A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations
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- by Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T. Nakao
- Math. Comp. 82 (2013), 1543-1557
- DOI: https://doi.org/10.1090/S0025-5718-2013-02676-2
- Published electronically: February 25, 2013
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Abstract:
This paper presents constructive a posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations (PDEs) on a bounded domain. This type of estimate plays an important role in the numerical verification of the solutions for boundary value problems in nonlinear elliptic PDEs. In general, it is not easy to obtain the a priori estimates of the operator norm for inverse elliptic operators. Even if we can obtain these estimates, they are often over estimated. Our proposed a posteriori estimates are based on finite-dimensional spectral norm estimates for the Galerkin approximation and expected to converge to the exact operator norm of inverse elliptic operators. This provides more accurate estimates, and more efficient verification results for the solutions of nonlinear problems.References
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Bibliographic Information
- Yoshitaka Watanabe
- Affiliation: Research Institute for Information Technology, Kyushu University, Fukuoka 812-8581, Japan
- Email: watanabe@cc.kyushu-u.ac.jp
- Takehiko Kinoshita
- Affiliation: Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan, Supported by GCOE ‘Fostering top leaders in mathematics’, Kyoto University
- Email: kinosita@kurims.kyoto-u.ac.jp
- Mitsuhiro T. Nakao
- Affiliation: Sasebo National College of Technology, Nagasaki 857-1193, Japan
- Email: mtnakao@post.cc.sasebo.ac.jp
- Received by editor(s): May 18, 2011
- Published electronically: February 25, 2013
- Additional Notes: This work was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan (No. 20224001, No. 21540134) and supported by Kyoto University Mathematics Global COE Program
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 82 (2013), 1543-1557
- MSC (2010): Primary 65N30, 35J25; Secondary 65N15, 35B45
- DOI: https://doi.org/10.1090/S0025-5718-2013-02676-2
- MathSciNet review: 3042574