Efficient Domain Decomposition Preconditioning for the p-version Finite Element
                          Method - The mass matrix.

                              Jonathan M. Smith

                             University of Durham
                     Department of Mathematical Sciences
                                Durham DH1 3LE
                                   England

                            j.m.smith@durham.ac.uk


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.