A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

Kazuaki Tanaka*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

This paper proposes a method for rigorously analyzing the sign-change structure of solutions of elliptic partial differential equations subject to one of the three types of homogeneous boundary conditions: Dirichlet, Neumann, and mixed. Given explicitly estimated error bounds between an exact solution u and a numerically computed approximate solution u^ , we evaluate the number of sign-changes of u (the number of nodal domains) and determine the location of zero level-sets of u (the location of the nodal line). We apply this method to the Dirichlet problem of the Allen–Cahn equation. The nodal line of solutions of this equation represents the interface between two coexisting phases.

Original languageEnglish
Pages (from-to)731-756
Number of pages26
JournalJapan Journal of Industrial and Applied Mathematics
Volume38
Issue number3
DOIs
Publication statusPublished - 2021 Sep

Keywords

  • Allen–Cahn equation
  • Computer-assisted proof
  • Elliptic differentical equations
  • Numerical verification
  • Sign-change structure
  • Verified numerical computation

ASJC Scopus subject areas

  • Engineering(all)
  • Applied Mathematics

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