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A mixed method for approximating Maxwell’s equations. (English) Zbl 0742.65091

The author analyzes a semi-mixed finite element approximation to the time-dependent Maxwell equations for the electric and magnetic fields on a bounded smooth domain. Assuming the existence of a solution of the initial-boundary value problem, a weak formulation is obtained so that the variational problem might be used to prove the existence of a weak solution. One advantage of weak formulation is that boundary conditions are enforced naturally which is also used to construct a numerical scheme.

The goal is to analyze discrete versions of the weak formulation to approximate the electric and magnetic fields. First of all the semidiscrete Maxwell system in weak form is considered and estimates for $\left(E-{E}^{h}\right)\left(t\right)$ and $\left(H-{H}^{h}\right)\left(t\right)$ in the ${L}^{2}\left({\Omega }\right)$ norm are proved. First an estimate for a general class of spaces are provided and then it is shown how the general results may be used to prove error estimates when approximations are constructed using discontinuous piece- wise polynomials for the electric field and curl conforming elements of Nédélec type for the magnetic field.

The first estimate is very general allowing curved or polygonal domains and the full Maxwell system. However, although the estimate is of optimal order in $h$, it is not optimal in the regularity required for a continuous solution. Subsequently a refined analysis under special circumstances is presented. In particular, it is assumed that the domain is smooth and convex, that $\epsilon$ and $\mu$ (dilectric constant and magnetic permeability, respectively) are constant, and that $\sigma =0$ (conductivity of the medium). These assumptions allow the use of a Helmholtz decomposition of the electric and magnetic field vectors. The refined error analysis allows a proof of convergence that is of almost optimal order, and almost optimal in the regularity assumptions for the continuous fields.

##### MSC:
 65Z05 Applications of numerical analysis to physics 65M15 Error bounds (IVP of PDE) 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 35Q60 PDEs in connection with optics and electromagnetic theory