Multiscale methods based on multigrid principles

The focus of this thesis is on two state-of-the-art multiscale methods that are based on multigrid concepts. The first is a space-time variational multigrid for bridging discrete scales with either coarse grained discrete or continuum scales in molecular dynamics simulations. The method consists of the waveform relaxation scheme aimed at capturing the high frequency response of the atomistic vibrations and a coarse scale solution in space and time aimed at resolving smooth features (in both space and time domains) of the discrete medium. The method is implicit in space and time, possesses superior stability properties and consequently enables larger time steps governed by accuracy considerations of coarse scale quantities of interest. Performance studies on polymer melts have shown significant speed-up compared to the classical explicit methods, in particular on parallel machines.; The second is a multiscale filter, termed as GGB or Generalized Global Basis, aimed at enhancing performance of multigrid solvers for difficult systems such as those arising from indefinite and nonsymmetric matrices. The GGB filter constructs an auxiliary coarse model from the largest eigenvalues of the iteration matrix. It projects these modes which would cause slow convergence to a coarse problem which is then used to capture these modes. The best performance is obtained when the filter is accelerated by GMRES and used for problems with multiple right hand sides. In addition it is demonstrated that GGB can enhance restarted GMRES strategies by retention of important subspace information. For nonlinear problems we propose two heuristic strategies intended to reduce the cost of eigen computations. The GGBalpha scheme enriches the filter operator with new eigenvectors while the modified method (MGGB) selectively reuses the same filter. Both methods use the criterion of principal angles between subspaces spanned between the previous and current filter operators. Numerical examples on a wide range of engineering applications, both in serial and parallel, clearly show that the multiscale filter significantly accelerates multigrid on those problems.