Leydecker, Florian; Maischak, Matthias; Stephan, Ernst P.; Teltscher, Matthias A \(p\)-hierarchical error estimator for a fem-bem coupling of an eddy current problem in \(\mathbb R^3\). (English) Zbl 1316.65103 J. Korean Soc. Ind. Appl. Math. 17, No. 3, 139-170 (2013). Summary: We extend a \(p\)-hierarchical decomposition of the second degree finite element space of Nédélec for tetrahedral meshes in three dimensions given in [R. Beck et al., in: ENUMATH 99. Numerical mathematics and advanced applications. Proceedings of the 3rd European conference, Jyväskylä, Finland, July 26–30, 1999. Singapore: World Scientific. 110–120 (2000; Zbl 0970.78006)] to meshes with hexahedral elements, and derive \(p\)-hierarchical decompositions of the second degree finite element space of Raviart-Thomas in two dimensions for triangular and quadrilateral meshes. After having proved stability of these subspace decompositions and requiring certain saturation assumptions to hold, we construct a local a posteriori error estimator for fem and bem coupling of a time-harmonic electromagnetic eddy current problem in \(\mathbb R^3\) . We perform some numerical tests to underline reliability and efficiency of the estimator and test its usefulness in an adaptive refinement scheme. Cited in 2 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs 78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory 78M15 Boundary element methods applied to problems in optics and electromagnetic theory Keywords:hierarchical based a posteriori error estimation; Maxwell’s equations; eddy currents; FEM-BEM coupling; Nédélec’s elements; Raviart-Thomas elements Citations:Zbl 0970.78006 Software:maiprogs PDFBibTeX XMLCite \textit{F. Leydecker} et al., J. Korean Soc. Ind. Appl. Math. 17, No. 3, 139--170 (2013; Zbl 1316.65103) Full Text: DOI