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21st GAMM-Seminar Leipzig on
Robust Fast Solvers

Max-Planck-Institute for Mathematics in the Sciences
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  21st GAMM-Seminar
January, 26th-28th, 2005
 
     
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  Abstract Boris Khoromskij, Thu, 15.00-15.30 Previous Contents Next  
  H-Matrix Preconditioning for Domain Decomposition with Brick-and-Mortar Coefficients
Boris Khoromskij (MPI Leipzig)

A class of hierarchical matrices (tex2html_wrap_inline13-matrices) allows the data-sparse approximation to integral and more general nonlocal operators (say, the Poincaré-Steklov operators) with almost linear cost. We consider the tex2html_wrap_inline15-matrix-based approximation to the Schur complement on the interface [2] corresponding to the FEM discretisation of an elliptic operator tex2html_wrap_inline17 with jumping coefficients in tex2html_wrap_inline19. As with the standard Schur complement domain decomposition methods, we split the elliptic inverse tex2html_wrap_inline21 as a sum of local inverses associated with subdomains (this can be implemented in parallel), and the corresponding Poincaré-Steklov operator on the interface.

Using the hierarchical formats based on either standard or weakened admissibility criteria (cf. [1]) we elaborate the approximate Schur complement inverse in an explicit form that is proved to have a linear-logarithmic cost tex2html_wrap_inline23, where tex2html_wrap_inline25 is the number of degrees of freedom on the interface. The tex2html_wrap_inline27-matrix-based preconditioner can be also applied.

Numerical tests confirm the almost linear cost of our parallel direct Schur complement method. In particular, we consider examples with the brick-and-mortar structure of coefficients arising in the skin modeling problem.

[1] W. Hackbusch, B.N. Khoromskij and R. Kriemann. Hierarchical Matrices Based on Weak Admissibility Criterion. Computing 73 (2004), 207-243.

[2] W. Hackbusch, B.N. Khoromskij and R. Kriemann. Direct Schur Complement Method by Domain Decomposition based on Hierarchical Matrix Approximation. Preprint MPI MIS No. 25, Leipzig, 2004 (submitted).
 

 
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28.01.2005 Impressum
 
Concept, Design and Realisation
[O->]Jens Burmeister (Uni Kiel), Kai Helms (MPI Leipzig)
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