Several difficulties arise when designing Multigrid (MG) solvers for the systems of linear equations encountered in lattice gauge theories (LGT). First, these systems become increasingly ill conditioned as one probes in greater detail the microscopic structures of these theories -- much more so than standard elliptic systems of partial differential equations (PDEs). In addition, the near nullspace of the system matrix is locally oscillatory and varies with fluctuations in the background gauge field, which is itself randomly prescribed. Classical MG methods assume explicit knowledge of the near nullspace in their design (typically that it is geometrically smooth in algebraic neighborhoods) and; hence, are not immediately applicable to these systems. This talk presents a complex-arithmetic adaptive MG solver for 2D LGT systems. The presented method uses a geometric blocking scheme to define the sparsity pattern of a local and sparse coarse space basis and an adaptive smoothed aggregation MG setup procedure to define its coefficients.

This is a joint work in collaboration with M. Brezina, J. Ruge, T. Manteuffel, and S. McCormick of CU Boulder, R. Falgout of CASC-LLNL, R. Brower, M. Clark, J. Osborne, and C. Rebbi of BU, S. MacLachlan and K. Oosterlee of T.U. Delft, and L. Zikatanov of PSU