Analysis of preconditioned Picard iterations for the Navier-Stokes equations

Mixed finite element formulations of fluid flow problems lead to large systems of equations of saddle-point type for which iterative solution methods are mandatory for reasons of efficiency. A successful approach in the design of solution methods takes into account the structure of the problem; in particular, it is well-known that an efficient solution can be obtained if the associated Schur complement problem can be solved efficiently and robustly. In this work we analyze a preconditioner for the Schur complement for the Oseen problem which was introduced in Numerical Analysis Report No. 99/06 We show that the spectrum of the preconditioned system is independent of the mesh parameter; moreover, we demonstrate that the number of GMRES iterations grows like the square-root of the Reynolds number for steady-state problems, while for time-dependent problems this dependence becomes negligible. In both the steady-state and time-dependent case the performance is mesh-independent

Constraint interface preconditioning for topology optimization problems

The discretization of constrained nonlinear optimization problems arising in the field of topology optimization yields algebraic systems which are challenging to solve in practice, due to pathological ill-conditioning, strong nonlinearity and size. In this work we propose a methodology which brings together existing fast algorithms, namely, interior-point for the optimization problem and a novel substructuring domain decomposition method for the ensuing large-scale linear systems. The main contribution is the choice of interface preconditioner which allows for the acceleration of the domain decomposition method, leading to performance independent of problem size.Comment: To be published in SIAM J. Sci. Com

A Green's function preconditioner for the steady-state Navier-Stokes equations

In this paper we present an efficient method for solving the sparse linear system of equations arising from the discretization of the linearised 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 based on Green's tensor for the Oseen operator. The preconditioner supersedes existing preconditioners for the Oseen problem in that it exhibits only a mild dependence on the viscosity (inverse Reynolds number) and, most importantly, improved performance with the size of the problem. This comes as no surprise, as preconditioners based on the continuous inverse are expected to perform better on discretizations which approximate well the continuous operator

Preconditioning the Advection-Diffusion Equation: the Green's Function Approach

We look at the relationship between efficient preconditioners (i.e., good approximations to the discrete inverse operator) and the generalized inverse for the (continuous) advection-diffusion operator -- the Green's function. We find that the continuous Green's function exhibits two important properties -- directionality and rapid downwind decay -- which are preserved by the discrete (grid) Green's functions, if and only if the discretization used produces non-oscillatory solutions. In particular, the downwind decay ensures the locality of the grid Green's functions. Hence, a finite element formulation which produces a good solution will typically use a coefficient matrix with almost lower triangular structure under a "with-the-flow" numbering of the variables. It follows that the block Gauss-Seidel matrix is a first candidate for a preconditioner to use with an iterative solver of Krylov subspace type

Truncation preconditioners for stochastic Galerkin finite element discretizations

Stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and non-affine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones.Comment: 27 pages, 6 table

A preconditioner for the 3D Oseen equations

We describe a preconditioner for the linearised incompressible Navier-Stokes equations (the Oseen equations) which requires as components only a preconditioner/solver for each of a discrete Laplacian and a discrete advection-diffusion operator. With this preconditioner, convergence of an iterative method such as GMRES is independent of the mesh size and depends only mildly on the viscosity parameter (the inverse Reynolds number). Thus when the component preconditioner/solvers are effective on their respective subproblems (as one expects with an appropriate multigrid cycle for instance) a fast Oseen solver results

A note on constraint preconditioning

We discuss and extend the results derived in Keller, Gould and Wathen [SIMAX, 21(4), 2000] for constraint preconditioning. In particular, we improve the existing results as well as generalise them to the case where the (1,1) block of the preconditioner has a non-trivial kernel. We also analyse the form of the preconditioner with negated constraints, which ensures that the preconditioned system is diagonalisable, while preserving the non-unit eigenvalues and negating some unit eigenvalues