A new and detailed analysis of the basic Uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady Stokes operator is presented. The paper focuses on the following new aspects: explicit construction of the Uzawa pressure-operator spectrum for a semiperiodic model problem; general relationship of the convergence rate of the Uzawa procedure to classical inf-sup di...

Abstract The paper is devoted to Krylov type modifications of the Uzawa method on operator level for Stokes problem in order accelerate convergence. First block preconditioners and their effect convergence are studied. Then it shown that a Krylov–Uzawa iteration produces superlinear smooth domains, estimation given its speed.

This paper concerns an adaptive finite element method for the elliptic obstacle problem.We consider the formulation of the problem as an elliptic variational inequation.The adaptive algorithm (modified Uzawa adaptive method)we construct is based on a combination of the Uzawa method associated with the corresponding multivalued operator and a convergent adaptive mehtod for the linear problem. As...

In this paper we use the eigenfunctions of the Laplacian to approximate the solution of some nonlinear equations which are used to model natural phenomena (as the Navier-Stokes flow equations, for instance). In this respect we propose a numerical algorithm, combining the Uzawa and Arrow-Hurwitz algorithms. The algorithm proposed here shares features from both algorithms, and it has the followin...

In this paper, we introduce a general multilevel gradient Uzawa algorithm for symmetric saddle point systems. We compare its performance with the performance of the standard Uzawa multilevel algorithm. The main idea of the approach is to combine a double inexact Uzawa algorithm at the continuous level with a gradient type algorithm at the discrete level. The algorithm is based on the existence ...

This paper studies convergence analysis of a preconditioned inexact Uzawa method for nondi erentiable saddle-point problems. The SOR-Newton method and the SOR-BFGS method are special cases of this method. We relax the Bramble– Pasciak–Vassilev condition on preconditioners for convergence of the inexact Uzawa method for linear saddle-point problems. The relaxed condition is used to determine the...

Several of the economically interesting aspects of the Uzawa-Lucas model are not really emphasized in the exposition in B & S, Section 5.2.2, which instead focusses on quite technical aspects such as the transitional dynamics of (a simplified version of) the model. Hence, in this lecture note I shall attempt to give an account of the Uzawa-Lucas model which closer to the presentation in Lucas (...

We present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove con...

The Stokes problem plays an important role in computational fluid dynamics since it is encountered in the time discretization of (incompressible) Navier-Stokes equations by operator-splitting methods [2, 3]. Space discretization of the Stokes problem leads to large scale ill-conditioned systems. The Uzawa (preconditioned) conjugate gradient method is an efficient method for solving the Stokes p...

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 the light of the new spectral results ...