Thu, 11 May 2006

14:00 - 15:00
Comlab

Diagonal scaling of discrete differential forms

Prof Mark Ainsworth
(University of Strathclyde)
Abstract

The use of discrete differential forms in the construction of finite element discretisations of the Sobolev spaces H^s, H(div) and H(curl) is now routinely applied by numerical analysts and engineers alike. However, little attention has been paid to the conditioning of the resulting stiffness matrices, particularly in the case of the non-uniform meshes that arise when adaptive refinement algorithms are used. We study this issue and show that the matrices are generally rather poorly conditioned. Typically, diagonal scaling is applied (often unwittingly) as a preconditioner. However, whereas diagonal scaling removes the effect of the mesh non-uniformity in the case of Sobolev spaces H^s, we show this is not so in the case of the spaces H(curl) and H(div). We trace the reason behind this difference, and give a simple remedy for curing the problem.