The Local Superconvergence of the Linear Finite Element Method for the Poisson Problem

Assume that n 2 . In this study, the Richardson extrapolation for the tensor-product block element and the linear finite element theory of the Green's function will be combined to study the local superconvergence of finite element methods for the Poisson equation in a bounded polytopic domain < subset of> R n (polygonal or polyhedral domain for n =2,3 ), where a family of tensor-product block partitions is not required or the solution need not have high global smoothness. We present a special family of partitions T h satisfying, for any e T h , e is a tensor-product block whenever ( e , ) h where ( e , ) denotes the distance between e and . By the linear finite element theory of the Green's function and the Richardson extrapolation for the tensor-product block element, we obtain the local superconvergence of the displacement for the linear finite element method over the special family of partitions T h . (c) 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 930-946, 2014