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Published in:
2005 | OriginalPaper | Chapter
Direct Schur Complement Method by Hierarchical Matrix Techniques
Authors : Wolfgang Hackbusch, Boris N. Khoromskij, Ronald Kriemann
Published in: Domain Decomposition Methods in Science and Engineering
Publisher: Springer Berlin Heidelberg
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The goal of this paper is the construction of a data-sparse approximation to the Schur complement on the interface corresponding to FEM and BEM approximations of an elliptic equation by domain decomposition. Using the hierarchical (
ℌ
-matrix) formats we elaborate the
approximate Schur complement inverse
in an explicit form. The required cost
$$\mathcal{O}$$
(
N
Γ
log
q
N
Γ
) is almost linear in
N
Γ
— the number of degrees of freedom on the interface. As input, we use the Schur complement matrices corresponding to subdomains and represented in the
ℌ
-matrix format. In the case of piecewise constant coefficients these matrices can be computed via the BEM representation with the cost
$$\mathcal{O}$$
(
N
Γ
log
q
N
Γ
), while in the general case the FEM discretisation leads to the complexity
O
(
N
Ω
log
q
N
Ω
).