TY - JOUR
T1 - Numerical verification method for positive solutions of elliptic problems
AU - Tanaka, Kazuaki
N1 - Funding Information:
We express our sincere thanks to Prof. Kazunaga Tanaka (Waseda University, Japan) for helpful advice and comments about this study, Prof. Michael Plum (Karlsruhe Institut für Technologie, Germany) for helping us to correct a mistake in Theorem 2.1, and Dr. Kohei Yatabe (Waseda University, Japan) for his contribution to improving English expressions of this paper. We also express our profound gratitude to two anonymous referees for their highly insightful comments and suggestions. This work is supported by JST CREST Grant Numbers JPMJCR14D4, and JSPS KAKENHI Grant Number JP17H07188 and JP19K14601, and Mizuho Foundation for the Promotion of Sciences, Japan.
Funding Information:
We express our sincere thanks to Prof. Kazunaga Tanaka (Waseda University, Japan) for helpful advice and comments about this study, Prof. Michael Plum (Karlsruhe Institut für Technologie, Germany) for helping us to correct a mistake in Theorem 2.1 , and Dr. Kohei Yatabe (Waseda University, Japan) for his contribution to improving English expressions of this paper. We also express our profound gratitude to two anonymous referees for their highly insightful comments and suggestions. This work is supported by JST CREST Grant Numbers JPMJCR14D4 , and JSPS KAKENHI Grant Number JP17H07188 and JP19K14601 , and Mizuho Foundation for the Promotion of Sciences, Japan .
Publisher Copyright:
© 2019 Elsevier B.V.
PY - 2020/5/15
Y1 - 2020/5/15
N2 - The purpose of this paper is to propose methods for verifying the positivity of a weak solution u of an elliptic problem assuming H0 1-error estimation ‖u−uˆ‖H0 1 ≤ρ given some numerical approximation uˆ and an explicit error bound ρ. We provide a sufficient condition for the solution to be positive and analyze the range of application of our method for elliptic problems with polynomial nonlinearities. We present numerical examples where our method is applied to some important problems.
AB - The purpose of this paper is to propose methods for verifying the positivity of a weak solution u of an elliptic problem assuming H0 1-error estimation ‖u−uˆ‖H0 1 ≤ρ given some numerical approximation uˆ and an explicit error bound ρ. We provide a sufficient condition for the solution to be positive and analyze the range of application of our method for elliptic problems with polynomial nonlinearities. We present numerical examples where our method is applied to some important problems.
KW - Computer-assisted proof
KW - Elliptic problems
KW - Newton's method
KW - Numerical verification
KW - Positive solutions
KW - Verified numerical computation
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U2 - 10.1016/j.cam.2019.112647
DO - 10.1016/j.cam.2019.112647
M3 - Article
AN - SCOPUS:85076103208
SN - 0377-0427
VL - 370
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
M1 - 112647
ER -