A numerical method to verify the elliptic eigenvalue problems including a uniqueness property (Q1818413)

A verification method proposed by \textit{K. Nagatou}, \textit{N. Yamamoto} and \textit{M. T. Nakao} [An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim. 20, No. 5-6, 543-565 (1959)] in the context of nonlinear boundary value problems, is extended to elliptic eigenvalue problems \(-\Delta u+ qu=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\) under the normalization \(\int_\Omega u^2dx= 1\). Here, \(\Delta\) denotes the Laplace operator on a bounded convex domain \(\Omega\subset \mathbb{R}^2\). Using accurate finite element approximations to eigensolutions \((u,\lambda)\in H^1_0(\Omega)\times \mathbb{R}\) together with certain a posteriori estimates, Banach's fixed point theorem can be applied, provided that some verification condition is satisfied. In this way the author obtains inclusion sets for locally unique solutions \((u,\lambda)\) and separately for \(\lambda\) as well. Two examples are treated and numerical results are presented.