A Adaptive Finite Element Method Based on a New Posteriori Error Estimator and Its Application

Abstract: Adaptive finite element methods in numerical solution of partial differential equations play an important role in engineering and science computations. It's aimed to achieve better accuracy with minimum degree of freedom. The foundation of the adaptive method is to construct a good a posteriori error estimator, which represent the global and local error between the numerical solution and the exact solution, and then use it to guide the mesh refinement.In this thesis, we consider the adaptive finite methods for elliptic problem in two dimensional bounded domain. On the aspect of a posteriori error estimates, by global refinement of the mesh, we get a auxiliary fine mesh and a better approximation of the true solution defined on this auxiliary mesh. We can prove that very little computation is needed to get the better approximation. Then we adapt the energy error between the better approximation and the finite element solution defined on the mesh we care about as a posteriori error estimator. On the aspect of mesh refinement and optimization, we use the CVT technique, which can produce a hight quality meshes, and then have some superconvergence properties. The numerical examples verified that our adaptive algorithm is effective.In this thesis, there are two points of innovation: First, to get the a posteriori error estimator, we use some Gauss-Seidel iterations to obtain a solution which can approximate the exact solution with a higher accuracy than the finite element solution, so it needs small additional computation. Second, in the process of adaptive finite element, we adapt the mesh refinement and optimization method based on the CVT, then we always maintain grid with a good quality for every mesh level, so that the numerical solution can approximate the exact solution with a high accuracy…
Key words: Finite element; a posteriori error estimates; CVT; Adaptive

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