Schur complements on Hilbert spaces and saddle point systems

Abstract

MSC:Keywords: Inexact Uzawa algorithms Saddle point system Multilevel methods Adaptive methods a b s t r a c t For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya-Babušca-Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca-Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions.

Uzawa methods for a class of block three‐by‐three saddle‐point problems

“…The Uzawa method is an effective method for solving the saddle‐point problem , which has the following scheme: where τ is a given parameter and Q is a symmetric positive definite matrix. Recently, there is a rapidly increasing literature, which is concerned with the inexact Uzawa methods because of the minimal memory requirements and easiness to be implemented; see other works for example …”

“…Recently, there is a rapidly increasing literature, which is concerned with the inexact Uzawa methods because of the minimal memory requirements and easiness to be implemented; see other works for example. [11][12][13][14][15][16][17][18]38,[40][41][42] In this section, we will employ the iteration scheme (6) to solve the block three-by-three saddle-point problem (1) by repartitioning its coefficient matrix  into the form of (2). By the notation…”

“…Therefore, it is necessary to study the numerical methods for the system (1) based on its specific structure. In this work, we extend the Uzawa method studied in other works [11][12][13]15,16,37,38 to solve the saddle-point problem (1). The direct extension of the Uzawa method requires the exact solution of a symmetric indefinite system of linear equations at each step.…”

“…In this work, we extend the Uzawa method studied in other works to solve the saddle‐point problem . The direct extension of the Uzawa method requires the exact solution of a symmetric indefinite system of linear equations at each step.…”

The corrected Uzawa method for solving saddle point problems

“…Because the matrices A and B are usually large and sparse, iterative methods become more attractive than direct methods for solving the saddle point problem , although the direct methods play an important role in the form of preconditioners embedded in an iterative framework. In the case that the coefficient matrix of Equation is nonsingular, many efficient iterative methods as well as their numerical properties have been studied in the literature, for example, Uzawa‐type methods , matrix splitting iterative methods , relaxation iterative methods , iterative null space methods , and the references therein. Golub, Wu, and Yuan proposed a successive overrelaxation (SOR)‐like method for solving nonsingular saddle point problems, studied the convergence rates, behavior of the spectral radius, and optimal parameters, and made some comparisons.…”