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

  key = {2001:zzz},
  author = {M.~Griebel and M.~A. Schweitzer},
  title = {A Particle-Partition of Unity Method---{P}art {II}:
		  {E}fficient Cover Construction and Reliable Integration},
  journal = {SIAM J. Sci. Comp.},
  year = {2002},
  optkey = {},
  volume = {23},
  number = {5},
  pages = {1655--1682},
  note = {},
  annote = {refereed article,256D},
  ps = { 1},
  pdf = { 1},
  abstract = { In this paper we present a meshfree discretization
		  technique based only on a set of irregularly spaced points
		  and the partition of unity approach. We focus on the cover
		  construction and its interplay with the integration problem
		  arising in a Galerkin discretization. We present a
		  hierarchical cover construction algorithm and a reliable
		  decomposition quadrature scheme. Here, we decompose the
		  integration domains into disjoint cells on which we employ
		  local sparse grid quadrature rules to improve computational
		  efficiency. The use of these two schemes already reduces
		  the operation count for the assembly of the stiffness
		  matrix significantly. Now, the overall computational costs
		  are dominated by the number of the integration cells. We
		  present a regularized version of the hierarchical cover
		  construction algorithm which reduces the number of
		  integration cells even further and subsequently improves
		  the computational efficiency. In fact, the computational
		  costs during the integration of the nonzeros of the
		  stiffness matrix are comparable to that of a finite element
		  method, yet the presented method is completely independent
		  of a mesh. Moreover, our method is applicable to general
		  domains and allows for the construction of approximations
		  of any order and regularity. }