Fast Uzawa algorithm for generalized saddle point problems
Uzawa algorithms for solving generalized saddle point problems \[ \left( \begin{matrix} A & B^\top \\ B & -C \end{matrix} \right) \binom xy= \binom fg \] are considered. At first, convergence results for the inexact linear and nonlinear Uzawa algorithm for solving saddle point problems [see \textit{J. H. Bramble}, \textit{J. E. Pasciak}, and \textit{A. T. Vassilev}, SIAM J. Numer. Anal. 34, 1072-1092 (1997; Zbl 0873.65031)] are extended to the corresponding algorithms for generalized saddle point problems. Then, a new nonlinear Uzawa algorithm is presented in which a nonlinear approximation of the inverse of the Schur complement \((B A^{-1} B^\top + C)\) is used to accelerate the convergence. The convergence of this nonlinear Uzawa algorithm is analyzed. The Uzawa algorithms are applied to solve the incompressible steady state Stokes problem discretized by means of the mixed finite element method. The presented results of numerical experiments show that the new algorithm converges fast.
scientific article; zbMATH DE number 1961607