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A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations


Authors: Yoshitaka Watanabe, Takehiko Kinoshita and Mitsuhiro T. Nakao
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
Published electronically: February 25, 2013
MathSciNet review: 3042574
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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.


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  • 1. M. Agueh, Gagliardo-Nirenberg inequalities involving the gradient $ L^2$-norm, C. R. Math. Acad. Sci., Paris, 346 (2008), 757-762. MR 2427077 (2009f:49049)
  • 2. F. Kikuchi and X. Liu, Estimation of interpolation error constants for the $ P_0$ and $ P_1$ triangular finite elements, Computer methods in applied mechanics and engineering, 196 (2007), 3750-3758. MR 2340000 (2008f:65214)
  • 3. S. Kimura and N. Yamamoto, On explicit bounds in the error for the $ H_0^1$-projection into piecewise polynomial spaces, Bulletin of Informatics and Cybernetics, 31 (1999), No. 2, 109-115. MR 1737441
  • 4. T. Kinoshita, K. Hashimoto and M.T. Nakao, On the $ L^2$ a priori error estimates to the finite element solution of elliptic problems with singular adjoint operator, Numerical Functional Analysis and Optimization, 30 (2009), no. 3-4, 289-305. MR 2514218 (2010d:65325)
  • 5. M.T. Nakao, N. Yamamoto and S. Kimura, On the Best Constant in the Error Bound for the $ H_0^1$-Projection into Piecewise Polynomial Spaces, Journal of Approximation Theory, 93 (1998), 491-500. MR 1624846 (99d:41035)
  • 6. M.T. Nakao, K. Hashimoto and Y. Watanabe, A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing, 75 (2005), 1-14. MR 2161437 (2006e:65098)
  • 7. M.T. Nakao and K. Hashimoto, Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications, Journal of Computational and Applied Mathematics, 218 (2008), no. 1, 106-115. MR 2431603 (2009e:65181)
  • 8. S. Oishi, Numerical verification of existence and inclusion of solutions for nonlinear operator equations, Journal of Computational and Applied Mathematics, 60 (1995), no. 1-2, 171-185. MR 1354654 (96g:65058)
  • 9. M. Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), no. 2-3, 419-442. MR 2589483 (2011c:35161)
  • 10. S.M. Rump, INTLAB-INTerval LABoratory, in Developments in Reliable Computing, Tibor Csendes, ed., pp. 77-104, Kluwer Academic Publishers, Dordrecht, (1999). http://www.ti3.tu-harburg.de/rump/intlab/

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Additional 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

DOI: https://doi.org/10.1090/S0025-5718-2013-02676-2
Keywords: Constructive a posteriori estimates, Galerkin method, linear elliptic PDEs
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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