An introduction to domain decomposition algorithms

Sunday, November 28, 2010 - 4:00pm - 5:30pm
Keller 3-180
Olof Widlund (New York University)
Variational formulation and piece-wise linear finite element approximations
of Poisson's problem. Dirichlet and Neumann boundary conditions and
Poincaré's and Friedrichs's inequalities. A word about linear elasticity.
Condition numbers of finite element matrices and the preconditioned conjugate
gradient method.

Domains and subdomains. Subdomain matrices as building blocks for domain
decomposition methods and the related Schur complements. The two-subdomain
case: the Neumann--Dirichlet and Schwarz alternating algorithms; they can
be placed in a unified framework and written in terms of Schur complements.
Extension to the case of many subdomains; coloring, the problems of singular
subdomain matrices, and the need to use a coarse, global problem. Three
assumptions and the basic result on the condition number of additive Schwarz

Classical and more recent two--level additive Schwarz methods. Remarks
on the effect of irregular subdomains. Extensions to elasticity problems
including the almost incompressible case.

Modern iterative substructuring methods: FETI–DP and BDDC. An introduction
in terms of block-Cholesky for problems only partially assembled. The
equivalence of the spectra. Results on elasticity including incompressible
Stokes problems.
MSC Code: