Abstract

A Numerical verification method for eigenvalue problems of second-order elliptic operators M.T. Nakao We consider a numerical technique to verify the exact eigenvalues and eigenfunctions of second-order elliptic operators in some neighborhood of their approximations. This technique is based on the numerical verification method for the solutions of second order nonlinear elliptic boundary value problems. Formulating the eigenvalue problem as a fixed point equation of compact operator in a certain Sobolev space, we apply Schauder's or Banach's fixed point theorems via the verified numerical computations in computer using the Newton-like method with error estimates for the $C^{0}$ finite element solution. We decompose the problem as two parts, one is the finite dimensional, i.e., directly computational part, and the other the infinite dimensional, i.e., could not be computable part by the usual numerical methods. While the former stands for the finite dimensional approximation by the finite element method, the latter can be evaluated by the constructive a priori error estimates. Combining them, the conditions of the fixed point theorem are validated in computer. In this talk, we introduce such a technique to enclose the eigenpairs including a uniqueness property of the following self-adjoint eigenvalue problem: \[ \left \{ \begin{array}{rcll} -\Delta u + qu & = & \lambda u & {\rm in}~\Omega,\\ u & = & 0 & {\rm on}~\partial \Omega, \end{array} \right. \] where $\Omega $ is a bounded convex domain in ${\rm R}^2$ and $q\in L^{\infty }(\Omega )$. Several numerical examples are presented. Particularly, we note that this technique can also be applied to the non-self-adjoint problems.

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