Conditioning of linear systems arising from penalty methods
William Layton, Shuxian Xu · ETNA - Electronic Transactions on Numerical Analysis · 2023
Penalizing incompressibility in the Stokes problem leads, under mildassumptions, to matrices with condition numbers $\kappa =\mathcal{O}(\varepsilon ^{-1}h^{-2})$, with $\varepsilon =$ penalty parameter $\ll1$ and$h= $ meshwidth $<1$. Although $\kappa =\mathcal{O}(\varepsilon^{-1}h^{-2}) $ is large, practical tests seldom report difficulty in solvingthese systems. In the SPD case, using the conjugate gradient method, thisis usually explained by spectral gaps occurring in the penalized coefficientmatrix. Herein we point out a second contributing factor. Since the solutionis approximately incompressible, solution components in the eigenspacesassociated with the penalty terms can be small. As a result, the effective condition number can be much smaller than the standard condition number.