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We consider a Johnson-N\'ed\'elec FEM-BEM coupling, which is a direct andnon-symmetric coupling of finite and boundary element methods, in order tosolve interface problems for the magnetostatic Maxwell's equations with themagnetic vector potential ansatz. In the FEM-domain, equations may benon-linear, whereas they are exclusively linear in the BEM-part to guaranteethe existence of a fundamental solution. First, the weak problem is formulatedin quotient spaces to avoid resolving to a saddle point problem. Second, weestablish in this setting well-posedness of the arising problem using theframework of Lipschitz and strongly monotone operators as well as a stabilityresult for a special type of non-linearity, which is typically considered inmagnetostatic applications. Then, the discretization is performed in theisogeometric context, i.e., the same type of basis functions that are used forgeometry design are considered as ansatz functions for the discrete setting. Inparticular, NURBS are employed for geometry considerations, and B-Splines,which can be understood as a special type of NURBS, for analysis purposes. Inthis context, we derive a priori estimates w.r.t. h-refinement, and point outto an interesting behavior of BEM, which consists in an amelioration of theconvergence rates, when a functional of the solution is evaluated in theexterior BEM-domain. This improvement may lead to a doubling of the convergencerate under certain assumptions. Finally, we end the paper with a numericalexample to illustrate the theoretical results, along with a conclusion and anoutlook.

The Johnson-N\'ed\'elec FEM-BEM Coupling for magnetostatic problems in the isogeometric framework