Defect Correction Methods: Theory and ApplicationsK. Böhmer, H.J. Stetter 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. |
Contents
| 1 | |
Defect Correction for Operator Equations | 33 |
BConvergence Properties of Certain IDEC | 39 |
Expansions Defect Corrections Through Projection Methods Direct | 57 |
Simultaneous Newtons Iteration for the Eigenproblem | 67 |
On Some Twolevel Iterative Methods | 75 |
Methods Conclusion | 85 |
Local Defect Correction Method and Domain Decomposition Techniques | 88 |
Theory for the Variational | 115 |
Mixed Defect Correction Iteration for the Solution of a Singular | 122 |
Solution of Linear and Nonlinear Algebraic Problems with Sharp | 147 |
Residual Correction and Validation in Functoids | 169 |
Defect Corrections in Applied Mathematics and Numerical Software | 193 |
Deferred Corrections Software and Its Application to Seismic Ray Tracing 211 | 210 |
Experiences in Designing PDE Software with | 227 |
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accuracy adaptive algorithm Anal analysis application approximate inverse approximate solution arithmetic assumptions asymptotic expansion Ax=b Banach space Böhmer boundary conditions boundary value problems bounded coarse grid components computation consider convergence corresponding D₁ defect correction defect correction method defect correction principle defect correction step defined denotes difference formulae discretization error discretization methods eigenvalues elliptic error estimates example finite element finite element method fixed point functoid given global discretization error global error Hackbusch IDeC inclusion interpolation iterated defect corrections iterative method Lemma linear systems m₁ mapping Math matrix MDCP mesh model problem multigrid method Newton methods norm numerical solution obtained operator ordinary differential equations parameter PDE's Pereyra polynomial Proof residual satisfies Section sequence SIAM singular perturbation smooth solved solver Stetter subspaces Theorem v₁ vector