Diagonal Scaling of Discrete Differential Forms

Mark Ainsworth
Mathematics Department, Strathclyde University, Glasgow.

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.