Singular perturbation problems frequently arise in large applications, e.g. in the area of computational fluid dynamics. They usually have solutions with shocks and layers, which require adaptive mesh refinement and fast basic solvers in order to be resolved efficiently. In addition the solvers should contain inherent parallelism to be run efficiently on distributed architectures.
It is well known that the efficient numerical solution of singular perturbation problems is extremely difficult. Standard multilevel methods lack their usual efficiency due to a convergence rate of \rho >= 0.5 for the two level method which causes convergence rates approaching 1.0 with an increasing number of levels. The situation is even worse on (massively) parallel machines since flow direction dependant relaxations result in poor parallel efficiency. Extensive and systematic numerical experiments have lead us to multigrid variants which show surprisingly nice convergence behaviour while being fully parallelizable. Numerical results for standard multilevel methods and our improved schemes will be presented.
The modified multilevel algorithms need additional artificial viscosity in the basic dicretization. Therefore, methods like defect correction (which will not be discussed here) and adaptive refinement become even more attractive. First results for singular perturbation problems solved by fast adaptive composite methods (FAC, AFAC) will be discussed. These algorithms are implemented on the basis of the P++ and AMR++ tools, which are presented in another talk on this conference.