We consider a boundary value problem (BVP) for a reduced system of time harmonic Maxwell equations in magnetized plasma. The dielectric tensor is strongly anisotropic and the system admits resonant solutions in the context of the limit absorption principle. In particular, in the vanishing viscosity limit the normal component of the electric field becomes infinite and non integrable at the resonant point, and the system becomes ill-posed. In this article we recast the problem in the framework of mixed variational problems and we propose a well-posed formulation that characterizes the singular limit solutions. A key tool is the method of manufactured solutions [7] to construct an integral variational characterization of the jump conditions at the resonance. The well posedness is demonstrated and basic numerical results illustrate the robustness of our approach.
The modelling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coefficient $\alpha $ and compact terms. The coefficient $\alpha $ changes sign. The region where $\{\alpha>0\}$ is propagative, and the region where $\{\alpha <0\}$ is non propagative and elliptic. The two models are coupled through the line $\varSigma =\{\alpha =0\}$. Generically, it is an ill-posed problem and additional information must be introduced to get a satisfactory treatment at $\varSigma $. In this work, we define the solution by relying on the limiting absorption principle ($\alpha $ is replaced by $\alpha +i0^+$) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular and a singular part, which originates at $\varSigma $, and on quasi-solutions. It leads to a new well-posed mixed variational formulation with coupling. As we design explicit quasi-solutions, numerical experiments can be carried out, which illustrate the good properties of this new tool for numerical computation.
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