| The numerical solution of the Dirac equation is the main computational bottleneck in the simulation of quantum electrodynamics (QED) and quantum chromodynamics (QCD). The Dirac equation is a first-order system of partial differential equations coupled with a random background gauge field. Traditional finite-difference discretizations of this system are sparse and highly structured, but contain random complex entries introduced by the background field. For even mildly disordered gauge fields the near kernel components of the system are highly oscillatory, rendering standard multi-level iterative methods ineffective. As such, the solution of such systems accounts for the vast majority of computation in the simulation of the theory.;In this thesis, two discretizations of a simplified model problem are introduced, based on least-squares finite elements. The first discretization is obtained by direct discretization of the governing equation using least-squares finite elements. The second is obtained by applying the same discretization methodology to a transformed version of the original system. It is demonstrated that the resulting linear systems satisfy several desirable physical properties of the continuum theory and agree spectrally with the continuum the operator. To date, these are the first discretizations to accomplish these goals without extending the theory to a costly extra dimension.;Finally, it is shown that the resulting linear systems are amenable to effective preconditioning by algebraic multigrid methods. Specifically, classical algebraic multigrid (AMG) and adaptive smoothed aggregation (alphaSA) multigrid are employed. The result is a solution process that is efficient and scalable as both the lattice size and the disorder of the background field is increased, and the simulated fermion mass is decreased. |
