An E-based mixed FEM and BEM coupling for a time-dependent eddy current problem

Well-posedness for a family of degenerate parabolic mixed equations

“…The abstract theory developed in Section 2 can be used to study the continuous case of the mixed formulations proposed for the eddy current model in [15,1,2,6], but we will only focus in the formulations studied in [1] and [6]. In both cases was deduced the formulation of the eddy current problem in terms of a time-primitive of the electric field, i.e., in terms of the unknown u given by u(x, t) := t 0 E(x, s)ds.…”

“…The idea of combining mixed variational formulation with time dependent evolution problem is not new, starting from the 1980s, for instance, [14,4], and more recently, to study a dynamics fluid problem [7] and electromagnetic applications [1,2,6,15]. Bernardi & Raugel [4] have introduced some abstract framework for an usual (non-degenerate) mixed parabolic equation inspired on a dynamics fluid model called the Stokes problem.…”

“…where X, Y and M are real Hilbert spaces with the embedding X ⊆ Y is continuous and dense, D ′ (0, T ; M ) is the space of M -valued distributions on [0, T ], (•, •) Y is the inner product on Y , a : X × X → R, b : X × M → R are continuous bilinear forms, u 0 ∈ Y and f ∈ L 2 (0, T ; X ′ ). However, it is not possible to apply this abstract theory to the formulations arising from electromagnetic problems studied in [1,2,6], because in these cases the first term inside of the timederivative is not an inner product in the whole space Y , namely that problems are degenerate. Similarly, that abstract theory can not be applied to the formulation analyzed in [15], because in this formulation the right-hand term of the second equation in (1.1) is non-zero.…”

Fully discrete finite element approximation for a family of degenerate parabolic mixed equations

“…For instance, in [9] the authors have introduced some abstract framework for that kind of problems. However, mixed formulations arising from electromagnetic problems (see, for instance [1,2,18]), can not fit in that aforementioned theory, because in these cases the first term inside of the time-derivative is not an inner product, which implies that the resulting problem is degenerate.…”

“…where X, Y and M are real Hilbert spaces with the imbedding X ⊆ Y is continuous and dense, R : Y → Y ′ , A : X → X ′ are linear and bounded operators, b : X × M → R is a bounded bilinear form, f ∈ L 2 (0, T ; X ′ ) and g ∈ L 2 (0, T ; M ′ ). This kind of problems appears in several applications, for instance, to the approximation of the heat equation by means of Raviart-Thomas (RT) elements (see [26]), to the fluid dynamic equations (see [9] and [10]), and to electromagnetic applications (see [1,2,18,28]). Sufficient conditions to the well-posedness of Problem (1.1) with an appropriate initial condition, are given in the recent paper [3].…”