Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications

Abstract

In the context of Tokamak fusion plasma, the gyrokinetic Poisson equation has to be solved on disk-like domains which correspond to the poloidal cross-section of the Tokamak geometry. In its simpliest form, this cross section takes a circular form but deformed geometries were found to be advantageous and are more realistic. We propose a taylored solver for the gyrokinetic Poisson equation on disklike geometries described by curvilinear coordinates. Our solver also copes with an anisotropic, locally refined mesh to model the edge part of the Tokamak and a rapidly dropping density profile. Multigrid methods can achieve optimal complexity for many problems and are among the most efficient solvers for elliptic model problems. Multigrid methods for geometries described by curvilinear coordinates are however less common. We present a taylored geometric multigrid algorithm using optimized line smoothers to enable parallel scalability. We propose an implicit extrapolation scheme for our algorithm to increase the order of convergence by using nonstandard numerical integration rules for P1 finite elements. We also use problem-specific finite differences, which numerically show the same order of convergence as their finite elements' counterpart, enabling a matrix-free implementation with a low memory footprint.