Bounds on Spectral Condition Numbers of Matrices Arising in the P-version of the Nite Element Method

We estimate condition numbers of p-version matrices for tensor product elements with two choices of reference element degrees of freedom. In one case (Lagrange elements) the condition numbers grow exponentially in p, whereas in the other (hierarchical basis functions based on Tchebychee poly-nomials) the condition numbers grow rapidly but only algebraically in p. We conjecture that regardless of the choice of basis the condition numbers grow like p 4d or faster, where d is the dimension of the spatial domain. Over the last several years, a series of papers has appeared concerning the p-version of the nite element method The label p denotes the degree of the piecewise{polynomial functions used in constructing the FEM trial space. In the p-version, improved approximate solutions are sought by increasing p while the triangulation of the domain, and thus the maximum triangle diameter h, is held xed. In the h-version, improved approximations are obtained by reening the triangulation of the domain, and thus reducing h, while p is held xed. Beyond this, there is no diierence in principle between these two versions of the FEM; a practitioner who solved a problem once for a particular p and h could not say whether he was using the p-version or the h-version. For the h-version as applied to second order elliptic problems, it is not diicult to show that the spectral condition number K(h) of the nite element matrix operator is O(h ?2), where h is the mesh parameter 21]. This is true for a great variety of elements and is independent of the dimension of the spatial domain of the problem. For the p-version, the spectral condition number K(p) follows no such simple rule. The condition number is heavily dependent on the choice of degrees of Numerische Mathematik Electronic Edition { page numbers may diier from the printed version page 333 of Numer.