  In a recent work, 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 $w=(I-\Delta)u$. 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 $H^2$ on any time interval $[0,T]$ for all 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 with a
nonrestrictive condition on the time step. 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.
