ROBUST MULTIGRID ALGORITHMS FOR 3D ELLIPTIC AND CONVECTION DIFFUSION PROBLEMS ON CELL-CENTERED

GRIDS

O.P.Iliev, LSTM, Univ. Erlangen-Nurnberg Cauerstrasse 4, D-91058 Erlangen, Germany and Inst. of Mathematics, Bulg. Acad. Sci., Acad. G.Bonchev str., bl.8, 1113 Sofia, Bulgaria e-mail iliev@lstm.uni-erlangen.de

Robust multigrid algorithms for 3D elliptic and convection diffusion problems on cell–centered grids are presented. The coarse grid is not nested in the fine grid in this case, but the coarsening is simple because any coarse volume is a sum of 8 (for 3D case) finite volumes. Considered equations are discretized by finite volume methods and this is essential in building coarse grid equations. The flux through a coarse finite volume surface is equal to sum of the fluxes through respective surfaces of finite volumes which compose this coarse finite volume. This allows the coarse grid matrix to be built in a ”black–box” manner in the case of orhogonal grids and absent of mixed derivatives. However, we use not only the fine grid matrix and right hand side, but also some additional information for the problem to be solved. For example, we use the value for Peclet number in the case of solving convection diffusion problems.

The robustness of the considered multigrid algorithms is based on careful discretization and on successfull combination of multigrid with Krylov subspace iterative methods. Sawtooth-type multigrid cycle is exploited. The choice of the used multigrid components is described below.

• The coarse grid matrix is built by fine grid one following above described ”flux” approach. In solving convection–diffusion equation we consider two approaches: i)coarsening the overall flux; ii)separate coarsening of convective and diffusive fluxes. It is shown that second approach is preferable for dominating convection problems.

• Two different robust relaxation methods are considered - ILU_{β} as a smoother, or conjugate gradients type method as a rougher. These two ralaxation methods are compared for several anysotropic elliptic problems.

Results from numerical experiments with several elliptic and convection diffusionproblems are demonstrated.