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), [1-8]. 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
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
partition of unity method. We have developed a cheap multilevel solver which uses (localized)
L^2-projections for the interlevel transfers and a block-smoother 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.289--314.
I. Babuska and J.M. Melenk,
The Partition of Unity Method, Int. J. Numer. Meth. Engrg., 40 (1997), pp. 727--758.
M. Griebel and M. A. Schweitzer, A Particle-Partition of
Unity Method for the Solution of Elliptic, Parabolic and
Hyperbolic PDE, SIAM J. Sci. Comput., 22 (2000), pp. 853--890.
A Particle-Partition of Unity Method-Part II: Efficient Cover
Construction and Reliable Integration, SIAM J. Sci. Comput., 23 (2002), pp. 1655--1682.
M. Griebel and M. A. Schweitzer, A Particle-Partition of Unity Method-Part III: A Multilevel
Solver, SIAM J. Sci. Comput., 24 (2002), pp. 377--409.
M. Griebel and M. A. Schweitzer,
A Particle-Partition of Unity Method-Part 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.~161--192.
M. Griebel and M. A. Schweitzer , A
Particle-Partition of Unity Method--Part V: Boundary
Conditions, in Geometric Analysis and Nonlinear Partial Differential Equations},
S. Hildebrandt and H. Karcher, eds., Springer,
2002, pp. 517--540.
M. A. Schweitzer, A Parallel Multilevel Partition of
Unity Method, vol. 29 of Lecture Notes in Computational Science and Engineering, Springer,