Abstract

It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary values problem does not degrade significantly as the mesh is refined locally, provided the mesh remains nondegenerate and a natural scaling of the basis functions is used. Bounds for the Euclidean condition number as a function of the number of degrees of freedom are derived in n >= 2 dimensions. When n>=3, the bound is the same as for the regular mesh case, but when n=2 a factor appears in the bound for the condition number that is logarithmic in the ratio of the maximum and minimum mesh sizes. Application of the results to the conjugate-gradient iterative method for solving such linear systems are given.