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Superconvergence and stability for boundary penalty techniques of finite difference methods
Author(s) -
Li ZiCai,
Huang HungTsai,
Huang Jin
Publication year - 2008
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.20302
Subject(s) - penalty method , mathematics , superconvergence , biharmonic equation , dirichlet boundary condition , boundary value problem , collocation (remote sensing) , norm (philosophy) , stability (learning theory) , mathematical analysis , finite element method , mathematical optimization , computer science , law , physics , machine learning , political science , thermodynamics
Abstract The finite difference method (FDM) is used for Dirichlet problems of Poisson's equation, and the Dirichlet boundary condition is dealt with by boundary penalty techniques. Two penalty techniques, penalty‐integrals and penalty‐collocations (i.e., fixing), are proposed in this paper. The error bounds in the discrete H 1 norm and the infinite norms are derived. The stability analysis is based on the new effective condition number (Cond_eff) but not on the traditional condition number (Cond). The bounds of Cond_eff are explored to display that both the penalty‐integral and the penalty‐collocation techniques have good stability; the huge Cond is misleading. Since the penalty‐collocation technique (i.e., the fixing technique) is simpler, it has been applied in engineering problem for a long time. It is worthy to point out that this paper is the first time to provide a theoretical justification for such a popular penalty‐collocation (fixing) technique. Hence the penalty‐collocation is recommended for dealing with the complicated constraint conditions such as the clamped and the simply support boundary conditions of biharmonic equations. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008