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

TL;DR

This paper develops a non-symmetric FEM-BEM coupling method for magnetostatic problems using isogeometric analysis with NURBS and B-Splines, providing well-posedness, stability, and convergence results.

Contribution

It introduces a novel isogeometric FEM-BEM coupling framework for magnetostatics, including theoretical analysis and improved convergence insights.

Findings

Well-posedness established using Lipschitz and monotone operator theory.

A priori error estimates derived for h-refinement.

Potential doubling of convergence rate for certain functionals in BEM.

Abstract

We consider a Johnson-N\'ed\'elec FEM-BEM coupling, which is a direct and non-symmetric coupling of finite and boundary element methods, in order to solve interface problems for the magnetostatic Maxwell's equations with the magnetic vector potential ansatz. In the FEM-domain, equations may be non-linear, whereas they are exclusively linear in the BEM-part to guarantee the existence of a fundamental solution. First, the weak problem is formulated in quotient spaces to avoid resolving to a saddle point problem. Second, we establish in this setting well-posedness of the arising problem using the framework of Lipschitz and strongly monotone operators as well as a stability result for a special type of non-linearity, which is typically considered in magnetostatic applications. Then, the discretization is performed in the isogeometric context, i.e., the same type of basis functions that are used for geometry design are considered as ansatz functions for the discrete setting. In particular, NURBS are employed for geometry considerations, and B-Splines, which can be understood as a special type of NURBS, for analysis purposes. In this context, we derive a priori estimates w.r.t. h-refinement, and point out to an interesting behavior of BEM, which consists in an amelioration of the convergence rates, when a functional of the solution is evaluated in the exterior BEM-domain. This improvement may lead to a doubling of the convergence rate under certain assumptions. Finally, we end the paper with a numerical example to illustrate the theoretical results, along with a conclusion and an outlook.