice of parameters which define a particular member of the family. For example, the MFD discretization scheme for quadrilateral meshes depends on three parameters. The correct choice of these parameters may guarantee that the resulting numerical scheme satisfy the DMP principle. The analysis of this strategy is based on the properties of M-matrices. The monotonicity limits of MFD method are investigated in several practically important cases including meshes generated using the Adaptive Mesh Refinement (AMR) strategy.
Host: Humberto C Godinez Vazquez, Mathematical Modeling and Analysis Theoretical Division, 5-9188