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Defect correction and Galerkin's method for second‐order boundary value problems
Author(s) -
Chibi AhmedSalah,
Moore Gerald
Publication year - 1992
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.1690080506
Subject(s) - mathematics , piecewise , sobolev space , boundary value problem , norm (philosophy) , galerkin method , finite element method , order (exchange) , discontinuous galerkin method , mathematical analysis , bilinear interpolation , bilinear form , statistics , physics , finance , political science , law , economics , thermodynamics
Abstract The defect correction technique, based on the Galerkin finite element method, is analyzed as a procedure to obtain highly accurate numerical solutions to second‐order elliptic boundary value problems. The basic solutions, defined over a rectangular region Ω, are computed using continuous piecewise bilinear polynomials on rectangles. These solutions are O ( h 2 ) accurate globally in the second‐order discrete Sobolev norm. Corrections to these basic solutions are obtained using higher‐order piecewise polynomials (Lagrange polynomials or splines) to form defects. An O ( h 2 ) improvement is gained on the first correction. The lack of regularity of the discrete problems (beyond the second‐order Sobolev norm) makes it impossible to retain this order of improvement, but for problems satisfying certain periodicity conditions, straightforward arbitrary accuracy is obtained, since these problems possess high‐order regularity. © 1992 John Wiley & Sons, Inc.