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

Watanabe, Yoshitaka; Kinoshita, Takehiko; Nakao, Mitsuhiro

doi:10.1090/S0025-5718-2013-02676-2

A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations
HTML articles powered by AMS MathViewer

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

PDF | Request permission

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

Martial Agueh, Gagliardo-Nirenberg inequalities involving the gradient $L^2$-norm, C. R. Math. Acad. Sci. Paris 346 (2008), no. 13-14, 757–762 (English, with English and French summaries). MR 2427077, DOI 10.1016/j.crma.2008.05.015
Fumio Kikuchi and Xuefeng Liu, Estimation of interpolation error constants for the $P_0$ and $P_1$ triangular finite elements, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 37-40, 3750–3758. MR 2340000, DOI 10.1016/j.cma.2006.10.029
Seiji Kimura and Nobito Yamamoto, On explicit bounds in the error for the $H^1_0$-projection into piecewise polynomial spaces, Bull. Inform. Cybernet. 31 (1999), no. 2, 109–115. MR 1737441
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, Numer. Funct. Anal. Optim. 30 (2009), no. 3-4, 289–305. MR 2514218, DOI 10.1080/01630560802679364
Mitsuhiro T. Nakao, Nobito Yamamoto, and Seiji Kimura, On the best constant in the error bound for the $H^1_0$-projection into piecewise polynomial spaces, J. Approx. Theory 93 (1998), no. 3, 491–500. MR 1624846, DOI 10.1006/jath.1998.3172
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), no. 1, 1–14. MR 2161437, DOI 10.1007/s00607-004-0111-1
Mitsuhiro T. Nakao and Kouji Hashimoto, Guaranteed error bounds for finite element approximations of noncoercive elliptic problems and their applications, J. Comput. Appl. Math. 218 (2008), no. 1, 106–115. MR 2431603, DOI 10.1016/j.cam.2007.04.036
Shin’ichi Oishi, Numerical verification of existence and inclusion of solutions for nonlinear operator equations, J. Comput. Appl. Math. 60 (1995), no. 1-2, 171–185. Linear/nonlinear iterative methods and verification of solution (Matsuyama, 1993). MR 1354654, DOI 10.1016/0377-0427(94)00090-N
Michael Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan J. Indust. Appl. Math. 26 (2009), no. 2-3, 419–442. MR 2589483
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/