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An Interface-Fitted Mesh Generator and Numerical Methods for Elliptic Interface Problems


In this thesis, we propose different numerical methods for solving elliptic interface problems in three dimensions and moving boundary problems in two dimensions. In the first part, a simple and efficient interface-fitted mesh algorithm which can produce a semi-unstructured interface-fitted mesh in two and three dimensions quickly is developed in this thesis. Elements in such interface- fitted meshes are not restricted to simplices but can be polygons or polyhedra. Especially in 3D, the polyhedra instead of tetrahedra can avoid slivers, which are the major difficulty in finite element methods. Virtual element methods are applied to solve elliptic interface problems with solutions and flux jump conditions. Algebraic multigrid solvers are used to solve the resulting linear al- gebraic system. In the second part, we impose the interface-fitted meshes to moving boundary problems and present the applications in biology simulation. Static interface problems with peri- odic conditions are considered at first, then the higher order accuracy algorithm is also developed. Finally moving interface problems are discussed. We couple the interface Poisson equation and the transport equation for the level set function together to simulate the tissue and tumor growth phenomenon. Numerical results are presented to illustrate the effectiveness of our methods.

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