Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients |
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Authors: | S A Sauter R Warnke |
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Affiliation: | 1. Institut für Mathematik, Universit?t Zürich, Winterthurerstr. 190, 8057, Zürich, Switzerland 2. msg systems ag, Robert-Bürkle-Stra?e 1, 85737, Ismaning, Germany
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Abstract: | In this paper, we will introduce composite finite elements for solving elliptic boundary value problems with discontinuous
coefficients. The focus is on problems where the geometry of the interfaces between the smooth regions of the coefficients
is very complicated.
On the other hand, efficient numerical methods such as, e.g., multigrid methods, wavelets, extrapolation, are based on a multi-scale
discretization of the problem. In standard finite element methods, the grids have to resolve the structure of the discontinuous
coefficients. Thus, straightforward coarse scale discretizations of problems with complicated coefficient jumps are not obvious.
In this paper, we define composite finite elements for problems with discontinuous coefficients. These finite elements allow
the coarsening of finite element spaces independently of the structure of the discontinuous coefficients. Thus, the multigrid
method can be applied to solve the linear system on the fine scale.
We focus on the construction of the composite finite elements and the efficient, hierarchical realization of the intergrid
transfer operators. Finally, we present some numerical results for the multigrid method based on the composite finite elements
(CFE–MG). |
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Keywords: | 35J20 65N15 65N30 |
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