Fast Uzawa algorithm for generalized saddle point problems. (English) Zbl 1032.65029

Cao, Zhi-Hao

Appl. Numer. Math. 46, No. 2, 157-171 (2003).

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 J. H. Bramble, J. E. Pasciak, and 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.

Reviewer: Michael Jung (Dresden)

MSC:

65F10	Iterative numerical methods for linear systems
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
35Q30	Navier-Stokes equations
65F35	Numerical computation of matrix norms, conditioning, scaling
76D07	Stokes and related (Oseen, etc.) flows
76M10	Finite element methods applied to problems in fluid mechanics
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs

Keywords:

iterative methods; indefinite systems of linear equations; generalized saddle point problem; Uzawa algorithm; preconditioning; Stokes problem; mixed finite element method; convergence acceleration; numerical experiments

Citations:

Zbl 0873.65031

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