The fall semester 2021 working group will be on preconditioners. We plan to study mainly multigrid and algebraic preconditioners and try to find connections between those.

Organizers are Noel Walkington, Yangwen Zhang and Franziska Weber.

The working group takes place on Tuesdays during the semester at 2.30pm in Wean Hall 7218 or alternatively over zoom (email us if you want the zoom link).

Schedule:

Sept 7, 2021: Organizational meeting

Sept 14, 2021: Introductory lecture by Noel Walkington

Sept 21, 2021: The V-cycle multigrid  and related methods, part I (Franziska Weber)

Sept 28, 2021: The V-cycle multigrid  and related methods, part II (Franziska Weber)

Oct 5, 2021: The V-cycle multigrid  and related methods, part III (Franziska Weber): We'll prove the technical lemma that results in the error estimate for the V-cycle multigrid method.

Oct 12, 2021: Domain decomposition methods: The foundations, part I (David Shirokoff): We study the paper by P.L. Lions: On the Schwarz Alternating Method III: A Variant for Nonoverlapping Subdomains.

Oct 19, 2021: Domain decomposition methods: The foundations, part II (David Shirokoff)

Oct 26, 2021: Domain decomposition methods: The foundations, part III (David Shirokoff)

Nov 9, 2021: Domain decomposition methods: The foundations, part IV (David Shirokoff)

Nov 16, 2021: Noel presents Susanne Brenner's paper `The condition number of the Schur complement in domain decomposition'

Nov 30, 2021: Noel continues to present Susanne Brenner's paper `The condition number of the Schur complement in domain decomposition'

References:

• McCormick: Multigrid methods https://epubs.siam.org/doi/book/10.1137/1.9781611971057

• A multigrid tutorial https://www.researchgate.net/publication/220690328_A_Multigrid_Tutorial_2nd_Edition 

• D. Braess, W. Hackbusch (1983). A new convergence proof for the multigrid method including the V-cycle. SIAM J. Numer. Anal. 20, 967-975.

• Spielman and Teng: Nearly linear time algorithms for graph partitioning, graph sparsification, and solving linear systems, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM Press, New York, NY, 2004, pp. 81–90.

• D. A. Spielman. Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices. http://www.cs.yale.edu/homes/spielman/PAPERS/icm10post.pdf

• Boman, E. , Hendrickson, B. , & Vavasis, S.  (2008). Solving Elliptic Finite Element Systems in Near-Linear Time with Support Preconditioners. SIAM J. Numer. Anal., 46, 3264-3284 

• P.L. Lions: On the Schwarz Alternating Method III: A Variant for Nonoverlapping Subdomains. Third international symposium on domain decomposition methods for partial differential equations. Vol. 6. SIAM Philadelphia, PA, 1990. http://www.ddm.org/DD03/On_the_Schwarz_Alternating_Method_III_A_Variant_for_Nonoverlapping_Subdomains_%28Lions%29.pdf

• Susanne Brenner: The condition number of the Schur complement in domain decomposition. Numer. Math. (1999) 83: 187-203 https://link.springer.com/content/pdf/10.1007/s002110050446.pdf