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

• Kazuaki Tanaka

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 $${\hat{u}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>u</mml:mi><mml:mo>^</mml:mo></mml:mover></mml:math> , 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.

Journal

• Japan Journal of Industrial and Applied Mathematics

  Japan Journal of Industrial and Applied Mathematics 38 (3), 731-756, 2021-01-27

  Springer Science and Business Media LLC

Keywords

• Applied Mathematics
• General Engineering

Details

• CRID: 1360008373653266176
• NII Article ID: 210000178465
• DOI: 10.1007/s13160-021-00456-0
• ISSN: 1868937X
• ISSN: 09167005
• Web Site: https://link.springer.com/content/pdf/10.1007/s13160-021-00456-0.pdf
• Web Site: https://link.springer.com/article/10.1007/s13160-021-00456-0/fulltext.html
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