B4

Long Chen. Lecture Notes: Subspace Correction and Multigrid Methods.

ABSTRACT:
This is a short introduction to the method of subspace correction [6] on solving algebraic equations. Many iterative methods including multigrid methods can be developed and analyzed in this framework. This notes will emphasis on the theoretical convergence analysis of iterative methods, especially multigrid methods. A new identity [10] for a successive subspace correction method is included to make the analysis of multigrid methods more straightforward. The presentation mainly follows Xu’s work [6, 8, 7, 9].

CONTENTS
1. Introduction 1
2. Basic linear iterative methods 6
3. Convergence analysis of basic iterative methods 8
4. Space decomposition 12
5. Interpolation spaces using multilevel subspaces 16
6. Subspace correction 22
7. PSC: BPX and Hierarchal preconditioner 24
8. SSC: X-Z identity 27
9. Multigrid method and its convergences 32
References 38
Appendix: Matrix Theory 39