Robuste und schnelle Mehrgitterverfahren

In this talk, we analyze multilevel algorithms without a smoothing on the whole finite element space. Instead, it is used a subspace correction on a complementary space. Here, the fine-grid space is a direct sum of the coarse-grid space and the complementary space. The convergence rate of such a multilevel algorithm depends on the constant in the strengthened Cauchy-Schwarz inequality between the coarse-grid space and the complementary space. Small constants imply a fast multilevel algorithm. Such constants can be obtained, if the complementary space is spanned by prewavelets or generalized prewavelets. Now, semi-coarsening and line relaxation lead to a fast and robust multilevel algorithm for a certain class of anisotropic elliptic equations and convection-diffusion equations. To obtain robustness with respect to discontinuities in x- and y-direction, one has to apply problem dependent generalized prewavelets. We explain how to construct such functions and present some numerical results.