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

  key = {2001:yyy},
  author = {M.~Griebel and M.~A. Schweitzer},
  title = {A Particle-Partition of Unity Method---{P}art {III}: {A}
		  Multilevel Solver},
  institution = {Sonderforschungsbereich 256, Institut f\"ur Angewandte
		  Mathematik, Universit\"at Bonn},
  journal = {SIAM J. Sci. Comp.},
  year = {2002},
  volume = {24},
  number = {2},
  pages = {377--409},
  note = {},
  annote = {refereed article,256D},
  ps = { 1},
  pdf = { 1},
  abstract = {In this paper we focus on the efficient solution of the
		  linear block-systems arising from a Galerkin discretization
		  of an elliptic partial differential equation of second
		  order with the partition of unity method (PUM). We present
		  a cheap multilevel solver for partition of unity
		  discretizations of any order. The shape functions of a PUM
		  are products of piecewise rational partition of unity (PU)
		  functions and higher order local approximation functions
		  (usually a local polynomial. Furthermore, they are
		  non-interpolatory. In a multilevel approach we not only
		  have to cope with non-interpolatory basis functions but
		  also with a sequence of nonnested spaces due to the
		  meshfree construction. Hence, injection or interpolatory
		  interlevel transfer operators are not available for our
		  multilevel PUM. Therefore, the remaining natural choice for
		  the prolongation operators are L2-projections. Here, we
		  exploit the partition of unity construction of the function
		  spaces and a hierarchical construction of the PU itself to
		  localize the corresponding projection problem. This
		  significantly reduces the computational costs associated
		  with the setup and the application of the interlevel
		  transfer operators. The second main ingredient for our
		  multilevel solver is the use of a block-smoother to treat
		  the local approximation functions simultaneously. The
		  results of our numerical experiments in two and three
		  dimensions show that the convergence rate of the proposed
		  multilevel solver is independent of the number of patches.
		  The convergence rate is slightly dependent on the local
		  approximation orders.}