198 1997 Johan Walden johan@tdb.uu.se Filter Bank Preconditioners for Finite Difference Discretizations of PDEs
Abstract
We study preconditioners that are based on filter bank methods. Filter
banks are more general than biorthogonal wavelets, and are easier to
adapt to boundaries. The filter bank transform is shown to efficiently
decompose operators coming from the discretization of PDEs with finite
difference methods into two parts; one that has a diagonal preconditioner,
and one small part that can be directly inverted. This feature is
shown both for problems with periodic and non-periodic boundary
conditions; the change being small due to the locality of the filter
bank transform. In contrast to earlier work on the topic, the
``right'' transform is chosen for each problem, meaning that both
non-periodicity, and higher-dimensional tensor product operators are
taken into account. This is shown to improve the method. For a
one-dimensional problem, the condition numbers of the problems
involved are shown to be bounded by a constant, independently of the
problem size. The algorithmic complexity of the method is also
analyzed.