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Numerical Methods for Nonlinear Elliptic Differential EquationsA Synopsis$
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Klaus Boehmer

Print publication date: 2010

Print ISBN-13: 9780199577040

Published to Oxford Scholarship Online: January 2011

DOI: 10.1093/acprof:oso/9780199577040.001.0001

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A general discretization theory

A general discretization theory

Chapter:
(p.173) 3 A general discretization theory
Source:
Numerical Methods for Nonlinear Elliptic Differential Equations
Author(s):

Klaus Böhmer

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199577040.003.0003

A new general discretization theory unifies the generalized Petrov-Galerkin method and one of the classical methods. Linearization is a main tool: the derivative of the operator in the exact solution has to be boundedly invertible. For quasilinear problems, in Sobolev spaces Wm'p(Ω), with 2 ≤ p 〈 ∞, this chapter obtains stability and convergence results with respect to discrete Hm(Ω) norms. This is complemented by the monotone approach for 1 ≤ p 〈 ∞ with Wm'p(Ω) convergence. Our approach allows a unified proof for stability, convergence and Fredholm results for the discrete solutions and their computation. A few well-known basic concepts from functional analysis and approximation theory are combined: coercive bilinear forms or monotone operators, their compact perturbations, interpolation, best approximation and inverse estimates for approximating spaces yield the classical “consistency and stability imply convergence”. The mesh independence principle is the key for an efficient solution for all discretizations of all nonlinear problems considered here.

Keywords:   new discretization theory, generalized Petrov-Galerkin method, classical methods, linearization, coercive bilinear forms, compact perturbations, computation of solutions, mesh independence principle

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