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Subgrid stabilization of Galerkin approximations of linear contraction semi‐groups of class C 0 in Hilbert spaces
Author(s) -
Guermond JeanLuc
Publication year - 2001
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 - 10.1002/1098-2426(200101)17:1<1::aid-num1>3.0.co;2-1
Subject(s) - mathematics , hilbert space , galerkin method , bilinear interpolation , discontinuous galerkin method , norm (philosophy) , nonlinear system , approximations of π , mathematical analysis , contraction (grammar) , bilinear form , finite element method , medicine , physics , thermodynamics , statistics , quantum mechanics , political science , law
This article presents a stabilized Galerkin technique for approximating linear contraction semi‐groups of class C 0 in a Hilbert space. The main result of this article is that this technique yields an optimal approximation estimate in the graph norm. The key idea is two‐fold. First, it consists in introducing an approximation space that is broken up into resolved scales and subgrid scales, so that the bilinear form associated with the generator of the semi‐group satisfies a uniform inf‐sup condition with respect to this decomposition. Second, the Galerkin approximation is slightly modified by introducing an artificial diffusion on the subgrid scales. Numerical tests show that the method applies also to nonlinear semi‐groups. © 2001 John Wiley & Sons., Inc. Numer Methods Partial Differential Eq 17: 1–25, 2001