In recent years we have observed both the increased popularity of the p-version finite element method and domain decomposition preconditioning techniques. Current work has given us theoretical and numerical results, showing that we can reduce the condition number of the stiffness matrix from polynomial to polynomial logarithmic in the number of degrees of freedom.
However in the p-version, we have an additional problem. We observe that heirachical bases, which while being very natural bases for the p-version finite element method, are very unnatural bases for the mass matrix. This can be seen by noting the growth in the condition number is exponential in p, the degree of the polynomial on each element.
Using methods based on those for the stiffness matrix, derived by Babuska, Craig, Mandel and Pitkaranta in 1988-89, it is possible to control this ill-conditioning; resulting in a bound on the relative condition independent of p, for relatively little work. In this paper we shall present present empirical results for both triangular and quadrilateral quasi-uniform meshes, and analytical bounds for quadrilateral meshes.