Defect Correction Methods

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.

Contenu

The Defect Correction Approach.- Historical Examples of Defect Correction.- General Defect Correction Principles.- Discretization of Operator Equations.- Defect Correction and Discretization.- Multi-level and Multi-grid Methods.- References.- Defect Correction for Operator Equations.- Defect Correction Algorithms for Stiff Ordinary Differential Equations.- Specification of the Algorithm.- Previous Results.- The Concept of B-Convergence.- B-Convergence Properties of Certain IDeC-Algorithms.- References.- On a Principle of Direct Defect Correction Based on A-Posteriori Error Estimates.- Basic Relations Between Defect Corrections and Asymptotic Expansions.- Defect Corrections Through Projection Methods.- Direct Defect Correction via Finite Element Methods for Singularly Perturbed Differential Equations.- References.- Simultaneous Newton's Iteration for the Eigenproblem.- The Sylvester Equation AX?XB= C.- A Quadratic Equation for the Invariant Subspace.- Newton's Method on (7).- Simplified Newton's Method.- Modified Newton's Methods.- Conclusion.- References.- On Some Two-level Iterative Methods.- The General Two-level Iterative Process.- Applications to Integral Equations.- Projection-iterative Methods.- Iterative Aggregation Methods.- Conclusion.- References.- Multi-grid Methods.- Local Defect Correction Method and Domain Decomposition Techniques.- Defect Correction Method.- Local Defect Correction (Algorithm; Numerical Examples of the Local Defect Correction; Error Estimates; Multi-grid Iteration with Local Defect Correction).- Domain Decomposition Methods.- References.- Fast Adaptive Composite Grid (FAC) Methods: Theory for the Variational Case.- Two-level Methods.- Remarks.- References.- Mixed Defect Correction Iteration for the Solution of a Singular Perturbation Problem.- Local Mode Analysis.- The Defect Correction Principle.- The Mixed Defect Correction Process (MDCP).- Local Mode Analysis for the MDCP Solution.- The Convergence of the MDCP Iteration.- Boundary Analysis of the MDCP Solutions.- Numerical Examples.- References.- Computation of Guaranteed High-accuracy Results.- Solution of Linear and Nonlinear Algebraic Problems with Sharp, Guaranteed Bounds.- Computer Arithmetic.- Inclusion Methods for Linear Systems.- Implementation of Inclusion Algorithms.- Nonlinear Systems.- Conclusion.- References.- Residual Correction and Validation in Functoids.- Functoids and Roundings.- Iterative Residual Correction.- References.- Defect Corrections in Applied Mathematics and Numerical Software.- Defect Corrections and Hartree-Fock Method.- Hartree-Fock Method.- Defect Corrections on Infinite Intervals.- Asymptotic Boundary Conditions and Defect Corrections with Changing Boundary Points.- References.- Deferred Corrections Software and Its Application to Seismic Ray Tracing.- Discontinuous Interfaces at Known Locations.- PASVA4, Part I.- Discontinuities at Unknown Locations.- PASVA4, Part II. Algebraic Parameters and Conditions.- Three-dimensional Two-point Ray Tracing.- Future Developments.- References.- Numerical Engineering: Experiences in Designing PDE Software with Selfadaptive Variable Step Size/Variable Order Difference Methods.- Estimate of the Truncation Error.- The Error Equation.- The Ordinary BVP.- 2- and 3-D Elliptic PDE's.- The Ordinary IVP.- Parabolic PDE's.- Three Examples.- Concluding Remarks.- References.