Iterative techniques for time dependent Stokes problems

AbstractIn this paper, we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent Stokes equations using a mixed finite element spatial discretization. At each time step, a two by two block system corresponding to a perturbed Stokes problem must be solved. Although there are a number of techniques for iteratively solving this type of block system, to be effective, they require a good preconditioner for the resulting pressure operator (Schur complement). In contrast to the time independent Stokes equations where the pressure operator is well conditioned, the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced (and/or the Reynolds number is increased). In this paper, we shall describe and analyze preconditioners for the resulting pressure systems. These preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and Reynolds number parameter k

New computational approach for the linearized scalar potential formulation of the magnetostatic field problem

The linearized scalar potential formulation of the magnetostatic field problem is considered. The approach involves a reformulation of the continuous problem as a parametric boundary problem. By the introduction of a spherical interface and the use of spherical harmonics, the infinite boundary condition can also be satisfied in the parametric framework. The reformulated problem is discretized by finite element techniques and a discrete parametric problem is solved by conjugate gradient iteration. This approach decouples the problem in that only standard Neumann type elliptic finite element systems on separate bounded domains need be solved. The boundary conditions at infinity and the interface conditions are satisfied during the boundary parametric iteration

Iterative algorithm for the volume integral method for magnetostatics problems

Volume integral methods for solving nonlinear magnetostatics problems are considered in this paper. The integral method is discretized by a Galerkin technique. Estimates are given which show that the linearized problems are well conditioned and hence easily solved using iterative techniques. Comparisons of iterative algorithms with the elimination method of GFUN3D shows that the iterative method gives an order of magnitude improvement in computational time as well as memory requirements for large problems. Computational experiments for a test problem as well as a double layer dipole magnet are given. Error estimates for the linearized problem are also derived

Parallel multilevel preconditioners

In this paper, we shall report on some techniques for the development of preconditioners for the discrete systems which arise in the approximation of solutions to elliptic boundary value problems. Here we shall only state the resulting theorems. It has been demonstrated that preconditioned iteration techniques often lead to the most computationally effective algorithms for the solution of the large algebraic systems corresponding to boundary value problems in two and three dimensional Euclidean space. The use of preconditioned iteration will become even more important on computers with parallel architecture. This paper discusses an approach for developing completely parallel multilevel preconditioners. In order to illustrate the resulting algorithms, we shall describe the simplest application of the technique to a model elliptic problem

A multilevel preconditioner for domain decomposition boundary systems

In this note, we consider multilevel preconditioning of the reduced boundary systems which arise in non-overlapping domain decomposition methods. It will be shown that the resulting preconditioned systems have condition numbers which be bounded in the case of multilevel spaces on the whole domain and grow at most proportional to the number of levels in the case of multilevel boundary spaces without multilevel extensions into the interior