11th GAMM-Workshop on

Multigrid and Hierarchic Solution Techniques


  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  
  Ulrich Langer : Algebraic Multigrid Preconditioners for Boundary Element Matrices

Efficient preconditioners for boundary element (BE) matrices are needed in many application. In the primal as well in the dual boundary element domain decomposition (DD) methods preconditioners for the discrete single layer potential integral operator, arising from the BE approximation of interior or exterior Dirichlet boundary value problems, are required. In the primal DD method, preconditioners for the assembled discrete hypersingular integral operator are needed. Geometric and algebraic multigrid techniques based on sparse approximations of the corresponding boundary element matrices are a powerful technique for the construction of robust and at least almost optimal preconditioners for these BE matrices. In this talk we present new algebraic multigrid (AMG) preconditioners for sparse boundary element matrices arising from the Adaptive-Cross-Approximation (ACA) of dense boundary element matrices. As model problem we consider the single layer potential integral equation resulting from the interior Dirichlet boundary value problem for the Laplace equation. The standard collocation, or Galerkin boundary element discretization leads to fully populated system matrices which require O(N2) complexity for the memory and the matrix-by-vector multiplication, where N denotes the number of boundary unknowns. Sparse approximations such as ACA reduce this complexity to O(N) up to some polylogarithmical factor. Since the single layer potential operator is a pseudodifferential operator of the order -1, the resulting boundary element matrices are ill-conditioned. Iterative solvers dramatically suffers from this property for growing N. Our AMG preconditioners avoid this dramatical grow in the number of iterations and lead to almost optimal solvers with respect to the total complexity. This behaviour is confirmed by our numerical experiments. We mention that our AMG preconditioners use only single grid information provided by the usual mesh data and by the ACA anyway. Analogous results have been obtained for the hypersingular operator.
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