Andrew J. Wathen

Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics

The intended readership includes graduate students and researchers in engineering, numerical analysis, applied mathematics and interdisciplinary scientific computing. The publisher describes the book as follows: * An excellent introduction to finite elements, iterative linear solvers and scientific computing * Contains theoretical problems and practical exercises * All methods and examples use freely available software * Focuses on theory and computation, not theory for computation * Describes approximation methods and numerical linear algebra

A Note on Preconditioning for Indefinite Linear Systems

Preconditioners are often conceived as approximate inverses. For nonsingular indefinite matrices of saddle-point (or KKT) form, we show how preconditioners incorporating an exact Schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues. Thus approximations of the Schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses.

Constraint Preconditioning for Indefinite Linear Systems

The problem of finding good preconditioners for the numerical solution of indefinite linear systems is considered. Special emphasis is put on preconditioners that have a 2 × 2 block structure and that incorporate the (1,2) and (2,1) blocks of the original matrix. Results concerning the spectrum and form of the eigenvectors of the preconditioned matrix and its minimum polynomial are given. The consequences of these results are considered for a variety of Krylov subspace methods. Numerical experiments validate these conclusions.

Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. Part I of this work described a conjugate gradient-like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems [A. J. Wathen and D. J. Silvester, SIAM J. Numer. Anal., 30 (1993), pp. 630–649]. Using simple arguments, estimates for the eigenvalue distribution of the discrete Stokes operator on which the convergence rate of the iteration depends are easily derived. Part I discussed the important case of diagonal preconditioning (scaling). This paper considers the more general class of block preconditioners, where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables. It is shown that, provid...

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.

Fast iterative solution of stabilised Stokes systems, part I: using simple diagonal preconditioners

Mixed finite element approximation of the classical Stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables. Iterative solution of such indefinite systems is feasible and is an attractive approach for large problems. The use of stabilisation methods for convenient (but unstable) mixed elements introduces stabilisation parameters. We show how these can be chosen to obtain rapid iterative convergence. We propose a conjugate gradient-like method (the method of preconditioned conjugate residuals) which is applicable to symmetric indefinite problems, describe the effects of stabilisation on the algebraic structure of the discrete Stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends. Here we discuss the simple case of diagonal preconditioning. Our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable. We demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete Laplacian are approachable for the Stokes problem.

A Preconditioner for the Steady-State Navier--Stokes Equations

We present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady-state Navier--Stokes equations (also known as the Oseen equations). The solver is an iterative method of Krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the Oseen operator. The preconditioner may also be conceived through a weaker heuristic argument involving differential operators. Computations indicate that convergence for the preconditioned discrete Oseen problem is only mildly dependent on the viscosity (inverse Reynolds number) and, most importantly, that the number of iterations does not grow as the mesh size is reduced. Indeed, since the preconditioner is motivated through analysis of continuous operators, the number of iterations decreases for smaller mesh size which accords with better approximation of these operators.

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case.

Efficient preconditioning of the linearized Navier—Stokes equations for incompressible flow

We outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of Navier–Stokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid V-cycle for the scalar convection–diffusion operator, and a multigrid V-cycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid, robust with respect to the time-step, and only deteriorates very slowly as the Reynolds number is increased.

Optimal Solvers for PDE-Constrained Optimization

Optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering, particularly in problems of design. The solution of such PDE-constrained optimization problems is usually a major computational task. Here we consider simple problems of this type: distributed control problems in which the 2- and 3-dimensional Poisson problem is the PDE. The large-dimensional linear systems which result from discretization and which need to be solved are of saddle-point type. We introduce two optimal preconditioners for these systems, which lead to convergence of symmetric Krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem. These preconditioners are block structured and involve standard multigrid cycles. The optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases. The theoretical proof indicates that these approaches may have much broader applicability for other PDEs.