Alfred H. Schatz

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.

Higher order local accuracy by averaging in the finite element method

This chapter describes the class of finite element subspaces and explains the main result on the accuracy of K h * u h ,where K h is a fixed function, u h represents local averages, and * denotes convolution. The function K h has the following properties: (1) K h has small support; (2) K h is independent of the specific choice of S h or the operator L; (3) K h * u h is easily computable from u h ; and (4) K h * u h approximates u to higher order than does u h . The chapter also discusses on notation, subspaces and the construction of K h .

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.

Some Convergence Estimates for Semidiscrete Galerkin Type Approximations for Parabolic Equations

In this paper we derive convergence estimates for certain semidiscrete methods used in the approximation of solutions of initial boundary value problems with homogeneous Dirichlet boundary conditions for parabolic equations. These methods contain the ordinary Galerkin method based on approximating subspaces with functions vanishing on the boundary of the basic domain, and also some methods without such restrictions. The results include

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

L_2

Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: part I. global estimates

estimates, maximum norm estimates, interior estimates for difference quotients and superconvergence estimates. Some proofs depend on known results for the associated elliptic problem. Several of these estimates are derived for positive time under weak assumptions on the initial data.

Pointwise Error Estimates and Asymptotic Error Expansion Inequalities for the Finite Element Method on Irregular Grids: Part II. Interior Estimates

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.

Some new error estimates for Ritz-Galerkin methods with minimal regularity assumptions

This part contains new pointwise error estimates for the finite element method for second order elliptic boundary value problems on smooth bounded domains in R N . In a sense to be discussed below these sharpen known quasi-optimal L∞ and W 1∼ ∞ estimates for the error on irregular quasi-uniform meshes in that they indicate a more local dependence of the error at a point on the derivatives of the solution u. We note that in general the higher order finite element spaces exhibit more local behavior than lower order spaces. As a consequence of these estimates new types of error expansions will be derived which are in the form of inequalities. These expansion inequalities are valid for large classes of finite elements defined on irregular grids in R N and have applications to superconvergence and extrapolation and a posteriori estimates. Part II of this series will contain local estimates applicable to non-smooth problems.

Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. Part 1: a smooth problem and globally quasi-uniform meshes

This part contains new interior pointwise error estimates for the finite element method for second order elliptic problems in

Maximum-norm interior estimates for Ritz-Galerkin methods

\mathbb R^N