Inexact Uzawa Research Articles

Inexact Barzilai‐Borwein method for saddle point problems

AbstractThis paper considers the inexact Barzilai–Borwein (BB) algorithm applied to saddle point problems. To this aim, we study the convergence properties of the inexact BB algorithm for symmetric positive definite linear systems. Suppose that gk and g̃k are the exact residual and its approximation of the linear system at the kth iteration, respectively. We prove the R‐linear convergence of the algorithm if ∥g̃k – gk∥⩽η∥g̃k∥ for some small η>0 and all k. To adapt the algorithm for solving saddle point problems, we also extend the R‐linear convergence result to the case when the right‐hand term ∥g̃k∥ is replaced by ∥g̃k–1∥. Although our theoretical analyses cannot provide a good estimate to the parameter η, in practice, we find that η can be as large as the one in the inexact Uzawa algorithm. Further numerical experiments show that the inexact BB algorithm performs well for the tested saddle point problems. Copyright © 2007 John Wiley & Sons, Ltd.

A finite element based level set method for two-phase incompressible flows

We present a method that has been developed for the efficient numerical simulation of two-phase incompressible flows. For capturing the interface between the phases the level set technique is applied. The continuous model consists of the incompressible Navier–Stokes equations coupled with an advection equation for the level set function. The effect of surface tension is modeled by a localized force term at the interface (so-called continuum surface force approach). For spatial discretization of velocity, pressure and the level set function conforming finite elements on a hierarchy of nested tetrahedral grids are used. In the finite element setting we can apply a special technique to the localized force term, which is based on a partial integration rule for the Laplace–Beltrami operator. Due to this approach the second order derivatives coming from the curvature can be eliminated. For the time discretization we apply a variant of the fractional step θ-scheme. The discrete saddle point problems that occur in each time step are solved using an inexact Uzawa method combined with multigrid techniques. For reparametrization of the level set function a new variant of the fast marching method is introduced. A special feature of the solver is that it combines the level set method with finite element discretization, Laplace–Beltrami partial integration, multilevel local refinement and multigrid solution techniques. All these components of the solver are described. Results of numerical experiments are presented.

Nonlinear Inexact Uzawa Algorithms for Linear and Nonlinear Saddle-point Problems

This paper proposes some nonlinear Uzawa methods for solving linear and nonlinear saddle-point problems. A nonlinear inexact Uzawa algorithm is first introduced for linear saddle-point problems. Two different PCG techniques are allowed in the inner and outer iterations of the algorithm. This algorithm is then extended for a class of nonlinear saddle-point problems arising from some convex optimization problems with linear constraints. For this extension, some PCG method used in the inner iteration needs to be carefully constructed so that it converges in a certain energy norm instead of the usual $l^2$-norm. It is shown that the new algorithm converges under some practical conditions and there is no need for any a priori estimates on the minimal and maximal eigenvalues of the two local preconditioned systems involved. The two new methods perform more efficiently than the existing methods in the cases where no good preconditioners are available for the Schur complements.

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.

Grad-div stablilization for Stokes equations

In this paper a stabilizing augmented Lagrangian technique for the Stokes equations is studied. The method is consistent and hence does not change the continuous solution. We show that this stabilization improves the well-posedness of the continuous problem for small values of the viscosity coefficient. We analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact Uzawa method.

Fast Uzawa algorithms for solving non‐symmetric stabilized saddle point problems

AbstractBramble et al. (Math. Comput. (2000); 69 (230) : 667–689) discussed the Uzawa type algorithms for iteratively solving nonsymmetric stable saddle point problems. In this paper, we consider the Uzawa type algorithms for iteratively solving generalized saddle point problems. Our main concern is how to accelerate the convergence of the inexact Uzawa type algorithms. The contributions of the paper are that a new non‐linear Uzawa type algorithm is presented and its convergence is analysed. Applications to Navier–Stokes equations by mixed finite element discretization with an unstable pair of approximation spaces, i.e. Q1‐P0 pair of approximation spaces, are discussed and the results of the numerical experiments of applying our new algorithm are presented. Copyright © 2003 John Wiley & Sons, Ltd.

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.

On Schwarz-type Smoothers for Saddle Point Problems

In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.

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.

An Iterative Method with Variable Relaxation Parameters for Saddle-Point Problems

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-type algorithms: it is always convergent without any a priori estimates on the spectrum of the preconditioned Schur complement matrix, which may not be easy to achieve in applications. The rate of the convergence of the inexact Uzawa method is analyzed. Numerical results of the algorithm applied for the Stokes problem and a purely linear system of algebraic equations are presented.

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.