Evaluation of the condition number in linear systems arising in finite element approximations

This paper derives upper and lower bounds for the ℓ^p- condition number of the stiffness matrix resulting from the finite element approximation of a linear, abstract model problem. Sharp estimates in terms of the meshsize h are obtained. The theoretical results are applied to finite element approximations of elliptic PDE's in variational and in mixed form, and to first-order PDE's approximated using the Galerkin-Least Squares technique or by means of a non-standard Galerkin technique in L¹(Ω). Numerical simulations are presented to illustrate the theoretical results.