Robust Multi-Grid Methods

In full multigrid methods for elliptic difference equations one works on a sequence of meshes where a number of pre- and/or postsmoothing steps are performed on each level. As is well known these methods can converge very fast on problems with a smooth solution and a regular mesh, but the rate of convergence can be severely degraded for problems with unisotropy or discontinuous coefficients unless some form of robust smoother is used. Also problems can arise with the increasingly coarser meshes because for some types of discretization methods, coercivity may be lost on coarse meshes and on massively parallel computers the computation cost of transporting information between computer processors devoted to work on various levels of the mesh can dominate the whole computing time. For discussions about some of these problems, see (11). Here we propose a method that uses only two levels of meshes, the fine and the coarse level, respec tively, and where the corrector on the coarse level is equal to a new type of preconditioner which uses an algebraic substructuring of the stiffness matrix. It is based on the block matrix tridiagonal structure one gets when the domain is subdivided into strips. This block-tridiagonal form is used to compute an approximate factorization whereby the Schur complements which arise in the recursive factorization are approximated in an indirect way, i. e.

Inhalt
A robust preconditioner based on algebraic substructuring and two-level grids.- Adaptive multigrid Solution of the convection-diffusion equation on the DIRMU multiprocessor.- Finite volume multigrid Solutions of the two-dimensional incompressible Navier-Stokes equations.- Concepts for a dimension independent application of multigrid algorithms to semiconductor device Simulation.- Algebraic multigrid methods and the Schur complement.- A multigrid method for steady Euler equations, based on flux-difference Splitting with respect to primitive variables.- Treatment of singular perturbation problems with multigrid methods.- The frequency decomposition multi-grid algorithm.- On global multigrid convergence for nonlinear problems.- Multigrid methods for the Solution of the compressible Navier-Stokes equations..- On multigrid methods of the first kind for Symmetrie boundary integral equations of nonnegative order.- Effective preconditioning for spectral multigrid methods.- Numerical Solution of transonic potential flow in 2d compressor cascades using multi-grid techniques.- Local mode smoothing analysis of various incomplete factorization iterative methods.- Multigrid and defect correction for the steady Navier-Stokes equations.- Towards multigrid acceleration of 2d compressible Navier-Stokes finite volume implicit schemes..- Multigrid with ILU-smoothing: systematic tests and improvements.- Multilevel preconditioning matrices and multigrid V-cycle methods.- Two remarks on multigrid methods.- On the robustness of ILU-smoothing.- List of participants.