Optimal finite element mesh for elliptic equation of divergence form

Abstract

We consider an optimal triangular mesh minimizing the condition number of the finite element stiffness matrix for an elliptic equation - Sigma(partial derivativexi)/(partial derivative)(a(ij) (partial derivativexi)/(partial derivativeu)) = f, u\(partial derivativeOmega) = g. Using a sharp bound for the condition number of the stiffness matrix, it is shown that the element of the optimal uniform triangular mesh is equilateral with respect to the metric which is the inverse of the coefficient matrix in the equation. It is verified by numerical examples that Such a mesh is really effective in reducing the condition number of the stiffness matrix. In addition, we suggest an algorithm generating a mesh in which every element is almost equilateral with respect to a metric.