A Guaranteed Bound of the Optimal Constant in the Error Estimates for Linear Triangular Elements

We consider a numerical method to get a guaranteed bound of the optimal constant in the error estimates of a finite element method with linear triangular elements in the plane. The problem is reduced to a kind of smallest eigenvalue problem for an elliptic operator in a certain function space on the reference triangle. In order to solve the problem, we formulate a numerical verification procedure based on finite element approximations and constuctive error estimates. Consequently, we obtain a sufficiently sharp bound of the desired constant by a computer assisted proof. In this paper, we provide the basic idea and outline the concept of verification procedures as well as show the final numerical result. The detailed description of procedures for actual computations will be presented in the forthcoming paper [11].