On an efficient line smoother for the p-multigrid γ-cycle

Abstract: As part of the development of a Poisson solver for the spectral element discretization used in the GeoFluid Object Workbench (GeoFLOW) code, we propose a solver for the linear system arising from a Gauss-Legendre-Lobatto global spectral method. We precondition using a p-multigrid {\gamma}-cycle with highly-vectorizable smoothers, that we refer to as line smoothers. Our smoothers are restrictions of spectral and finite element discretizations to low-order one-dimensional problems along lines, that are solved by a reformulation of cyclic reduction as a direct multigrid method. We illustrate our method with numerical experiments showing the apparent boundedness of the iteration count for a fixed residual reduction over a range of moderately deformed domains, right hand sides and Dirichlet boundary conditions.