A numerical approach to the proof of existence of solutions for elliptic problems

Mitsuhiro T. Nakao

Research output: Contribution to journalArticle

58 Citations (Scopus)

Abstract

In this paper, we describe a method which proves by computers the existence of weak solutions for linear elliptic boundary value problems of second order. It is shown that we can constitute the computing procedures to verify the existence, uniqueness and inclusion set of a solution based on Schauder's fixed point theorem. Using the finite element approximations for some simple Poisson's equations and the results of error estimates, we generate iteratively a set sequence composed of functions and attempt to construct automatically the set including the exact solution. Further, the conditions of verifiability by this method are considered and some numerical examples of verification are presented.

Original languageEnglish
Pages (from-to)313-332
Number of pages20
JournalJapan Journal of Applied Mathematics
Volume5
Issue number2
DOIs
Publication statusPublished - 1988 Jun
Externally publishedYes

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Keywords

  • boundary value problems
  • error estimates
  • finite element method
  • fixed point theorem

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

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