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-1curl 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-1curl uδ + δεuδ = f, to completely ignore the divergence-free constraint div εu = 0. We show 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, a quasioptimal error bound in the H(curl ;Ω) norm is obtained between u and uδ,h, including a uniform (with respect to δ) stability of uδ,h in the 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, curl u € (Hr(Ωj ))3 for some 0 < r < 1, where r may not be 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 confirm the theory. © 2012 Society for Industrial and Applied Mathematics.