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BIT Numerical Mathematics
Erschienen in: BIT Numerical Mathematics 3/2012

01.09.2012

Stable multilevel splittings of boundary edge element spaces

verfasst von: Ralf Hiptmair, Shipeng Mao

Erschienen in: BIT Numerical Mathematics | Ausgabe 3/2012

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Abstract

We establish the stability of nodal multilevel decompositions of lowest-order conforming boundary element subspaces of the trace space \({\boldsymbol{H}}^{-\frac {1}{2}}(\operatorname {div}_{\varGamma },{\varGamma })\) of \({\boldsymbol{H}}(\operatorname {\bf curl},{\varOmega })\) on boundaries of triangulated Lipschitz polyhedra. The decompositions are based on nested triangular meshes created by uniform refinement and the stability bounds are uniform in the number of refinement levels.
The main tool is the general theory of P. Oswald (Interface preconditioners and multilevel extension operators, in Proc. 11th Intern. Conf. on Domain Decomposition Methods, London, 1998, pp. 96–103) that teaches, when stability of decompositions of boundary element spaces with respect to trace norms can be inferred from corresponding stability results for finite element spaces. \({\boldsymbol{H}}(\operatorname {\bf curl},{\varOmega })\)-stable discrete extension operators are instrumental in this.
Stable multilevel decompositions immediately spawn subspace correction preconditioners whose performance will not degrade on very fine surface meshes. Thus, the results of this article demonstrate how to construct optimal iterative solvers for the linear systems of equations arising from the Galerkin edge element discretization of boundary integral equations for eddy current problems.
Vorheriger Artikel Resolvent Krylov subspace approximation to operator functions
Nächster Artikel GMRES and the Arioli, Pták, and Strakoš parametrization

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Fußnoten
1
The phrase that a constant “depends on shape-regularity” means that this constant may be a function of C s from (4.1).
 
2
The phrase that a constant “depends on quasi-uniformity” means that this constant may be a function of C u from (4.2).
 
3
Computations done in MATLAB using dense matrices and the cond() command. For want of matrix compression no further refinement was possible.
 
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Metadaten
Titel
Stable multilevel splittings of boundary edge element spaces
verfasst von
Ralf Hiptmair
Shipeng Mao
Publikationsdatum
01.09.2012
Verlag
Springer Netherlands
Erschienen in
BIT Numerical Mathematics / Ausgabe 3/2012
Print ISSN: 0006-3835
Elektronische ISSN: 1572-9125
DOI
https://doi.org/10.1007/s10543-012-0369-1

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