190 1996 Elisabeth Larsson bette@tdb.uu.se A domain decomposition method for the Helmholtz equation in a multilayer domain
Abstract
The two-dimensional Helmholtz equation for problems where the physical
domain consists of layers with different material properties is studied.
An efficient preconditioner for iterative solution of the problem is
constructed.
The problem is discretized with fourth-order accurate finite difference
operators. For the construction of the radiation boundary conditions
also a fourth-order finite element method is used.
The large, sparse, complex, indefinite, and ill-conditioned system of
equations that arises is solved with preconditioned restarted GMRES.
A domain decomposition method is used, where the preconditioning is based
on the Schur complement algorithm with "fast Poisson type" preconditioners
for the subdomains.
The memory requirements for the preconditioner are nearly linear in the
number of unknowns. The arithmetic complexity for one iteration is low
${\cal O}(m_2\sum_d m_1^{(d)}\log_2 m_1^{(d)})$, where the sum is taken
over the subdomains, and $m_1$ and $m_2$ are the number of gridpoints in the
two coordinate directions.
As a model problem electromagnetic wave propagation in a three-layered
waveguide is used. Numerical experiments show that convergence
is achieved in a few iterations. Compared with banded Gaussian elimination,
which is a standard solution method, the iterative method shows
significant gain in both memory requirements and arithmetic complexity.
Furthermore, the relative gain grows when the problem size increases.