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.
Funding Information:
We express our sincere thanks to Prof. Kazunaga Tanaka (Waseda University, Japan) for his helpful advice, Prof. Mitsuhiro T.?Nakao (Kyushu University / Waseda University, Japan) for insightful comments, and Taisei Asai (Waseda University, Japan) for helping to create easy-to-read figures and tables. We also express our profound gratitude to an anonymous referee for highly insightful comments and suggestions.
Publisher Copyright:
© 2021, The Author(s).
PY - 2021/9
Y1 - 2021/9
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
VL - 38
SP - 731
EP - 756
JO - Japan Journal of Industrial and Applied Mathematics
JF - Japan Journal of Industrial and Applied Mathematics
SN - 0916-7005
IS - 3
ER -