A CELL-CENTRED MULTIGRID ALGORITHM FOR ALL GRID SIZES
Thor Gjesdal
Christian Michelsen Research AS
P.O. Box 3, 5036 Fantoft, Norway
phone: +47-55 57 40 40, fax: +55-57 40 41
email: Thor.Gjesdal@cmr.no
Multigrid methods are optimal; that is, their rate of convergence is
independent of the number of grid points, because they use a nested sequence
of coarse grids to represent different scales of the solution. This nesting
does however usually lead to certain restrictions of the permissible size of
the discretised problem. In cases where the modeller is free to specify the
whole problem, such constraints are of little importance because they can be
taken into consideration from the outset. We are here considering the
situation where there are other competing constraints on the resolution.
These restrictions may stem from the physical problem, for example if the
discretised operator contains experimental data measured on a fixed grid; or
from the need to avoid limitations set by the hardware. In this paper we
discuss a modification to the cell-centred multigrid algorithm, so that it can
be used for problems with any resolution. We discuss in particular a
coarsening strategy and choice of intergrid transfer operators that can handle
grids with both an even or odd number of cells. The method is described and
applied to linear equations obtained by discretisation of two- and
three-dimensional second-order elliptic PDEs.