Optimal defect corrections on composite nonmatching finite-element meshes

Ahmed-Salah Chibi ^{In this paper, we analyse the ‘defect-correction’technique on a general smooth region, via composite finite-elementmeshes (a Cartesian mesh and a polar mesh) on two overlappingsubdomains (a rectangle and an annulus). Boundary interpolatorymappings of higher degree are used, in the Schwarz method, topass from one mesh to another. An explicit relation is givenbetween the degree of these mappings and the number of optimalcorrections to be computed. Optimal convergence results forthe discrete bilinear basic solution, in higher-order discreteSobolev norms, are obtained on the subdomains. Because the successof the defect-correction technique is based on the uniformityof the discretization and the regularity of the exact solution,the defects are computed on the subdomains in the same way asfor the basic solution. Optimal O(h2) improvement per correctionis obtained. Numerical results are presented to support thetheory.}     

Defect correction and Galerkin's method for second-order boundary value problems

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(h²) 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(h²) 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.