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Superconvergence of Ritz‐Galerkin finite element approximations for second order elliptic problems
Author(s) -
Wang Chunmei
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22231
Subject(s) - superconvergence , mathematics , finite element method , discontinuous galerkin method , mathematical analysis , mixed finite element method , galerkin method , interpolation (computer graphics) , piecewise , boundary value problem , animation , physics , computer graphics (images) , computer science , thermodynamics
In this paper, the author derives an O ( h 4 ) ‐superconvergence for the piecewise linear Ritz‐Galerkin finite element approximations for the second‐order elliptic equation − ∇ · ( A ∇ u ) = f equipped with Dirichlet boundary conditions. This superconvergence error estimate is established between the finite element solution and the usual Lagrange nodal point interpolation of the exact solution, and thus the superconvergence at the nodal points of each element. The result is based on a condition for the finite element partition characterized by the coefficient tensor A and the usual shape functions on each element, called A ‐equilateral assumption in this paper. Several examples are presented for the coefficient tensor A and finite element triangulations which satisfy the conditions necessary for superconvergence. Some numerical experiments are conducted to confirm this new theory of superconvergence.