p-version Finite Element Method - The mass matrix.

University of Durham

Department of Mathematical Sciences

Durham DH1 3LE

England

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.

Contributed May 15, 1992.