Mathematical and numerical problems of some wave phenomena appearing in magnetic plasmas
This dissertation investigates mathematical and numerical aspects of some wave phenomena appearing in magnetic plasmas. In order to model a probing technique for fusion plasmas, called reflectometry, a particular form of Maxwell's equations is studied. In the model, the dielectric tensor presents vanishing eigenvalues and diagonal terms. The study of the dispersion relation evidences two kinds of phenomena: cut-offs and resonances if the wave number goes either to zero or to infinity. Part I of the thesis gathers the theoretical results. The main novelty consists in the definition of a resonant solution. Indeed, because of a smooth vanishing sign-changing coefficient, the solution may be singular: one of its components may be non-integrable. However, using a limit absorption principle, a resonant solution is explicitly obtained by studying the integrable solutions of the regularized system plus a limiting process. The theoretical expression of the singularity is validated by numerical tests concerning the regularized system as the regularizing term goes to zero. Part II focuses on the numerical results. It includes the design of a new numerical method adapted to smooth coefficients. The method is based on the Ultra Weak Variational Formulation but requires specific shape functions, designed as local approximations of the adjoint equation. The convergence analysis of the method is performed in one dimension, for two dimensions the design procedure and the interpolation property of the shapes functions are detailed. The resulting high order method numerically tackles the approximation of cut-offs while the approximation of resonant solutions is still very challenging.