A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems Authors
First Online: 24 March 2005 Received: 03 November 2003 Revised: 18 June 2004 DOI:
Cite this article as: Nakao, M., Hashimoto, K. & Watanabe, Y. Computing (2005) 75: 1. doi:10.1007/s00607-004-0111-1 Abstract.
In this paper, we propose a numerical method to verify the invertibility of second-order linear elliptic operators. By using the projection and the constructive
a priori error estimates, the invertibility condition is formulated as a numerical inequality based upon the existing verification method originally developed by one of the authors. As a useful application of the result, we present a new verification method of solutions for nonlinear elliptic problems, which enables us to simplify the verification process. Several numerical examples that confirm the actual effectiveness of the method are presented. AMS Subject Classifications: 35J25 35J60 65N25 Keywords Numerical verification unique solvability of linear elliptic problem finite element method References
Alefeld, G.: On the convergence of some interval-arithmetic modifications of Newton’s method. SIAM J. Numer. Anal.
21, 363–372 (1984).
Nagatou, K., Yamamoto, N., Nakao, M. T.: An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optimiz.
20, 543–565 (1999).
Nakao, M. T.: A numerical approach to the proof of existence of solutions for elliptic problems. Japan J. Appl. Math.
5, 313–332 (1988).
Nakao, M. T.: Solving nonlinear elliptic problems with result verification using an
H -1 residual iteration. Computing ( Suppl. 9), 161–173 (1993).
Nakao, M. T., Yamamoto, N.: Numerical verification of solutions for nonlinear elliptic problems using
L ∞ residual method. J. Math. Anal. Appl. 217, 246–262 (1998).
Nakao, M. T.: Numerical verification methods for solutions of ordinary and partial differential equations. Numer. Funct. Anal. Optimiz.
22(3–4), 321–356 (2001).
Nakao, M. T., Watanabe, Y.: An efficient approach to the numerical verification for solutions of elliptic differential equations. Numer. Algorith. 37. Special issue for Proceedings of SCAN 2002, 311–323 (2004).
Plum, M.: Explicit
H 2-estimates and pointwise bounds for solutions of second-order elliptic boundary value problems. J. Math. Anal. Appl. 165, 36–61 (1992).
Plum, M.: Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems. Computing
49, 25–44 (1992).
Plum, M.: Computer-assisted enclosure methods for elliptic differential equations. J. Linear Algebra Appl.
327, 147–187 (2001).
Rump S. M.: INTLAB-INTerval LABoratory, a Matlab toolbox for verified computations, version 4.1.2. Inst. Infomatik, Technical University of Hamburg – Hamburg.
Rump, S. M.: Solving algebraic problems with high accuracy. In: A new approach to scientific computation (Kulisch, U., and Miranker, W. L., eds.). New York: Academic Press 1983.
Yamamoto, N., Nakao, M. T.: Numerical verifications for solutions to elliptic equations using residual iterations with higher order finite element. J. Comput. Appl. Math.
60, 271–279 (1995).
Yamamoto, N.: A numerical verification method for solutions of boundary value problems with local uniqueness by Banach’s fixed point theorem. SIAM J. Numer. Anal.
35, 2004–2013 (1998). Copyright information
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