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Supraconvergence and supercloseness of a scheme for elliptic equations on nonuniform grids. (English) Zbl 1102.65109

Elliptic equations of second order are solved here by a finite difference method on anisotropic rectangular meshes. The finite difference method is considered as equivalent to a finite element method with a quadrature rule. Error estimates are derived via superconvergence.
The domain is assumed to be polygonal, but also \(H^3\)-regularity is supposed which can be verified only for boundaries with a smoothness that is not encountered with polygonal domains. In a previous paper, even \(H^4\)-regularity was assumed, and a figure in the paper contains even a domain with a reentrant corner. Here we have even not \(H^2\)-regularity in the generic case.

MSC:

65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N06 Finite difference methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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