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 {\bf \Large
 Domain Decomposition and Boundary Elements
 }\\[3mm]
 O.~Steinbach, \underline{W.~L.~Wendland}  \\
 Mathematisches Institut A, Universit\"at Stuttgart \\
 Pfaffenwaldring 57, 70569 Stuttgart, Germany \\
 e--mail: {\tt wendland@mathematik.uni-stuttgart.de}
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 Domain decomposition methods are mainly based on appropriate discretizations
 of the local Steklov--Poincar\'e operators. Since these mappings can
 be described by boundary integral equations boundary element methods
 are well suited for their discretization. Beside the well known
 symmetric approach we will focus in our talk on mixed or hybrid methods
 using single and double layer potentials only.
 Appropriate stability conditions and convergence results will be given.

 Since the discrete approximations of the Steklov--Poincar\'e operators
 are spectrally equivalent to the exact Galerkin stiffness matrix, we
 are able to construct efficient preconditioners to solve the discrete
 system by a Krylov subspace method in nearly optimal order.

 Combining the domain decomposition algorithm with a parallelization
 of the discretization process we can balance the numerical effort of
 numerical integration and the iterative solution of mixed boundary
 value problems.

 Applications will be given for three--dimensional problems involving
 a hybrid coupling with finite element methods based on Trefftz elements.

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