TY - JOUR
T1 - A posteriori verification of the positivity of solutions to elliptic boundary value problems
AU - Tanaka, Kazuaki
AU - Asai, Taisei
N1 - Funding Information:
This work is supported by JSPS KAKENHI Grant Number 19K14601 and JST CREST Grant Number JPMJCR14D4. All data generated or analyzed during this study are included in this published article.
Publisher Copyright:
© 2021, The Author(s).
PY - 2022/2
Y1 - 2022/2
N2 - The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither H2-regularity nor L∞-error estimation, but only H01-error estimation. In (J Comput Appl Math 370:112647, 2020), we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require L∞-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.
AB - The purpose of this paper is to develop a unified a posteriori method for verifying the positivity of solutions of elliptic boundary value problems by assuming neither H2-regularity nor L∞-error estimation, but only H01-error estimation. In (J Comput Appl Math 370:112647, 2020), we proposed two approaches to verify the positivity of solutions of several semilinear elliptic boundary value problems. However, some cases require L∞-error estimation and, therefore, narrow applicability. In this paper, we extend one of the approaches and combine it with a priori error bounds for Laplacian eigenvalues to obtain a unified method that has wide application. We describe how to evaluate some constants required to verify the positivity of desired solutions. We apply our method to several problems, including those to which the previous method is not applicable.
KW - Computer-assisted proofs
KW - Elliptic boundary value problems
KW - Error bounds
KW - Numerical verification
KW - Positive solutions
UR - http://www.scopus.com/inward/record.url?scp=85126292498&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85126292498&partnerID=8YFLogxK
U2 - 10.1007/s42985-021-00143-2
DO - 10.1007/s42985-021-00143-2
M3 - Article
AN - SCOPUS:85126292498
SN - 2662-2963
VL - 3
JO - Partial Differential Equations and Applications
JF - Partial Differential Equations and Applications
IS - 1
M1 - 9
ER -