On finite element schemes for Vlasov-Maxwell system and Schrödinger equation

Doctoral thesis, 2017

This thesis treats finite element schemes for two kind of problems, the Valsov-Maxwellsystem and the nonlinear Schrödinger equation. We study streamline diffusion schemes applied for numerical solution of the one and one-half dimensional relativistic Vlasov-Maxwell system. The study is made both in a priori and a posteriori settings. In the a priori setting we derive stability estimates and prove optimal convergence rates, due to the maximal available regularity of the exact solution. In addition to this we also prove existence and uniqueness of the numerical solution. In the a posteriori setting we use dual problems to prove error estimates in L_\infty (H^{-1}) norm. For the Maxwell equation we also prove error estimates in H^{-1} (H^{-1}) norms. Further more we study a hp-version of the streamline diffusion scheme for thethree dimensional Vlasov-Maxwell system in an a priori setting. A Nitsche type scheme is also introduced and analyzed for Maxwell's equations. For the nonlinear Schrödinger equation a two level time discretization is used. Here we derive a priori error estimates both in L_2 and H^1 norms.