Adaptive Multigrid Algorithm for Lattice QCD

We present a new multigrid solver that is suitable for the Dirac operator in the presence of disordered gauge fields. The key behind the success of the algorithm is an adaptive projection onto the coarse grids that preserves the near null space. The resulting algorithm has weak dependence on the gauge coupling and exhibits very little critical slowing down in the chiral limit. Results are presented for the Wilson-Dirac operator of the 2D U(1) Schwinger model.

Figure 1

Number of Dirac operator applications of standard CG versus MG CG techniques as a function of the fermion mass gap at $\beta =6$ with topological number $Q=0$ and $Q=4$ (average over 10 gauge fields, relative solver residual tolerance $|r|={10}^{-14}$).

Figure 2

Number of Dirac operator applications of standard CG versus MG CG as a function of the fermion mass gap at $\beta =10$ with topological number $Q=0$ and $Q=4$ (average over 10 gauge fields, relative solver residual tolerance $|r|={10}^{-14}$).