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A coercive mixed formulation for the generalized Maxwell problem. (English) Zbl 1477.65208

Summary: A coercive mixed variational formulation on \(H_0(\mathbf{curl};\Omega)\times H(\operatorname{div};\Omega)\) is proposed for the generalized Maxwell problem which typically arises from computational electromagnetism. The mixed variables are the electric field and a pseudo electric displacement field. The well-posedness of the mixed variational problem is proven in the general settings (multiply connected domain of Lipschitz-continuous boundary with a number of connected components, filling with discontinuous, anisotropic and inhomogeneous media); more importantly, the coercivity is established. A conforming finite element discretization is further proposed, where the electric field is approximated by \(H(\mathbf{curl};\Omega)\)-conforming edge element while the pseudo electric displacement field by \(H(\operatorname{div};\Omega)\)-conforming flux element. Error estimates are obtained, and in particular, the method produces an \(L^2\) curl-convergent approximation and more importantly, an \(L^2\) div-convergent approximation for the solution.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
35Q61 Maxwell equations
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