A finite-element model for the Hasegawa–Mima wave equation

In a recent work Karakazian and Nassif (2021)[1], two of the authors have formulated the non-linear space-time Hasegawa–Mima plasma equation as a coupled system of two linear PDEs, a solution of which is a pair (u,w), with w=(I−Δ)u. The first equation is of hyperbolic type and the second of elliptic type. Variational frames for obtaining weak solutions to the initial value Hasegawa–Mima problem with periodic boundary conditions were also derived. Using the Fourier basis in the space variables, existence of solutions were obtained. Implementation of algorithms based on Fourier series leads to systems of dense matrices. In this paper, we use a finite element space-domain approach to semi-discretize the coupled variational Hasegawa–Mima model, obtaining global existence of solutions in H2 on any time interval [0,T],∀T. In the sequel, full-discretization using an implicit time scheme on the semi-discretized system leads to a nonlinear full space-time discrete system. For the existence proof, a condition on the time step is imposed that is independent from the space mesh size h. For the uniqueness proof, a CFL-type of condition on the time step is imposed. Tests on a semi-linear version of the implicit nonlinear full-discrete system are conducted for several initial data, assessing the efficiency of our approach.