Defect Correction Methods

1. Front Matter

   Pages I-IX

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2. Introduction

   1. The Defect Correction Approach

      • K. Böhmer, P. W. Hemker, H. J. Stetter

      Pages 1-32

3. Defect Correction for Operator Equations

   1. Defect Correction Algorithms for Stiff Ordinary Differential Equations

      • R. Frank, J. Hertling, H. Lehner

      Pages 33-41

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

      • H.-J. Reinhardt

      Pages 43-66

   3. Simultaneous Newton’s Iteration for the Eigenproblem

      • Françoise Chatelin

      Pages 67-74

   4. On Some Two-level Iterative Methods

      • J. Mandel

      Pages 75-87

4. Multi-grid Methods

   1. Local Defect Correction Method and Domain Decomposition Techniques

      • W. Hackbusch

      Pages 89-113

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

      • S. McCormick

      Pages 115-121

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

      • P. W. Hemker

      Pages 123-145

5. Computation of Guaranteed High-accuracy Results

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

      • S. M. Rump

      Pages 147-168

   2. Residual Correction and Validation in Functoids

      • E. Kaucher, W. L. Miranker

      Pages 169-192

6. Defect Corrections in Applied Mathematics and Numerical Software

   1. Defect Corrections and Hartree-Fock Method

      • K. Böhmer, W. Gross, B. Schmitt, R. Schwarz

      Pages 193-209

   2. Deferred Corrections Software and Its Application to Seismic Ray Tracing

      • V. Pereyra

      Pages 211-226

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

      • W. Schönauer, E. Schnepf, K. Raith

      Pages 227-242

7. Back Matter

   Pages 243-243

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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.

• algorithms
• calculus
• differential equation
• finite element method
• mathematics
• complexity

• Fachbereich Mathematik, Universität Marburg, Federal Republic of Germany

  Klaus Böhmer

• Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Austria

  Hans J. Stetter

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