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Robust coupling of DPG and BEM for a singularly perturbed transmission problem. (English) Zbl 1405.65148

A transmission problem is considered. Precisely, a singularly perturbed reaction diffusion equation is given on a bounded domain and the Laplacian is given in the exterior domain. Both domains are coupled by standard transmission conditions. The authors establish a DPG (discontinuous Petrov-Galerkin method with optimal test functions) which is coupled with a BEM (Galerkin boundary element method). Further, they show the robustness for the field variables in the so-called balanced norms. Their coupling scheme depends on the one given in the paper [Math. Comput. 86, No. 307, 2261–2284 (2017; Zbl 1364.65248)] by T. Führer et al. and is adapted to the singularly perturbed case by using the scheme of N. Heuer and M. Karkulik [“DPG method with optimal test functions for a transmission problem”, Comput. Math. Appl. 70, No. 5, 1504–1518 (2015; doi:10.1016/j.camwa.2015.06.032)]. It is shown that the optimal test functions in the new method have to be computed only locally. Various numerical experiments in two dimensions are given to support the theory.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N38 Boundary element methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35B25 Singular perturbations in context of PDEs
35J25 Boundary value problems for second-order elliptic equations

Citations:

Zbl 1364.65248

Software:

HILBERT
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

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