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: curlμ−1curlu = 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: curlμcurluδ + δεuδ = f to completely ignore the divergence-free constraint div εu = 0. It is shown that uδ converges to u in H(curl ; Ω) norm as δ → 0. The edge finite element method is then analyzed for solving uδ. With the finite element solution uδ,h, quasi-optimal error bound in H(curl ; Ω) norm is obtained between u and uδ,h, including a uniform (with respect to δ) stability of uδ,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 Ωj with u, curlu ∈ (H(Ωj)) 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 Kh ellipticity (related to mixed methods) and a new theory for the Fortin interpolation operator. Numerical results presented confirm the theory.