ABSTRACT:
Numerical simulation of electromagnetic phenomena is of critical
importance in a number of practical applications. In many simulation
codes, Maxwell's equations are reduced to a second-order PDE system
involving one of the vector fields or a new vector potential. The
definite Maxwell equations, for example, arise after discretization in
time, while magnetostatics with a vector potential leads to the
semidefinite Maxwell problem.
In this talk, I will present some recent work on algebraic
preconditioners for unstructured finite element discretizations of the
above problems. Such discretizations result in large sparse linear
systems that could not be solved efficiently by the previously
existing algebraic multigrid (AMG) solvers.
The first preconditioning scheme that I will discuss uses an auxiliary
problem posed on a related mesh, which is more amenable for
constructing optimal order multigrid methods. In combination with a
domain embedding technique, this method can precondition a problem
defined on a very complicated mesh by a standard geometric multigrid
on a box. The main tool in the theory and the implementation of the
above approach is the interpolation operator, which maps auxiliary
functions into the (edge) finite element space. Recently, Hiptmair
and Xu showed that one could use an auxiliary space that is conforming
and is defined on the same mesh. I will present a number of numerical
experiments with a parallel version of their method, which utilizes
two internal AMG V-cycles: one for a scalar and one for a vector
Poisson-like matrix. The results demonstrate very good scalability on
unstructured 3D grids, including on problems with variable
coefficients and zero conductivity. |