Multilevel additive Schwarz method for the ℎ-𝑝 version of the Galerkin boundary element method

Heuer, Norbert; Stephan, Ernst; Tran, Thanh

doi:10.1090/S0025-5718-98-00926-0

Multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method

Authors: Norbert Heuer, Ernst P. Stephan and Thanh Tran
Journal: Math. Comp. 67 (1998), 501-518
MSC (1991): Primary 65N55, 65N38
DOI: https://doi.org/10.1090/S0025-5718-98-00926-0
MathSciNet review: 1451325
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Abstract: We study a multilevel additive Schwarz method for the $h$-$p$ version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the $h$-$p$ version with geometric meshes converges exponentially fast in the energy norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns $M$. We prove that the condition number $\kappa (P)$ of the multilevel additive Schwarz operator behaves like $O(\sqrt {M}\log ^2M)$. As a direct consequence of this we also give the results for the $2$-level preconditioner and also for the $h$-$p$ version with quasi-uniform meshes. Numerical results supporting our theory are presented.

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