Local Fourier k-grid (k=1,2,3) analysis for Navier-Stokes-type systems
R. Wienands and
C.W. Oosterlee
GMD - Institute for Algorithms and Scientific Computing (SCAI)
Schloss Birlinghoven, D-53754 Sankt Augustin, Germany
Abstract
Local Fourier one- (smoothing) and two-grid analysis (LFA) [1,2,6] are
well-known tools for the quantitative analysis and the design of
efficient multigrid methods for several problems.
The two-grid analysis is the basis
for classical asymptotic multigrid convergence estimates [6].
Moreover, it is the main analysis tool for nonsymmetric problems.
For several multigrid components or cycle variants, however, the
overall multigrid convergence cannot be approximated accurately
by two-grid factors. For example, if one is interested in the varying
performance of V- and W-cycles, of pre- and post-smoothing, of
different smoothers on different grids or of different discretizations
on different grids one has to consider at least one additional grid
leading to a three-grid analysis [8].
Although LFA is often applied to scalar equations,
two-grid values for systems of equations are rarely found
in the literature and three-grid values are completely missing.
To close this gap we developed a freely available, general Fourier
analysis program - LFA00_2D -
which will be presented in the first part of the talk.
LFA00_2D is a set of Fortran77 subroutines to perform Fourier k-grid (k=1,2,3)
analysis for two-dimensional systems of PDEs yielding measures of
h-ellipticity [1], smoothing factors, two- and three-grid convergence
factors and norms of the two- and three-grid operators.
Several discretizations of well-known
systems, like the biharmonic system, the Stokes equations, the Oseen
equations (linearized Navier-Stokes equations) have already been
implemented. Furthermore, an option to analyze your own linearized
system is available.
LFA00_2D supports a large variety of multigrid components including
- Coarsening strategy: standard (full), semi coarsening,
- Coarse grid discretization: direct PDE based, Galerkin-type,
- Prolongation: bilinear, cubic interpolation, matrix-dependent
interpolation (of Dendy[3]- and de Zeeuw[9]-type),
- Restriction: full, half weighting, injection,
- Relaxation: Jacobi, Gauss-Seidel, red-black-type smoothers,
point- and line-wise.
It is, furthermore, possible to estimate convergence factors if the
two- or three-grid operators are used as preconditioners for GMRES [7].
In the second part of the talk, we evaluate several non-staggered
discretizations for Navier-Stokes-type systems (w.r.t. an efficient
multigrid treatment) in order to demonstrate the usefulness of the
analysis program. Here, we focus on second order discretizations.
In particular, the Stokes and the Oseen equations are investigated,
discretized by central differences with an artificial pressure correction
term in the continuity equation (for low Reynolds numbers) or by higher
order upwind schemes like Dick's flux difference splitting [4]
using van Leer's kappa-scheme [5] (for high Reynolds numbers).
The Fourier results are confirmed by numerical experiments with the
incompressible Navier-Stokes equations.
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