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Über dieses Buch

Ten years ago, the term "defect correction" was introduced to characterize a class of methods for the improvement of an approximate solution of an operator equation. This class includes many well-known techniques (e.g. Newton's method) but also some novel approaches which have turned out to be quite efficient. Meanwhile a large number of papers and reports, scattered over many journals and institutions, have appeared in this area. Therefore, a working conference on "Error Asymptotics and Defect Corrections" was organized by K. Bohmer, V. Pereyra and H. J. Stetter at the Mathematisches Forschungsinstitut Oberwolfach in July 1983, a meeting which aimed at bringing together a good number of the scientists who are active in this field. Altogether 26 persons attended, whose interests covered a wide spectrum from theoretical analyses to applications where defect corrections may be utilized; a list of the participants may be found in the Appendix. Most of the colleagues who presented formal lectures at the meeting agreed to publish their reports in this volume. It would be presumptuous to call this book a state-of-the-art report in defect corrections. It is rather a collection of snapshots of activities which have been going on in a number of segments on the frontiers of this area. No systematic coverage has been attempted. Some articles focus strongly on the basic concepts of defect correction; but in the majority of the contributions the defect correction ideas appear rather as instruments for the attainment of some specified goal.




The Defect Correction Approach

This is an introductory survey of the defect correction approach which may serve as a unifying frame of reference for the subsequent papers on special subjects.
K. Böhmer, P. W. Hemker, H. J. Stetter

Defect Correction for Operator Equations

Defect Correction Algorithms for Stiff Ordinary Differential Equations

The application of suitable defect correction algorithms to stiff differential equations is analyzed. The B-convergence properties of such algorithms are discussed.
R. Frank, J. Hertling, H. Lehner

On a Principle of Direct Defect Correction Based on A-Posteriori Error Estimates

A combination of an iterative procedure with realistic a-posteriori error estimates allows the approximate solution of functional equations where an error improvement can be achieved which is controlled by a direct defect correction (or: residual improvement). The underlying mathematical theory is presented which is essentially based on the Inverse Function Theorem. As applications, defect corrections via projection methods for linear problems as well as for nonlinear problems are analyzed. For a linear model problem of a singularly perturbed one-dimensional boundary value problem, computational results are presented where one defect correction step is performed using a self-adaptive finite element method.
H.-J. Reinhardt

Simultaneous Newton’s Iteration for the Eigenproblem

For an ill-conditioned eigenproblem (close eigenvalues and/or almost parallel eigenvectors) it is advisable to group some eigenvalues and to compute a basis of the corresponding invariant subspace. We show how Newton’s method may be used for the iterative refinement of an approximate invariant subspace.
Françoise Chatelin

On Some Two-level Iterative Methods

Multigrid methods for boundary value problems and integral equations of the second kind, projection-iterative methods for operator equations, and iterative aggregation methods for systems of linear equations are shown to be particular cases of a unifying framework based on the defect correction principle. Several convergence proofs using contraction arguments are given.
J. Mandel

Multi-grid Methods

Local Defect Correction Method and Domain Decomposition Techniques

For elliptic problems a local defect correction method is described. A basic (global) discretization is improved by a local discretization defined in a subdomain. The convergence rate of the local defect correction iteration is proved to be proportional to a certain positive power of the step size. The accuracy of the converged solution can be described. Numerical examples confirm the theoretical results. We discuss multi-grid iterations converging to the same solution.
The local defect correction determines a solution depending on one global and one or more local discretizations. An extension of this approach is the domain decomposition method, where only (overlapping) local problems are combined. Such a combination of local subproblems can be solved efficiently by a multi-grid iteration. We describe a multi-grid variant that is suited for the use of parallel processors.
W. Hackbusch

Fast Adaptive Composite Grid (FAC) Methods: Theory for the Variational Case

The subject of this paper is the fast adaptive composite grid (FAC) method for solving variationally posed differential boundary value problems. Related to local defect corrections (LDC) and multilevel adaptive techniques (MLAT), an important difference is that FAC forces the user to specify the discrete problem on the finest composite grid — it is not rather an implicit result of the adaptive process. The advantages are that FAC can more readily meet practical objectives, does not suffer from some of the practical limitations of (“natural”) versions of LDC and MLAT, and lends itself to a simple theory. The latter is the subject of this paper.
S. McCormick

Mixed Defect Correction Iteration for the Solution of a Singular Perturbation Problem

We describe a discretization method (mixed defect correction) for the solution of a two-dimensional elliptic singular perturbation problem. The method is an iterative process in which two basic discretization schemes are used: one with and one without artificial diffusion. The resulting method is stable and yields a 2nd order accurate approximation in the smooth parts of the solution, without using any special directional bias in the discretization. The method works well also for problems with interior or boundary layers.
P. W. Hemker

Computation of Guaranteed High-accuracy Results

Solution of Linear and Nonlinear Algebraic Problems with Sharp, Guaranteed Bounds

In this paper new methods for solving algebraic problems with high accuracy are described. They deliver bounds for the solution of the given problem with an automatic verification of the correctness. Examples of such problems are systems of linear equations, over- and underdetermined systems of linear equations, algebraic eigenvalue problems, nonlinear systems, polynomial zeros, evaluation of arithmetic expressions, linear, quadratic and convex programming and others. The new methods apply for these problems over the space of real numbers, complex numbers as well as real intervals and complex intervals.
S. M. Rump

Residual Correction and Validation in Functoids

We combine four recently developed methodologies to provide a computational basis for function space problems (e. g. differential equations, integral equations …).
E. Kaucher, W. L. Miranker

Defect Corrections in Applied Mathematics and Numerical Software

Defect Corrections and Hartree-Fock Method

In the context of Hartree-Fock methods for the Schrödinger equation a special class of EVPs for ODEs on infinite intervals is shown to play a crucial role in the computation time. The usual discretization is combined with two very efficient ways to choose the finite boundary conditions. Then two kinds of defect corrections are applied.
K. Böhmer, W. Gross, B. Schmitt, R. Schwarz

Deferred Corrections Software and Its Application to Seismic Ray Tracing

We give first a historical account of the various stages of development of iterated deferred corrections software, mainly for ordinary two-point boundary value problems, but mentioning also some work on partial differential equations. Then we describe the latest code on the PASVA series (No. 4), which extends the earlier one to problems with discontinuous data and mixed systems of differential and algebraic conditions. Finally, an example of application of this code to two-point ray tracing on piece-wise continuous media is given.
V. Pereyra

Numerical Engineering: Experiences in Designing PDE Software with Selfadaptive Variable Step Size/Variable Order Difference Methods

The basic ideas in designing software for the numerical solution of nonlinear systems of elliptic and parabolic PDE’s with variable step size/variable order difference methods are presented. The error is estimated by the difference of difference formulae, using members of families of difference formulae. Basic solution methods are developed for the solution of the BVP and the IVP for ODE’s. These methods are extended and combined to solution methods for elliptic and parabolic PDE’s. The nonlinear equations are solved by a robust Newton-Raphson method. The method tends to balance all the relevant errors according to a prescribed relative tolerance. For the final solution an estimate of the error of the solution is computed which means e. g. a global error for the IBVP’s.
W. Schönauer, E. Schnepf, K. Raith


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