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For the first time, a nonlinear interface problem on an unbounded domain with nonmonotone set-valued transmission conditions is analyzed. The investigated problem involves a nonlinear monotone partial differential equation in the interior domain and the Laplacian in the exterior domain. Such a scalar interface problem models nonmonotone frictional contact of elastic infinite media. The variational formulation of the interface problem leads to a hemivariational inequality, which lives on the unbounded domain, and so cannot be treated numerically in a direct way. By boundary integral methods the problem is transformed and a novel hemivariational inequality (HVI) is obtained that lives on the interior domain and on the coupling boundary, only. Thus for discretization the coupling of finite elements and boundary elements is the method of choice. In addition smoothing techniques of nondifferentiable optimization are adapted and the nonsmooth part in the HVI is regularized. Thus we reduce the original variational problem to a finite dimensional problem that can be solved by standard optimization tools. We establish not only convergence results for the total approximation procedure, but also an asymptotic error estimate for the regularized HVI. [ABSTRACT FROM AUTHOR]

We propose and analyze a new coupling procedure for the Hybridizable Discontinuous Galerkin Method with Galerkin Boundary Element Methods based on a double layer potential representation of the exterior component of the solution of a transmission problem. We show a discrete uniform coercivity estimate for the non-symmetric bilinear form and prove optimal convergence estimates for all the variables, as well as superconvergence for some of the discrete fields. Some numerical experiments support the theoretical findings. [ABSTRACT FROM AUTHOR]

We consider a transmission problem consisting of a singularly perturbed reaction diffusion equation on a bounded domain and the Laplacian in the exterior, connected through standard transmission conditions. We establish a DPG scheme coupled with Galerkin boundary elements for its discretization, and prove its robustness for the field variables in so-called balanced norms . Our coupling scheme is the one from Führer et al., (2016), adapted to the singularly perturbed case by using the scheme from Heuer and Karkulik (2015). Essential feature of our method is that optimal test functions have to be computed only locally. We report on various numerical experiments in two dimensions. [ABSTRACT FROM AUTHOR]

We propose and analyze a numerical method to solve an elliptic transmission problem in full space. The method consists of a variational formulation involving standard boundary integral operators on the coupling interface and an ultra-weak formulation in the interior. To guarantee the discrete inf–sup condition, the system is discretized by the DPG method with optimal test functions. We prove that the principal unknowns are approximated quasi-optimally. Numerical experiments for problems with smooth and singular solutions confirm optimal convergence orders. [ABSTRACT FROM AUTHOR]

Operator preconditioning offers a general recipe for constructing preconditioners for discrete linear operators that have arisen from a Galerkin approach. The key idea is to employ matching Galerkin discretizations of operators of complementary mapping properties. If these can be found, the resulting preconditioners will be robust with respect to the choice of the bases for trial and test spaces. I survey the application of operator preconditioning to finite elements and boundary elements. [Copyright &y& Elsevier]

Abstract: We analyze a least squares formulation for the numerical solution of second-order lineartransmission problems in two and three dimensions, which allow jumps on the interface. In a bounded domain the second-order partial differential equation is rewritten as a first-order system; the part of the transmission problem which corresponds to the unbounded exterior domain is reformulated by means of boundary integral equations on the interface. The least squares functional is given in terms of Sobolev norms of order -1 and of order 1/2. These norms are computed by approximating the corresponding inner products using multilevel preconditioners for a second-order elliptic problem in a bounded domain Ω and for the weakly singular integral operator of the single layer potential on its boundary ∂Ω. As preconditioners we use both multigrid and BPX algorithms, and the preconditioned system has bounded or mildly growing condition number. Numerical experiments confirm our theoretical results. [Copyright &y& Elsevier]

Abstract: This paper deals with the symmetric linear systems of equations arising in the Galerkin boundary element method. In particular, we consider the application of several variants of the conjugate-gradient method to these systems and present some numerical results which shows that a simple preconditioning strategy can lead to significant improvements in solution times. [Copyright &y& Elsevier]

The dual reciprocity method (DRM) is a technique to transform the domain integrals that appear in the boundary element method into equivalent boundary integrals. In this approach, the nonlinear terms are usually approximated by an interpolation applied to the convective terms of the Navier-Stokes equations. In this paper, we introduce a radial basis function interpolation scheme for the velocity field, that satisfies the continuity equation (mass conservative). The proposed method performs better than the classical interpolation used in the DRM approach to represent such a field. The new scheme together with a subdomain variation of the dual reciprocity method allows better approximation of the nonlinear terms in the Navier-Stokes equations. [Copyright &y& Elsevier]