Chapter

Finite difference methods

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations

Published in print October 2010 | ISBN: 9780199577040
Published online January 2011 | e-ISBN: 9780191595172 | DOI: http://dx.doi.org/10.1093/acprof:oso/9780199577040.003.0008

Series: Numerical Mathematics and Scientific Computation

Finite difference methods

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Difference methods are, in most earlier approaches, either based upon the strong forms or studied as external approximation schemes. This chapter's new variational approach is based upon the weak form. The results are formulated for linear to quasilinear equations and systems of orders 2m, m ≥ 1, and for order 2 for natural boundary conditions. The discretization errors are estimated with respect to discrete Sobolev norms of order m for a problem of order 2m, similar to our FEMs results. This allows efficient defect correction methods as an appropriate strategy for formulating high order methods. Particularly interesting are symmetric forms. For special domains they yield convergence of order 2 and 2k for k defect corrections, compared to order 1 and k for the unsymmetric forms. The chapter finishes with curved boundaries, asymptotic expansions, and, combined with extrapolation methods, with the von Kármán equation.

Keywords: (un‐)symmetric difference equations; weak form; quasilinear equations; convergence in discrete Sobolevnorms; defect corrections; von Kármán equation

Chapter.  18354 words.  Illustrated.

Subjects: Mathematical and Statistical Physics

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