168 1995 Sverker Holmgren and Kurt Otto sverker@tdb.uu.se A framework for polynomial preconditioners based on fast transforms II: PDE applications
Abstract
The solution of systems of equations arising from systems of
time-dependent partial differential equations (PDEs) is considered.
Primarily, first-order PDEs are studied, but second-order derivatives
are also accounted for. The discretization is performed using a general
finite difference stencil in space and an implicit method in time.
The systems of equations are solved by a preconditioned Krylov subspace method.
The preconditioners exploit optimal and superoptimal approximations by
polynomials in a normal basis matrix, associated with a fast trigonometric
transform. Numerical experiments for high-order accurate discretizations are
presented. The results show that preconditioners based on fast transforms
yield efficient solution algorithms, even for large quotients between
the time and space steps. Utilizing a spatial grid ratio less than one,
the arithmetic work per grid point is bounded by a constant as the number
of grid points increases.