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SUMMARYWe consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier-Stokes equations. These systems are of the so-called saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For medium-sized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software.

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SUMMARYIn this paper, we discuss various techniques for solving the system of linear equations that arise from the discretization of the incompressible Stokes equations by the finite-element method. The proposed solution methods, based on a suitable approximation of the Schur-complement matrix, are shown to be very effective for a variety of problems. In this paper, we discuss three types of iterative methods. Two of these approaches use the pressure mass matrix as preconditioner (or an approximation) to the Schur complement, whereas the third uses an approximation based on the ideas of least-squares commutators (LSC). We observe that the approximation based on the pressure mass matrix gives h-independent convergence, for both constant and variable viscosity.

[1] The development of scalable robust solvers for unstructured finite element applications related to viscous flow problems in earth sciences is an active research area. Solving high-resolution convection problems with order of magnitude 10 8 degrees of freedom requires solvers that scale well, with respect to both the number of degrees of freedom as well as having optimal parallel scaling characteristics on computer clusters. We investigate the use of a smoothed aggregation (SA) algebraic multigrid (AMG)-type solution strategy to construct efficient preconditioners for the Stokes equation. We integrate AMG in our solver scheme as a preconditioner to the conjugate gradient method (CG) used during the construction of a block triangular preconditioner (BTR) to the Stokes equation, accelerating the convergence rate of the generalized conjugate residual method (GCR). We abbreviate this procedure as BTA-GCR. For our experiments, we use unstructured grids with quadratic finite elements, making the model flexible with respect to geometry and topology and O(h 3 ) accurate. We find that AMG-type methods scale linearly (O(n)), with respect to the number of degrees of freedom, n. Although not all parts of AMG have preferred parallel scaling characteristics, we show that it is possible to tune AMG, resulting in parallel scaling characteristics that we consider optimal, for our experiments with up to 100 million degrees of freedom. Furthermore, AMG-type methods are shown to be robust methods, allowing us to solve very illconditioned systems resulting from strongly varying material properties over short distances in the model interior.