HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-981030-A

Introduction

In the recent years, growing computer resources have allowed to perform three-dimensional electro-magnetic field calculations [15,16,18]. One of the essential problems in 3D electromagnetics is the necessity of gauging. An additional gauging condition has to be imposed on the magnetic vector potential to ensure its uniqueness. There are different gauging strategies considered (see, e.g., [12,13,14]). Sometimes, even an ungauged formulation (that may result in an indefinite system of linear equations) is solved by iterative methods [11]. Another strategy uses a well-defined iteration process to approximate the solution of an ungauged formulation [17]. >From the mathematical point of view, we have to choose appropriate spaces. In [9], the first author proposed a mixed variational formulation in $\left( H^1_0(\Omega)^3, L^2(\Omega) \right)$. Indeed, existence and uniqueness results as well as discretization error estimates for classes of simple nodal elements (e.g. the Mini element) have been proved. Unfortunately, the mixed variational formulation does not correspond completely with the magnetostatic problem. Moreover, it has been stated in [5] that magnetic field problems should rather be formulated in ${H}({\rm rot}) \cap {H}({\rm div})$ instead of $H^1$. Therefore, the aim of the present paper is to establish a mixed variational formulation for linear and nonlinear magnetostatic problems in the space ${H}({\rm rot}) \cap {H}({\rm div})$. Here, we formulate the flux-surface boundary condition as a Dirichlet condition with respect to the tangential components of the vector potential only, and a Neumann condition with respect to its normal component. The Dirichlet condition has been proposed, e.g., in [2], whereas the Neumann condition was not explicitly stated therein, but implicitly contained in the formulation. Thus, we set up the problem in the space ${H_0}({\rm rot}) \cap {H}({\rm div})$. Other boundary conditions can be formulated in a similar manner. As in the 2D case, cf. [8], additional assumptions with respect to the coefficient allow a rigorous analysis of nonlinear problems. In particular, the coefficient depends on the solution itself. Assumptions which are physically meaningful and that guarantee unique solveability are stated. The rest of the paper is organized as follows. In Section 2, we discuss the physical model. We present the variational formulation for linear magnetostatic problems in Section 3 and give analytical results. Section 4 is devoted to the results for nonlinear magnetostatic problems. Finally, we add some concluding remarks in Section 5.

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-981030-A