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**1 - 2**of**2**### Two-level Schwarz Methods for Indefinite Integral Equations

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"... this paper we consider additive Schwarz preconditioners for indefinite linear systems arising from the h-version of the boundary element method (BEM) for solving Helmholtz problems. Here we extend the approach introduced by Cai and Widlund [CW92] for finite element discretizations to boundary elemen ..."

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this paper we consider additive Schwarz preconditioners for indefinite linear systems arising from the h-version of the boundary element method (BEM) for solving Helmholtz problems. Here we extend the approach introduced by Cai and Widlund [CW92] for finite element discretizations to boundary element discretizations. We report on two-level methods applied to the h-version of the Galerkin method for weakly singular and hypersingular integral equations of the first kind on the interval \Gamma = (\Gamma1; 1). The Neumann problem for the Helmholtz equation in IR 2 n \Gamma leads to the hypersingular integral equation D k v(x) := \Gamma i 2 @ @n x Z \Gamma @ @n y [H 1 0 (kjx \Gamma yj)]v(y) ds y = g 1 (x); x 2 \Gamma: (1.1) Correspondingly the Dirichlet problem leads to the weakly singular integral equation V k /(x) = Z \Gamma H 1 0 (kjx \Gamma yj)/(y) ds y = g 2 (x); x 2 \Gamma: (1.2) There H 1 0 is the Hankel function of the first kind and of order zero, Imk 0, k 6= 0 and @ @n denotes the normal derivative on \Gamma

### Schwarz methods for graded meshes in 2D-BEM: Numerical results

"... this paper we present various numerical results for the additive and multiplicative Schwarz methods for the hypersingular integral equation of the first kind on nonuniform meshes, i.e. for the h-version of the Galerkin method on an algebraically graded mesh and the h-p-version on a geometrically ref ..."

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this paper we present various numerical results for the additive and multiplicative Schwarz methods for the hypersingular integral equation of the first kind on nonuniform meshes, i.e. for the h-version of the Galerkin method on an algebraically graded mesh and the h-p-version on a geometrically refined mesh. We consider the hypersingular integral equation Dv(x) := \Gamma