On solution uniqueness of elliptic boundary value problems

Authors:
Zi-Cai Li;Qing Fang;Hung-Tsai Huang;Yimin Wei
Affiliations:
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 80424, Taiwan and Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Ta ...;Department of Mathematical Sciences, Faculty of Science, Yamagata University, Yamagata 990-8560, Japan;Department of Applied Mathematics, I-Shou University, Kaohsiung County, Taiwan;Department of Mathematics, Fudan University and Key Laboratory of Mathematics for Nonlinear Science, Ministry of Education, Shanghai, 200433, PR China
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

In this paper, we consider the problem of solution uniqueness for the second order elliptic boundary value problem, by looking at its finite element or finite difference approximations. We derive several equivalent conditions, which are simpler and easier than the boundedness of the entries of the inverse matrix given in Yamamoto et al., [T. Yamamoto, S. Oishi, Q. Fang, Discretization principles for linear two-point boundary value problems, II, Numer. Funct. Anal. Optim. 29 (2008) 213-224]. The numerical experiments are provided to support the analysis made. Strictly speaking, the uniqueness of solution is equivalent to the existence of nonzero eigenvalues in the corresponding eigenvalue problem, and this condition should be checked by solving the corresponding eigenvalue problems. An application of the equivalent conditions is that we may discover the uniqueness simultaneously, while seeking the approximate solutions of elliptic boundary equations.