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Pointwise superconvergence of the gradient for the linear tetrahedral element
Author(s) -
Goodsell George
Publication year - 1994
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.1690100511
Subject(s) - superconvergence , pointwise , mathematics , finite element method , mathematical analysis , tetrahedron , piecewise , piecewise linear function , function (biology) , constant (computer programming) , boundary (topology) , approximation error , geometry , physics , evolutionary biology , biology , thermodynamics , computer science , programming language
We consider the finite element approximation to the solution of a self‐adjoint, second‐order elliptic boundary value problem in three dimensions over a fully uniform mesh of piecewise linear tetrahedral elements. Although the resulting approximation to the gradient is optimal for functions from the approximating space, it is, however, only O ( h ). We show how this can be improved by the recovery , from the finite element solution, of an approximation to the gradient, which is pointwise of a higher order of accuracy than that of the gradient of the finite element approximation. This approximation, termed a recovered gradient function , is, thus, superconvergent . The major task of our analysis is the establishing of an (almost) constant bound on the W 1 1seminorm of the finite element approximation to a smoothed derivative Green's function. © 1994 John Wiley & Sons, Inc.