Conditioning of linear systems arising from penalty methods

Penalizing incompressibility in the Stokes problem leads, under mildassumptions, to matrices with condition numbers $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2})$, $\varepsilon =$ penalty parameter $\ll1$ and$h= $ meshwidth $<1$. Although =\mathcal{O}(\varepsilon^{-1}h^{-2}) is large, practical tests seldom report difficulty solvingthese systems. In SPD case, using conjugate gradient method, thisis usually explained by spectral gaps occurring penalized coefficientmatrix. Herein we point out a second contributing factor. Since solutionis approximately incompressible, solution components eigenspacesassociated terms can be small. As result, effective number much smaller than standard number.