Inexact Uzawa Algorithm Research Articles

A Unified Approach for Uzawa Algorithms

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 convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite‐ or infinite‐dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition.

Large‐scale simulations of acoustic‐structure interaction using the fast multipole BEM

AbstractFor the simulation of acoustic‐structure interaction problems, the coupled field equations must be solved. The structure is commonly discretized using finite elements, whereas for the acoustic field the boundary element method (BEM) is favorable. A mortar BEM‐FEM coupling algorithm is developed that allows the combination of non‐conforming meshes. The high flexibility for the choice of discretizations offers a high efficiency, since specialized shape functions and adaptive mesh refinement can be used in the subdomains. The mortar coupling algorithm yields a saddle point problem that is solved using a preconditioned inexact Uzawa algorithm. The iterative solver enables the use of the fast multipole BEM and thus coupled simulations on large boundary element models.

A new nonlinear Uzawa algorithm for generalized saddle point problems

In [J.H. Bramble, I.E. Pasciak, A.T. Vassilev, Analysis of the inexact Uzawa algorithm for saddle point problems, SIAM J. Numer. Anal. 34 (1997) 1072–1092], a nonlinear Uzawa algorithm for solving generalized saddle point problems iteratively was considered. In [Z. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003) 157–171], Cao gave another nonlinear Uzawa algorithm in order to accelerate convergence. In this paper, a new nonlinear Uzawa method, which is defined by two nonlinear approximate inverses, is proposed and its convergence result is deduced. At the same time, we analyze convergence results of three nonlinear Uzawa methods. The results of numerical experiments are presented when we apply them to Stokes equations discretized by mixed finite element.

Convergence of iterative methods for stabilized saddle-point problems

The convergence of two classes of iterative methods for stabilized saddle-point problems (inexact Uzawa algorithms and a class of methods with symmetric preconditioners) are studied using Zulerhner’s unified approach.

Fast Uzawa algorithm for generalized saddle point problems

In SIAM J. Numer. Anal. 34 (1997) 1072, the authors considered the so-called “inexact Uzawa” algorithm for iteratively solving saddle point problems. In this paper, we consider the same algorithm for iteratively solving generalized saddle point problems. Our main concern is how to accelerate the convergence of the inexact Uzawa algorithm. The new contributions of the paper are that a new nonlinear Uzawa algorithm is derived and its convergence is analyzed. Applications to Stokes equations by mixed finite element discretization with an unstable pair of approximation spaces, i.e., Q 1− P 0 pair of approximation spaces, are discussed and the results of the numerical experiments are presented.

Inexact Uzawa algorithms for variational inequalities of the second kind

In this paper we discuss inexact Uzawa algorithms and inexact non-linear Uzawa algorithms to solve discretized variational inequalities of the second kind. We prove convergence results for the algorithms. Numerical examples are included to show the effectiveness of the algorithms.

Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems

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 Uzawa method with the classical steepest descent method and conjugate gradient method respectively. The two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved, including the preconditioned Schur complement. Numerical results of the algorithms applied for the Stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms.

Krylov-Based Uzawa Algorithms for the Solution of the Stokes Equations Using Discontinuous-Pressure Tetrahedral Finite Elements

The objective of this work is to develop an efficient and robust Krylov-based Uzawa algorithm for the solution of the 3D steady-state and unsteady-state Stokes equations with discontinuous-pressure tetrahedral finite elements. To this end, a class of preconditioned conjugate gradient Uzawa algorithms are presented and compared to those of the literature for other finite element types. A comparison is also made with an inexact Uzawa algorithm for the steady-state case. An analysis of the results obtained for various problems leads to an iterative scheme that is well adapted to the solution of Stokes problems of all sizes and complexities such as those found in industrial applications.

A sufficient condition for the convergence of the inexact Uzawa algorithm for saddle point problems

In this paper, the convergence property of the inexact Uzawa algorithm for solving symmetric indefinite linear systems is studied. A simple sufficient condition for the convergence of the inexact Uzawa algorithm is obtained. Two examples and numerical experiments illustrating the conclusion are provided.

Analysis of iterative methods for saddle point problems: a unified approach

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

The inexact Uzawa algorithm for saddle point problem

In this paper, a bound of rate for a slight modification of the inexact Uzawa algorithm for solving saddle point problems presented in a recent paper of Bramble, Passciak and Vassilev [1] is obtained.

On the Nonlinear Inexact Uzawa Algorithm for Saddle-Point Problems

This paper discusses convergence behavior of the nonlinear inexact Uzawa algorithm for solving saddle-point problems presented in a recent paper of Bramble, Pasciak, and Vassilev ( SIAM J. Numer. Anal., 34 (1997), pp. 1072--1092). It is shown that this algorithm converges under a condition weaker than that stated in their paper.

Direct Simulations of 2D Fluid-Particle Flows in Biperiodic Domains

We propose a method to simulate the motion of 2D rigid particles in a viscous, incompressible fluid. Within the arbitrary Lagrangian Eulerian framework, momentum equations for both the fluid and the particles are discretized, and a coupled variational formulation is established. By introducing an appropriate finite element approximation, a symmetric linear system is obtained. This system is solved by an inexact Uzawa algorithm. The main interest of such simulations lies in the average behaviour of a high number of particles. We therefore introduced a biperiodic formulation of the problem, which makes it possible to represent many-body mixtures at a reasonable computational cost. In order to model realistic situations, an extra term must be added to the pressure. This extra term is shown to be the lagrange multiplier associated with the vertical volume conservation constraint. We developed an appropriate unstructured mesh generator, so that the biperiodicity of the fields can be treated in a strong sense. The question of particle contact is addressed, and a simple technique to overcome numerical problems is proposed. The influence of periodic lengths is investigated through different simulations. The same case is simulated for different sizes of the window, and the behaviour of some space-averaged quantities is studied.

Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems

In this paper, we consider the so-called "inexact Uzawa" algorithm for iteratively solving linear block saddle point problems. Such saddle point problems arise, for example, in finite element and finite difference discretizations of Stokes equations, the equations of elasticity, and mixed finite element discretization of second-order problems. We consider both the linear and nonlinear variants of the inexact Uzawa iteration. We show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled. Bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned Uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left-hand block. In the case of nonlinear iteration, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left-hand block is of sufficient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear Uzawa algorithm. Applications to the Stokes equations and mixed finite element discretization of second-order elliptic problems are discussed and, finally, the results of numerical experiments involving the algorithms are presented.

Fluid-particle flow: a symmetric formulation

In this Note, we present a numerical method to simulate the motion of solid particles in a moving viscous fluid. The fluid is supposed to be Newtonian and incompressible. The Arbitrary Lagrangian Eulerian formulation of the Navier-Stokes equations is discretized at the first order in time, as are the equations for the solid bodies. The advection term is taken into account by a method of characteristics. The variational formulation of the coupled problem is then established, and the boundary integrals expressing the hydrodynamical forces are eliminated. By introduction of an appropriate Finite Element approximation, a symmetric linear system is obtained. This system is solved by an inexact Uzawa algorithm, preconditionned by a Laplace operator with Neumann boundary conditions on the pressure. Numerical results are presented, for 2 and 100 particles: The Reynolds number in both cases is of the order of 100.

Inexact and Preconditioned Uzawa Algorithms for Saddle Point Problems

Variants of the Uzawa algorithm for solving symmetric indefinite linear systems are developed and analyzed. Each step of this algorithm requires the solution of a symmetric positive-definite system of linear equations. It is shown that if this computation is replaced by an approximate solution produced by an arbitrary iterative method, then with relatively modest requirements on the accuracy of the approximate solution, the resulting inexact Uzawa algorithm is convergent, with a convergence rate close to that of the exact algorithm. In addition, it is shown that preconditioning can be used to improve performance. The analysis is illustrated and supplemented using several examples derived from mixed finite element discretization of the Stokes equations.