Ph.D. Dissertation Defense: Christopher A. Leibs
First-Order Systems Least-Squares Finite Element Methods and Nested Iteration for Electromagnetic Two-Fluid Kinetic- Based Plasma Models
Christopher A. Leibs
Applied Mathematics,
Date and time:
Tuesday, November 18, 2014 - 9:30am
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Grandview Conference Room, 1320 Grandview Ave.
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Efforts are currently being directed towards a fully-implicit, electromag-
netic, Jacobian-free Newton Krylov kinetic solver, motivating the neces-
sity of developing a suitable fluid-based, electromagnetic, precondition-
ing strategy. The two-fluid plasma Darwin model is an ideal approxi-
mation to the kinetic Jacobian and is the subject of this thesis. The two-
fluid plasma (TFP) model couples both an ion fluid and an electron fluid
with Maxwell’s equations. The ion and electron fluid equations consist
of the conservation of momentum and conservation of number density.
A Darwin approximation of Maxwell is used to eliminate spurious light
waves from the model in order to make coupling to non-relativistic parti-
cle models feasible. We analyze the TFP-Darwin system in the context of
a stand-alone solver with consideration of preconditioning a kinetic-JFNK
approach.
The TFP-Darwin system is addressed numerically by use of nested it-
eration (NI) and a First-Order Systems Least Squares (FOSLS) discretiza-
tion. An important goal of NI is to produce an approximation that is
within the basis of attraction for Newton’s method on a relatively coarse
mesh, and thus, on all subsequent meshes. After scaling and modifica-
tion, the TFP-Darwin model yields a nonlinear, first-order system of equa-
tions whose Frechet derivative is shown to be uniformly ́ H1
neighborhood of the exact solution. H1 ellipticity yields optimal finite ele-
ment performance and linear systems amenable to solution with Algebraic
Multigrid (AMG). In order to efficiently focus computational resources, an
adaptive mesh refinement scheme, based on the accuracy per computa-
tional cost, is leveraged. Numerical tests demonstrate the efficacy of the
approach, yielding an approximate solution within discretization error in
a relatively small number of computational work units.