11th GAMM-Workshop on

Multigrid and Hierarchic Solution Techniques

 

     
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  A. Almendral  
  M. Bader  
  R. Bank  
  M. Bebendorf  
  S. Beuchler  
  D. Braess  
  C. Douglas  
  L. Grasedyck  
  B. Khoromskij  
  R. Kornhuber  
  B. Krukier  
  U. Langer  
  C. Oosterlee  
  G. Pöplau  
  A. Reusken  
  J. Schöberl  
  M.A. Schweitzer  
  S. Serra Capizzano  
  B. Seynaeve  
  D. Smits  
  O. Steinbach  
  R. Stevenson  
  M. Wabro  
  R. Wienands  
 
     
  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.
  1. 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.
  2. I. Babuska and J.M. Melenk, The Partition of Unity Method, Int. J. Numer. Meth. Engrg., 40 (1997), pp. 727--758.
  3. 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.
  4. A Particle-Partition of Unity Method-Part II: Efficient Cover Construction and Reliable Integration, SIAM J. Sci. Comput., 23 (2002), pp. 1655--1682.
  5. 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.
  6. 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.
  7. 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.
  8. M. A. Schweitzer, A Parallel Multilevel Partition of Unity Method, vol. 29 of Lecture Notes in Computational Science and Engineering, Springer, 2003.

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