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 |
Keywords:
Maxwell problem; mixed formulation; edge element; flux element; coercivity; \(L^2\) div-convergenceReferences:
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