Study of Superconvergence by a Computer-Based Approach. Superconvergence of the Gradient in Finite Element Solutions of Laplace's and Poisson's Equations
MARYLAND UNIV COLLEGE PARK INST FOR PHYSICAL SCIENCE AND TECHNOLOGY
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This paper addresses the problem of existence of the superconvergence points by a computer based proof. We prove a basic mathematical theorem that the superconvergence point exists if and only if it can be found by certain numerical algorithm. We address the problem of the superconvergence points for the gradient of the finite element solution of the Laplace and Poisson equations. Our study shows that the sets of superconvergence points are very different for these two cases. We consider triangular as well as quadrilateral elements of degree p, 1 or p or 7. In the case of quadrilateral elements we analyze, among others, the tensor-product and the serendipity elements.
- Numerical Mathematics