Conditioning of linear systems arising from penalty methods

TL;DR

This paper investigates the conditioning of matrices from penalty methods in the Stokes problem, revealing that the effective condition number can be smaller than the theoretical estimate due to spectral gaps and small solution components.

Contribution

It identifies a second factor affecting the conditioning of penalized systems, beyond the known spectral gaps, related to small solution components in eigenspaces.

Findings

Effective condition number can be much smaller than the standard estimate.

Spectral gaps contribute to easier solution of penalized systems.

Small solution components in eigenspaces reduce the impact of large condition numbers.

Abstract

Penalizing incompressibility in the Stokes problem leads, under mild assumptions, to matrices with condition numbers $\kappa =\mathcal{O} (\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $<<1$, and $ h= $ mesh width $<1$. Although $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solving these systems. In the SPD case, using the conjugate gradient method, this is usually explained by spectral gaps occurring in the penalized coefficient matrix. Herein we point out a second contributing factor. Since the solution is approximately incompressible, solution components in the eigenspaces associated with the penalty terms can be small. As a result, the effective condition number can be much smaller than the standard condition number.