TY - JOUR

T1 - A posteriori verification for the sign-change structure of solutions of elliptic partial differential equations

AU - Tanaka, Kazuaki

N1 - Funding Information:
This work was supported by JSPS KAKENHI Grant Number 19K14601, JST CREST Grant Number JPMJCR14D4, Mizuho Foundation for the Promotion of Sciences, and The Okawa Foundation for Information and Telecommunications Grant Number 20-01.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Allen–Cahn equation

KW - Computer-assisted proof

KW - Elliptic differentical equations

KW - Numerical verification

KW - Sign-change structure

KW - Verified numerical computation

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U2 - 10.1007/s13160-021-00456-0

DO - 10.1007/s13160-021-00456-0

M3 - Article

AN - SCOPUS:85099832428

JO - Japan Journal of Industrial and Applied Mathematics

JF - Japan Journal of Industrial and Applied Mathematics

SN - 0916-7005

ER -