Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  author = {M. Griebel and F. Kiefer},
  title = {Generalized Hierarchical Basis Multigrid Methods for
		  Convection-Diffusion Problems},
  institution = {Sonderforschungsbereich 256, Institut f\"ur Angewandte
		  Mathematik, Universit\"at Bonn},
  year = {2001},
  type = {{SFB} {P}reprint 720},
  abstract = {We consider the efficient solution of discrete
		  convection-diffusion problems with dominating convection.
		  It is well known that the standard hierarchical basis
		  multigrid method (HBMG) leads neither to optimal (w.r.t.
		  mesh size) nor to robust solvers for discrete operators
		  arising from singularly perturbed convection-diffusion
		  problems. Its performance is strongly dependent on the
		  coefficients in the differential equation (e.g. strength of
		  convection) {\em and}\, the mesh size. (Pre-)Wavelet
		  splittings of the underlying function spaces allow for
		  efficient algorithms which can be viewed as generalized
		  HBMG methods. They can be interpreted as ordinary multigrid
		  methods which employ a special kind of multiscale smoother
		  and show an optimal convergence behavior for the respective
		  non-perturbed equations similar to classical multigrid.
		  Here, we present a general Petrov--Galerkin multiscale
		  approach which makes use of problem-dependent coarsening
		  strategies known from robust multigrid techniques
		  (matrix-dependent prolongations, algebraic coarsening)
		  together with certain (pre-)wavelet-like and hierarchical
		  multiscale decompositions of the trial and test spaces on
		  the finest grid. The presented numerical results show that
		  by this choice generalized HBMG methods can be constructed
		  which result in robust yet efficient solvers.},
  ps = { 1},
  annote = {report,multigrid}