A wirebasket preconditioner for the mortar boundary element method
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A wirebasket preconditioner for the mortar boundary element method 1 · Norbert Heuer1 ¨ Thomas Fuhrer
Received: 1 June 2015 / Accepted: 31 March 2017 © Springer Science+Business Media New York 2017
Abstract We present and analyze a preconditioner of the additive Schwarz type for the mortar boundary element method. As a basic splitting, on each subdomain we separate the degrees of freedom related to its boundary from the inner degrees of freedom. The corresponding wirebasket-type space decomposition is stable up to logarithmic terms. For the blocks that correspond to the inner degrees of freedom standard preconditioners for the hypersingular integral operator on open boundaries can be used. For the boundary and interface parts as well as the Lagrangian multiplier space, simple diagonal preconditioners are optimal. Our technique applies to quasiuniform and non-uniform meshes of shape-regular elements. Numerical experiments on triangular and quadrilateral meshes confirm theoretical bounds for condition and MINRES iteration numbers. Keywords Non-conforming boundary elements · Hypersingular operator · Domain decomposition · Mortar method · Preconditioner · Additive Schwarz method Mathematics Subject Classification (2010) 65N38 · 65N55 · 65F08
1 Introduction In recent years, different variants of the non-conforming boundary element method (BEM) have been developed. The underlying boundary integral equation is of the first Communicated by: Karsten Urban Thomas F¨uhrer
[email protected] 1
Facultad de Matem´aticas, Pontificia Universidad Cat´olica de Chile, Avenida Vicu˜na Mackenna 4860, Santiago, Chile
T. F¨uhrer, N. Heuer
kind with hypersingular operator. Non-conformity refers to the presence of discontinuous basis functions. (Note that, in the case of integral equations of the second kind or first kind equations with weakly-singular operator, conforming basis functions can be discontinuous.) The first paper on non-conforming BEM considers a Lagrangian multiplier to deal with the homogeneous boundary condition on open surfaces [8]. This technique was extended in [9] to domain decomposition approximations, and is usually referred to as mortar method. First results for non-conforming approximation of indefinite hypersingular operators can be found in [12]. In this paper we study preconditioners for the mortar BEM presented in [9]. These are the first results on preconditioning techniques for linear systems stemming from non-conforming boundary elements. Advantages of non-conforming BEM include the more flexible discretization for complicated surfaces, which also holds for the p-version, see [11], and a more flexible adaptivity, see [4, 10]. The analysis of non-conforming BEM methods becomes partially simpler, because the discrete trace spaces are defined in L2 . This is also helpful when analyzing preconditioners, as illustrated in this paper. In particular, the use of Lagrangian multipliers allows for a simple preconditioning of the additional terms (L2 bilinear forms), i.e., a diagonal scaling. On the other han
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