Block triangular preconditioners for nonsymmetric saddle point problems: Field-of-values analysis
The problem considered is to construct a preconditioner \({\mathcal B}\) for the system \( {\mathcal A} x=f \) where \[ {\mathcal A}=\left(\begin{matrix} F&B^T\\ B&0 \end{matrix} \right) \quad \text{and}\quad {\mathcal B}=\left (\begin{matrix} F&B^T\\ 0&-S \end{matrix} \right) \] with \(F=\nu A+N\), \(A\) symmetric and \(N\) skew-symmetric, and \(S=BF^{-1}B^T\) is a Schur complement. Systems of this form arise for example in the solution of Navier-Stokes equations using finite elements. Since \({\mathcal B}^{-1}\) is needed, this requires the inverses of \(F\) and \(S\) which can only be computed approximately (inexact solves). This corresponds to using an approximate preconditioner \(\widehat{\mathcal B}\). Convergence results for GMRES both with the exact preconditioner \({\mathcal B}\) and with the approximate preconditioner \(\widehat{\mathcal B}\) are obtained. The upper bounds for the convergence rates are independent of \(h\) (the discretization parameter) and it depends only mildly on the viscosity \(\nu\).
scientific article; zbMATH DE number 1271905