Daniel B Szyld

Temple University, Philadelphia


title : 
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An optimal block iterative method and preconditioner
for banded matrices with applications to PDEs on irregular domains

Abstract
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Classical Schwarz methods and preconditioners subdivide the domain of a
partial differential equation into subdomains and use Dirichlet or Neumann
transmission conditions at the artificial interfaces.
Optimizable Schwarz methods use Robin (or higher order) transmission
conditions instead, and the Robin parameter can be optimized so that the
resulting iterative method has an optimal convergence rate.  The usual
technique used to find the optimal parameter is Fourier
analysis; but this is only applicable to certain domains, for example, a
rectangle.
In this talk,  we present a completely algebraic view of Optimizable
Schwarz methods, including an algebraic approach to find the optimal
operator or a sparse approximation thereof. This approach allows us to
apply this method to any banded or block banded linear system of
equations, and in particular to discretizations of partial
differential equations in two and three dimensions on irregular
domains.  With the computable optimal operator, we prove that the
Optimizable Schwarz method converges in two iterations for the case of two
subdomains.  Similarly, we prove that when we use an Optimizable Schwarz
preconditioner with this optimal parameter, the underlying minimal
residual Krylov subspace method (e.g., GMRES) converges in two
iterations.
Very fast convergence is attained even when the optimal operator is
approximated by a sparse transmission matrix. Numerical examples
illustrating these results are presented.