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The local superconvergence of the linear finite element method for the poisson problem
Author(s) -
He Wenming,
Cui JunZhi,
Zhu Qiding,
Wen Zhongliang
Publication year - 2014
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.21842
Subject(s) - superconvergence , mathematics , finite element method , tensor product , mathematical analysis , mixed finite element method , extrapolation , extended finite element method , pure mathematics , physics , thermodynamics
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 Ω ⊂ ℜ n(polygonal or polyhedral domain for n = 2 , 3 ), where a family of tensor‐product block partitionsis not required or the solution need not have high globalsmoothness. We present a special family of partitionsT hsatisfying, for any e ∈ T h , e is a tensor‐product block whenever ρ ( e , ∂ Ω ) ≥ h whereρ ( e , ∂ Ω )denotes the distance between e and ∂ Ω. By the linear finite elementtheory of the Green's function and the Richardson extrapolation forthe tensor‐product block element, we obtain the localsuperconvergence of the displacement for the linear finite elementmethod over the special family of partitionsT h . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 930–946, 2014