A Numerical Method to Verify the Invertibility of Linear Elliptic Operators with Applications to Nonlinear Problems

Authors:
M. T. Nakao;K. Hashimoto;Y. Watanabe
Affiliations:
Faculty of Mathematics, Kyushu University, 812-8581, Fukuoka, Japan;Graduate School of Mathematics, Kyushu University, 812-8581, Fukuoka, Japan;Computing and Communications Center, Kyushu University, 812-8581, Fukuoka, Japan
Venue:
Computing - Editorial: Special issue on GAMM – Workshop on Guaranteed Error-bounds for the Solution of Nonlinear Problems in Applied Mathematics
Year:
2005

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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.