Although adaptive finite element methods (FEMs) are recognized as powerful techniques for solving mixed variational problems of fluid mechanics, usually they are not even proven to converge. Only recently, in [SINUM, 40 (2002), pp.1207-1229] Bänsch, Morin and Nochetto introduced an adaptive Uzawa FEM for solving the Stokes problem, and showed its convergence. In their paper, numerical experimen...

The Uzawa equivalence theorem [H. Uzawa, Walras’s Existence Theorem and Brouwer’s Fixed Point Theorem, Economic Studies Quarterly 8 (1962) 59–62] showed (classically) that the existence of Walrasian equilibrium in an economy with continuous excess demand functions is equivalent to Brouwer’s fixed point theorem, that is, the existence of a fixed point for any continuous function from an n-dimens...

The parameterized Uzawa preconditioners for saddle point problems are studied in this paper. The eigenvalues of the preconditioned matrix are located in (0, 2) by choosing the suitable parameters. Furthermore, we give two strategies to optimize the rate of convergence by finding the suitable values of parameters. Numerical computations show that the parameterized Uzawa preconditioners can lead ...

In this paper, we introduce and analyze Uzawa algorithms for non-symmetric saddle point systems. Convergence for the algorithms is established based on new spectral results about Schur complements. A new Uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework. Numerical results supporting the efficiency of the algorithms ar...

We compare three different time discretization schemes in combination with an augmented Lagrangian method to simulate the motion of an inextensible beam. The resulting saddle-point problem is solved with an Uzawa-DouglasRachford algorithm. The three schemes are tested on a benchmark with an analytical solution and on a more challenging application. We found that in order to obtain optimal conve...

In this paper, we propose an inexact Uzawa method with variable relaxation parameters for iteratively solving linear saddle-point problems. The method involves two variable relaxation parameters, which can be updated easily in each iteration, similar to the evaluation of the two iteration parameters in the conjugate gradient method. This new algorithm has an advantage over most existing Uzawa-t...

This paper focuses on the convergence of the modified nonlinear inexact Uzawa algorithm (MNIU) for solving the saddle point problem. We improve the sufficient conditions for convergence and the convergence rate shown in Bramble et al. [J. Bramble, J. Pasciak, and A. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal., 34 (1997), pp. 1072–1092] and Y...

The Gauge-Uzawa FEM is a new first order fully discrete projection method which combines advantages of both the Gauge and Uzawa methods within a variational framework. A time step consists of a sequence of d+1 Poisson problems, d being the space dimension, thereby avoiding both the incompressibility constraint as well as dealing with boundary tangential derivatives as in the Gauge Method. This ...

In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first give an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzaw...