A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

Abstract

To deal with the divergence-free constraint in a double curl problem: curl µ −1 curl u = f and div εu = 0 in Ω, where µ and ε represent the physical properties of the materials occupying Ω, we develop a δ-regularization method: curl µ −1 curl u δ + δεu δ = f to completely ignore the divergence-free constraint div εu = 0. It is shown that u δ converges to u in H(curl ; Ω) norm as δ → 0. The edge finite element method is then analyzed for solving u δ. With the finite element solution u δ,h , quasi-optimal error bound in H(curl ; Ω) norm is obtained between u and u δ,h , including a uniform (with respect to δ) stability of u δ,h in H(curl ; Ω) norm. All the theoretical analysis is done in a general setting, where µ and ε may be discontinuous, anisotropic and inhomogeneous, and the solution may have a very low piecewise regularity on each material subdomain Ω j with u, curl u ∈ (H r (Ω j)) 3 for some 0 < r < 1, where r may be not greater than 1/2. To establish the uniform stability and the error bound for r ≤ 1/2, we have respectively developed a new theory for the K h ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results presented confirm the theory.

Edge Element Method for Optimal Control of Stationary Maxwell System with Gauss Law

“…which holds due to the connectedness of Γ (see, e.g., [1]). Also, the Ladyzhenskaya-Babuška-Brezzi (LBB) condition (7) sup…”

“…This work is mainly motivated by two numerical challenges we have discussed above. In order to treat these two numerical difficulties, a novel edge element method was proposed recently for solving the stationary Maxwell equations (1) in [7] (for the case ρ = 0) and in [5] (for the general charge density). In contrast to most existing edge element schemes (see, e.g., [4,6,9,18]), the new method does not involve any saddlepoint structure.…”

“…Instead, it requires only the resolution of a symmetric and positive definite system, which can be solved by efficient and nearly optimal preconditioners, including [8,10,13,22]. More importantly, the new edge element method ensures the optimal convergence rate [5,7] and strong convergence of the Gauss law in some proper norm [5]. It is natural to ask whether the edge element method [5,7] with all its advantages can be extended and transferred to the optimal control problem (P).…”

“…More importantly, the new edge element method ensures the optimal convergence rate [5,7] and strong convergence of the Gauss law in some proper norm [5]. It is natural to ask whether the edge element method [5,7] with all its advantages can be extended and transferred to the optimal control problem (P). This is exactly the main objective of the current work.…”

Some remarks on a class of inverse problems related to the parabolic approximation to the Maxwell equations: a controllability approach

“…The operator M is obtained as a variational operator defined by the quadratic form a : V V ! R, defined by a.u, v/ D R O 1 curlu curlvdx, which is symmetric, because is a symmetric matrix and satisfies the V-coercivity condition (see [14]) on account of the Poincaré-Friedrichs inequality (see [24]) jjujj .…”