On the Conditioning of Finite Element Equations with Highly Refined Meshes

It is proven that the condition number of the linear system representing a finite element discretization of an elliptic boundary value 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 f 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. Applications of the results to the conjugate-gradient iterative method for solving such linear systems are given.