

Michael Griebel and Marc Alexander Schweitzer : A multilevel Solver for Partition of Unity Methods
In this talk we focus on the efficient solution of linear systems arising from the discretization of an
elliptic partial differential equation using a partition of unity method (PUM), [18]. A PUM is a
generalization of the finite element method (FEM) which does not rely on the availabilty of an
appropriate mesh and is hence referred to as a meshfree method .
The shape functions of a partition of unity method are products of piecewise rational partition of
unity functions phi_i with supp(phi_i)=\omega_i and higher order local approximation
functions \psi_i^n (usually a local polynomial of degree < p_i). Furthermore, they are
noninterpolatory. In a multilevel approach we not only have to cope with noninterpolatory 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
partition of unity method. We have developed a cheap multilevel solver which uses (localized)
L^2projections for the interlevel transfers and a blocksmoother to treat the local approximation
functions \psi_i^n for all n simultaneously. The convergence rate $\rho$ of this multilevel
solver is independent of the number and the distribution of the discretization points; yet $\rho$ is
slightly dependent on the local approximation orders p_i.

J.M. Melenk and I. Babuska, The Partition of Unity Finite
Element Method: Basic Theory and Applications, Comput. Meth. Appl.
Mech. Engrg., 139 (1996), pp.289314.

I. Babuska and J.M. Melenk,
The Partition of Unity Method, Int. J. Numer. Meth. Engrg., 40 (1997), pp. 727758.

M. Griebel and M. A. Schweitzer, A ParticlePartition of
Unity Method for the Solution of Elliptic, Parabolic and
Hyperbolic PDE, SIAM J. Sci. Comput., 22 (2000), pp. 853890.

A ParticlePartition of Unity MethodPart II: Efficient Cover
Construction and Reliable Integration, SIAM J. Sci. Comput., 23 (2002), pp. 16551682.

M. Griebel and M. A. Schweitzer, A ParticlePartition of Unity MethodPart III: A Multilevel
Solver, SIAM J. Sci. Comput., 24 (2002), pp. 377409.

M. Griebel and M. A. Schweitzer,
A ParticlePartition of Unity MethodPart IV: Parallelization,
in Meshfree Methods for Partial Differential Equations, M. Griebel
and M. A. Schweitzer, eds., vol. 26 of Lecture Notes in Computational Science
and Engineering, Springer, 2002, pp.~161192.

M. Griebel and M. A. Schweitzer , A
ParticlePartition of Unity MethodPart V: Boundary
Conditions, in Geometric Analysis and Nonlinear Partial Differential Equations},
S. Hildebrandt and H. Karcher, eds., Springer,
2002, pp. 517540.

M. A. Schweitzer, A Parallel Multilevel Partition of
Unity Method, vol. 29 of Lecture Notes in Computational Science and Engineering, Springer,
2003.
