Joseph E. Pasciak

The construction of preconditioners for elliptic problems by substructuring. I

In earlier parts of this series of papers, we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems. The resulting algorithms are well suited for implementation on computers with parallel architecture. In this paper, we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners. The iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns. These preconditioners involve an incomplete Chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work. Theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods.

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 preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems

This paper provides a preconditioned iterative technique for the solution of saddle point problems. These problems typically arise in the numerical approximation of partial differential equations by Lagrange multiplier techniques and/or mixed methods. The saddle point problem is reformulated as a symmetric positive definite system, which is then solved by conjugate gradient iteration. Applications to the equations of elasticity and Stokes are discussed and the results of numerical experiments are given.

Convergence estimates for product iterative methods with applications to domain decomposition

In this paper, we consider iterative methods for the solution of symmetric positive definite problems on a space % which are defined in terms of products of operators defined with respect to a number of subspaces. The simplest algorithm of this sort has an error-reducing operator which is the product of orthogonal projections onto the complement of the subspaces. New normreduction estimates for these iterative techniques will be presented in an abstract setting. Applications are given for overlapping Schwarz algorithms with many subregions for finite element approximation of second-order elliptic problems.

A least-squares approach based on a discrete minus one inner product for first order systems

The purpose of this paper is to develop and analyze a least-squares approximation to a first order system. The first order system represents a reformulation of a second order elliptic boundary value problem which may be indefinite and/or nonsymmetric. The approach taken here is novel in that the least-squares functional employed involves a discrete inner product which is related to the inner product in H -1 (Ω) (the Sobolev space of order minus one on Ω). The use of this inner product results in a method of approximation which is optimal with respect to the required regularity as well as the order of approximation even when applied to problems with low regularity solutions. In addition, the discrete system of equations which needs to be solved in order to compute the resulting approximation is easily preconditioned, thus providing an efficient method for solving the algebraic equations. The preconditioner for this discrete system only requires the construction of preconditioners for standard second order problems, a task which is well understood.

An iterative method for elliptic problems on regions partitioned into substructures

Some new preconditioners for discretizations of elliptic boundary problems are studied. With these preconditioners, the domain under consideration is broken into subdomains and preconditioners are defined which only require the solution of matrix problems on the subdomains. Analytic estimates are given which guarantee that under appropriate hypotheses, the preconditioned iterative procedure converges to the solution of the discrete equations with a rate per iteration that is independent of the number of unknowns. Numerical examples are presented which illustrate the theoretically predicted iterative convergence rates.

New estimates for multilevel algorithms including the V-cycle

The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm discussed recently together with J. Xu and the standard V-cycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels, even on nonconvex domains which do not provide full elliptic regularity. For example, the theory applies to the standard multigrid Vcycle on the L-shaped domain, or a domain with a crack, and yields a uniform convergence rate. We also prove uniform convergence rates for the multigrid V-cycle for problems with nonuniformly refined meshes. Finally, we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid V-cycle algorithms.

A preconditioning technique for the efficient solution of problems with local grid refinement

Abstract We develop a new preconditioning method for elliptic problems which allows for dynamic local grid refinement. The majority of the computation in the implementation of our method involves solution procedures on mesh domains with regular geometry. Accordingly, the resulting algorithms can be effectively vectorized. It seems feasible to incorporate these ideas into existing large-scale simulators without a complete redesign of the simulator.

On the stability of the L 2 projection in H 1 (Ω)

We prove the stability in H1(Ω) of the L2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the L2 projection in H1(Ω) holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.

Iterative techniques for time dependent Stokes problems

Abstract In 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 .