Magnetostatic and Electrostatic Problems in Inhomogeneous Anisotropic Media with Irregular Boundary and Mixed Boundary Conditions

Abstract

Magnetostatic and electrostatic problems with mixed boundary conditions are studied. The medium can have a nonsmooth boundary and very irregular physical properties due to inhomogeneity and anisotropy. The topological assumptions are general enough to meet the requirements of the engineering applications. Necessary and sufficient conditions for solvability are found and the set of the solutions is characterized. Moreover, uniqueness is recovered by means of a finite number of supplementary conditions which are equivalent to prescribing a finite number of suitably chosen fluxes or potentials. A functional framework in which other important problems of electromagnetics naturally fit is developed.

Mixed discontinuous Galerkin approximation of the Maxwell operator: The indefinite case

“…Let (u, p) be the analytical solution of (6)- (9), and let (u h , p h ) be the solution of (14) obtained with α ≥ α min > 0 and γ > 0. Then we have that…”

“…In this section we present a series of numerical experiments to highlight the practical performance of the mixed DG method introduced and analyzed in this article for the numerical approximation of the indefinite timeharmonic Maxwell equations (6)- (9). For simplicity, we restrict ourselves to two-dimensional model problems; additionally, we note that throughout this section we select the constants appearing in the interior penalty stabilization functions defined in (15) as follows: α = 10 2 and γ = 1.…”

Electrowetting of a 3D drop: numerical modelling with electrostatic vector fields

“…Indeed, the domain Ω under consideration is simply connected but, on the other hand, its boundary is not connected (it has two connected components Γ 0 and Γ L ). Following [15], E Ω can be expressed as…”

“…But, in Section 2.2, we provided a new approach for the characterization of the electrostatic field E Ω (compared to [15]). …”

Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements

“…, r, such that r k=1Ω k =Ω and all the coefficients of and µ (which may be globally discontinuous) are Lipschitz continuous in each of them. Notice that these assumptions will allow us to use results in [19] and [13].…”