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Convergence of Discretizations of Nonlinear Problems. A General Approach
Author(s) -
Ansorge R.
Publication year - 1993
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.19930731002
Subject(s) - discretization , mathematics , convergence (economics) , numerical analysis , symbolic convergence theory , nonlinear system , norm (philosophy) , weak convergence , compact convergence , rate of convergence , calculus (dental) , computer science , mathematical analysis , medicine , physics , computer security , dentistry , quantum mechanics , key (lock) , political science , law , economics , asset (computer security) , economic growth , computer network , channel (broadcasting)
In 1948 Kantorovich published his famous paper on connections between Numerical Mathematics and Functional Analysis. Particularly for Numerical Analysis, the paper [31] of Lax and Richtmyer (1956) on the numerical treatment of (linear) partial initial value problems made Functional Analysis an instrument definitively adopted by numerical analysts. A more general theory for the description and investigation of discretization methods – influenced by the Lax‐Richtmyer Theory and its extensions but not restricted to PDEs – was aroused by papers of Stummel in the 70s. A well developped Functional Analysis of Discretization Algorithms (Vainikko) came into being. After a brief historical summary, some modern aspects of the theory will be discussed, e.g. a general access to the problem of convergence of approximate solutions to weak and not necessarily unique solutions of operator equations, discretization of entropy conditions, extension of the theory to problems where convergence of the numerical solutions to weak solutions can not be guaranteed in the sense of a norm topology but only in the sense of weak convergence or weak set convergence etc. Concrete examples demonstrate the applicability of the theory.