The local superconvergence of the linear finite element method for the poisson problem

• Published in: Numerical methods for partial differential equations Vol. 30; no. 3; pp. 930 - 946
• Main Authors: He, Wen-ming, Cui, Jun-Zhi, Zhu, Qi-ding, Wen, Zhong-liang
• Format: Journal Article
• Language: English
• Published: New York Blackwell Publishing Ltd 01.05.2014; Wiley Subscription Services, Inc
• Subjects: Algorithms; Blocking; Displacement; Extrapolation; Finite element analysis; Finite element method; Green's function; Green's functions; local superconvergence; Mathematical analysis; Nonlinear programming; Partitions; Poisson equation; Richardson extrapolation; tensor-product block
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.21842

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 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 ρ ( e , ∂ Ω ) ≥ h where ρ ( e , ∂ Ω ) denotes the distance between e and ∂ Ω . 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 . © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 930–946, 2014