## Analysis of the discontinuous Galerkin method applied to collisionless plasma physics

##### Abstract

Two discontinuous Galerkin methods (DG), the discontinuous flow upwind Galerkin (DFUG)
and discontinuous flow upwind Galerkin-Nonsymmetric Interior Penalty Galerkin (DFUGNIPG)
methods, are proposed to approximate the Vlasov system for a perturbed flow
and the Vlasov-Poisson system, respectively. These methods are chosen due to their local
nature, local conservation properties, approximation properties, and their potential for
hp-refinement and parallelizability.
A new optimal inverse inequality is proved for polynomials in this dissertation. Using this
new inequality, an hp-optimal error estimate is proved for the NIPG method. Moreover, an
error estimate is derived for the Poisson equation satisfying a Dirichlet boundary condition,
where the righthandside of the equation is defined by a perturbed source term.
A new method, the DFUG method, for the Vlasov equation in six dimensional phase-space
is formulated such that the method is well-defined for flows that are discontinuous across
the mesh faces. Stability and h−optimal convergence results are proved for the method.
An error estimate is proved for the error between a solution to the Vlasov system that is
defined by a given flow and a solution to the Vlasov system that is defined by a perturbed
flow. Explicit conditions are given as to how well a perturbed flow must approximate a
given flow in order to achieve an optimal error estimate.
A new method, the DFUG-NIPG method, is proposed to approximate to the Vlasov-Poisson
system in six dimensional phase-space. In the case that a discrete solution resulting from the
DFUG-NIPG formulation exists, a partial hp-error estimate is proved for the error between
the true solution to the Vlasov-Poisson system and the discrete solution.
DG methods are applied to three benchmark examples and a fourth experimental example.
The first two benchmarks are to compute numerical solutions to the Vlasov-Poisson system
that is linearized about the Maxwellian distribution, in the first example, and the Lorentzian
distribution, in the second example, in order to verify that the correct Landau damping
decay rates for the electric field waves are obtained up to two digit decimal accuracy. The
third benchmark is to compute a numerical solution to the Vlasov-Poisson-Fokker-Planck
system to check that the results correspond with currently existing results obtained using
other numerical approaches. The third example is to compute a numerical solution to the
Vlasov-Poisson system that is subjected to an external force field function for a fixed amount
of time to determine if any BGK-like modes are present in the numerical solution.