Robust Algebraic Multigrid Methods and their Parallelization
  - Results

Boundary Element Methods

  • Collocation method: Resulting system matrices are dense and non-symmetric.
  • Galerkin method: System matrices symmetric but also dense.

  • Advantage: Only the boundary of the compuational domain has to be discretized - that reduces the number of unknowns.

       
     

  • Single layer potential operator: Pseudo-differential operator of order minus one -> common smoothing techniques are not appropriate.

  • Idea: [1] Bramble, Leyk and Pasciak: The Analysis of Multigrid Algorithms for Pseudo-Differential Operators of Order Minus One. Math. Comp. Vol. 63, Issue 208, 1994, p.461-478.
  • Hypersingular operator: large eigenvalues correspond to highly oscillating eigenfunctions, therefore common smoothing methods can be applied.
  • Approximating of dense BEM-matrices by the Adaptive Cross Approximation method (ACA)

    The domain will be subdivided into clusters and the interaction of one cluster to another will be classified into a far-field and a near-field region. Depending on this distance condition the corresponding matrix block entries will be calculated directly (green) or will be approximated by some low-rank matrices (red).
  • The total cost of the AMG algorithm will be reduced to almost O(N_h) arithmetical operations.
  • Memory demand also grows in O(N_h), which is important for the realization of finer discretization.

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    Numerical experiments in 2D confirm the efficiency of our approach very accurately. In the case of 3D domains it is a pretty difficult task to realize the Laplace-Beltrami operator on the surface, which is essential for constructing a proper smoother according to Bramble, Leyk and Pasciak.

    Publications


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