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Kobayashi, Kenta

A constructive a priori error estimation for finite element discretizations in a non-convex domain using singular functions. (English) Zbl 1186.65143

Japan J. Ind. Appl. Math. 26, No. 2-3, 493-516 (2009).
This contribution provides constructively a priori error estimates for the finite element solution of elliptic problems, in particular Poisson’s equation, in domains which are bounded by orthogonal line segments and additionally have re-entrant corners. The rate of convergence of the solution of such problems is improved by adding singular functions to the usual interpolating basis. Explicit a priori \(H_0^1\) and \(L^2\) error estimates of \(O(h)\) and \(O(h^2)\), respectively, are derived for the finite element solution of such problems with \(\pi/2\) or \(3\pi/2\) corners, whereas all constants required are determined numerically. Two examples (L- and H-shaped domains) to confirm the validity of the error estimates conclude this clearly structured and well-written paper.
Reviewer: Gunnar Possart (Erlangen)

Cited in 7 Documents

MSC:

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

finite element method; a priori error estimation; Poisson equation; singular functions; numerical examples; re-entrant corners; convergence; L-shaped domains; H-shaped domains
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\textit{K. Kobayashi}, Japan J. Ind. Appl. Math. 26, No. 2--3, 493--516 (2009; Zbl 1186.65143)
Full Text: DOI Euclid

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