Finite element methods for quantum electrodynamics using a Helmholtz decomposition of the gauge field
SUMMARY The Dirac equation of quantum electrodynamics (QED) describes the interaction between electrons and photons. Large-scale numerical simulations of the theory require repeated solution of the two-dimensional Dirac equation, a system of two first-order partial differential equations coupled to a background U(1) gauge field. Traditional 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 multilevel methods ineffective. We consider an alternate formulation of the governing equations obtained by a transformation of the continuum operator that decouples the system into separate scalar diffusion-like equations. We discretize the transformed system using least-squares finite elements and use adaptive smoothed aggregation multigrid (αSA) to solve the resulting linear system. We present numerical results and discuss implications of the transformed formulation in terms of the physical theory.