We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schoberl for elasticity problems to nonsymmetric problems Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity We present experimental evidence for both isoP2-P0 and isoP2-P1 finite elements in support of our conclusions We also show results of a comparison with two state-of-the-art preconditioners, showing the competitiveness of our approach

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An Augmented Lagrangian-Based Approach to the Oseen Problem

Iterative Methods for Linear Systems offers a mathematically rigorous introduction to fundamental iterative methods for systems of linear algebraic equations. The book distinguishes itself from other texts on the topic by providing a straightforward yet comprehensive analysis of the Krylov subspace methods, approaching the development and analysis of algorithms from various algorithmic and mathematical perspectives, and going beyond the standard description of iterative methods by connecting them in a natural way to the idea of preconditioning. Audience: The book supplements standard texts on numerical mathematics for first-year graduate and advanced undergraduate courses and is suitable for advanced graduate classes covering numerical linear algebra and Krylov subspace and multigrid iterative methods. It will be useful to researchers interested in numerical linear algebra and engineers who use iterative methods for solving large algebraic systems.

Iterative Methods for Linear Systems: Theory and Applications

We consider anisotropic second order elliptic boundary value problems in two dimensions, for which the anisotropy is exactly aligned with the coordinate axes. This includes cases where the operator features a singular perturbation in one coordinate direction, whereas its restriction to the other direction remains neatly elliptic. Most prominently, such a situation arises when polar coordinates are introduced.

Analysis of tensor product multigrid

The topic of this paper is motivated by the Navier--Stokes equations in rotation form. Linearization and application of an implicit time stepping scheme results in a linear stationary problem of Oseen type. In well-known solution techniques for this problem such as the Uzawa (or Schur complement) method, a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion-reaction type must be solved to update the velocity vector field. In this paper we analyze a standard finite element method for the discretization of this coupled system, and we introduce and analyze a multigrid solver for the discrete problem. Both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed. We prove stability results and discretization error bounds for the Galerkin finite element method. We present a convergence analysis of the multigrid method which shows the robustness of the solver. Results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver.

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Navier--Stokes Equations in Rotation Form: A Robust Multigrid Solver for the Velocity Problem

https://isc.tamu.edu/resources/preprints/2007/2007-01.pdf

Robust smoothers for high-order discontinuous Galerkin discretizations of advection-diffusion problems

Due to the vanishing ellipticity, multigrid methods for convection-dominated problems tend to loose their good convergence properties. With a suited discretisation, an appropriate ordering of the unknowns and, if necessary, a special block partitioning, the smoothing property of the Gaus-Seidel iteration and therefore good convergence can be regained. This paper deals with such an ordering strategy and block partitionings.

Probst Thomas

Papers

Downwind Gauß-Seidel-Smoothing for Convection-Dominated Problems