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
Introduction | 1 |
Multigrid Methods | 24 |
Defect Correction for Operator Equations | 33 |
Copyright | |
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a-posteriori error estimates accuracy adaptive algorithms Anal analysis application approximate inverse approximate solution assume assumptions asymptotic expansions B-convergence Banach space Böhmer boundary conditions boundary value problems coarse grid Comp computation consider convergence corresponding D₁ defect correction defect correction method defect correction principle defined denotes Discrete Newton methods discretization error discretization methods eigenvalues elliptic example finite element method fixed point function given global discretization error global error Hackbusch Hemker IDeC IDeC-algorithms initial approximation interpolation interval invariant subspace iterated defect corrections Lemma linear equations mapping Math matrix MDCP mesh MMGM model problem multigrid method Newton's method norm numerical solution obtained ordinary differential equations Pereyra polynomial Proof residual satisfies Section sequence SIAM singular perturbation small parameter smooth solved space Stetter subintervals Theorem u₁ Ueberhuber v₁ v₂ vector