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After briefly reviewing the interrelation between Riesz-bases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It is based on direct and inverse estimates for a certain scale of spaces induced by the underlying multiresolution sequence. Furthermore, we highlight some properties of these spaces pertaining to duality, interpolation, and applications to norm equivalences for Sobolev spaces. AMS Subject Classification: 41A17, 41A65, 46A35, 46B70, 46E35 Key Words: Riesz bases, biorthogonality, stability, projectors, interpolation theory, K-method, duality, Jackson, Bernstein inequalities 1 Background and Motivation A standard framework for approximately recapturing a function v in some infinite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equ...

CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental Two-Level Estimates 7 4 A Posteriori Error Estimates 16 5 Two-Level Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the Office of Naval Research under contract N00014-89J-1440. 2 Randolph E. Bank formulation of iterative methods for solving the large sparse sets of linear equations arising from the finite element discretization. Our goal is that the development should be largely self-contained, but at the same time accessible and interest

. We consider two level overlapping Schwarz domain decomposition methods for solving the finite element problems that arise from discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Standard finite element interpolation from the coarse to the fine grid may be used. Our theory requires no assumption on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be non-nested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasi-uniform. In addition, the domains defined by the fine and coarse grid need not be identical. The one important constraint is that the closure of the coarse grid must cover any portion of the fine grid boundary for which Neumann boundary conditions are given. In this general setting, our algorithms have the same optimal convergence rate of the usual ...

Iterative methods for linear systems of algebraic equations arising from the finite element discretization of nonsymmetric and indefinite elliptic problems are considered. Methods previously known to work well for positive definite, symmetric problems are extended to certain nonsymmetric problems, which can also have some eigenvalues in the left half plane. We first consider an additive Schwarz method applied to linear, second order, symmetric or nonsymmetric, indefinite elliptic boundary value problems in two and three dimensions. An alternative linear system, which has the same solution as the original problem, is derived and this system is then solved by using GMRES, an iterative method of conjugate gradient type. In each iteration step, a coarse mesh finite element problem and a number of local problems are solved on small, overlapping subregions into which the original region is subdivided. We show that the rate of convergence is independent of the number of degrees of freedom and the number of local problems if the coarse mesh is fine enough. The performance of the method in two dimensions is illustrated by results of several numerical experiments. We also consider two other iterative method for solving the same class of elliptic problems in two dimensions. Using an observation of Dryja and Widlund, we show that the rate of convergence of certain iterative substructuring methods deteriorates only quite slowly when the local problems increase in size. A similar result is established for Yserentant’s hierarchical basis method.

. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...

. This paper begins with an introduction to additive and multiplicative Schwarz methods. A two-level method is then reviewed and a new result on its rate of convergence is established for the case when the overlap is small. Recent results by Xuejun Zhang, on multi-level Schwarz methods, are formulated and discussed. The paper is concluded with a discussion of recent joint results with Xiao-Chuan Cai on nonsymmetric and indefinite problems. Key Words. domain decomposition, Schwarz methods, finite elements, nonsymmetric and indefinite elliptic problems AMS(MOS) subject classifications. 65F10, 65N30 1. Introduction. Over the last few years, a general theory has been developed for the study of additive and multiplicative Schwarz methods. Many domain decomposition and certain multigrid methods can now be successfully analyzed inside this framework. Early work by P.-L. Lions [23], [24] gave an important impetus to this effort. The additive Schwarz methods were then developed by Dryja and ...

Abstract. We propose a substructuring preconditioner for solving threedimensional elliptic equations with strongly discontinuous coefficients. The new preconditioner can be viewed as a variant of the classical substructuring preconditioner proposed by Bramble, Pasiack and Schatz (1989), but with much simpler coarse solvers. Though the condition number of the preconditioned system may not have a good bound, we are able to show that the convergence rate of the PCG method with such substructuring preconditioner is nearly optimal, and also robust with respect to the (possibly large) jumps of the coefficient in the elliptic equation. 1.

We propose a robust interpolation for multigrid based on the concepts of energy minimization and approximation. The formulation is general; it can be applied to any dimensions. The analysis for one dimension proves that the convergence rate of the resulting multigrid method is independent of the coefficient of the underlying PDE, in addition to being independent of the mesh size. We demonstrate numerically the effectiveness of the multigrid method in two dimensions by applying it to a discontinuous coefficient problem and an oscillatory coefficient problem. We also show using a one-dimensional Helmholtz problem that the energy minimization principle can be applied to solving elliptic problems that are not positive definite.

. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...