Multigrid for chiral lattice fermions: Domain wall

The phenomena of critical slowing down in the iterative solution of the Dirac equation presents a major challenge to further applications of lattice field theory in the approach to the continuum solution. We propose a new multigrid approach for chiral fermions, applicable to both the 5D domain wall or 4D overlap operator. The central idea is to directly coarsen the 4D Wilson kernel, giving an effective domain wall or overlap operator on each level. We provide here an explicit construction for the Shamir domain wall formulation with numerical tests for the 2D Schwinger prototype, demonstrating near ideal multigrid scaling. The framework is designed for a natural extension to 4D lattice QCD chiral fermions, such as the Möbius, Zolotarev or Borici domain wall discretizations or directly to a rational expansion of the 4D overlap operator. For the Shamir operator, the effective overlap operator is isolated by the use of a Pauli-Villars preconditioner in the spirit of the Kähler-Dirac spectral map used in a recent staggered multigrid algorithm [R. C. Brower, E. Weinberg, M. A. Clark, and A. Strelchenko, Phys. Rev. D 97, 114513 (2018)].