# Least-Squares Finite Element Methods for Quantum Electrodynamics

- 2010

#### Abstract

A significant amount of the computational time in large Monte Carlo simulations of lattice field theory is spent inverting the discrete Dirac operator. Unfortunately, traditional covariant finite difference discretizations of the Dirac operator present serious challenges for standard iterative methods. For interesting physical parameters, the discretized operator is large and ill-conditioned, and has random coefficients. More recently, adaptive algebraic multigrid (AMG) methods have been shown to be effective preconditioners for Wilson's discretization [1] [2] of the Dirac equation. This paper presents an alternate discretization of the 2D Dirac operator of Quantum Electrodynamics (QED) based on least-squares finite elements. The discretization is systematically developed and physical properties of the resulting matrix system are discussed. Finally, numerical experiments are presented that demonstrate the effectiveness of adaptive smoothed aggregation (αSA) multigrid as a preconditioner for the discrete field equations. 1. Introduction. Quantum Chromodynamics (QCD) is the leading theory in the Standard Model of particle physics of the strong interactions between color charged particles (quarks) and the particles that bind them (gluons). Analogous to the way that electrically charged particles exchange photons to create an electromagnetic field, quarks exchange gluons to form a very strong color force field. Contrary to the electromagnetic force, the strong force binding quarks does not get weaker as the particles get farther apart. As such, at long distances (low energies), quarks have not been observed independently in experiment and, due to their strong coupling, perturbative techniques, which have been so successful in describing weak interactions in Quantum Electrodynamics (QED), diverge for the low-energy regime of QCD. Instead, hybrid Monte Carlo (HMC) simulations are employed in an attempt to numerically predict physical observables in accelerator experiments [6]. A main computational bottleneck in an HMC simulation is computation of the so-called fermion propagator, another name for the inverse of the discrete Dirac operator. This process accounts for a large amount of the overall simulation time. For realistic physical parameter values, the Dirac operator has random coefficients and is extremely ill-conditioned. The two main parameters of interest are the temperature (β) of the background gauge field and the fermion mass (m). For small temperature values (β < 5), the entries in the Dirac matrix become extremely disordered. Moreover, as the fermion mass approaches its true physical value, performance of the community standard Krylov solvers degrades; a phenomenon known as critical slowing down. As a result, the development of sophisticated preconditioners for computing …

**DOI:**10.1137/080729633