Unstructured finite element solvers in gyrokinetic turbulence simulations of burning plasmas

Many plasma physics simulation applications utilize an unstructured finite element method (FEM) discretization of an elliptic operator, of the form -∇²u + αu = f; when α is equal to zero a "pure" Laplacian or Poisson equation results and when α is greater than zero a Helmholtz equation is produced. Discretized equations of this form, often 2D, occur in many tokamak fusion plasma, or burning plasma, applications - from MHD to Gyrokinetic codes. This report investigates the performance characteristics of basic classes of linear solvers (ie, direct, one-level iterative, and multilevel iterative methods) on 2D unstructured FEM problems of the form -∇²u + αu = f, with both α = 0 and α ≠ 0. The purpose of this work is to provide computational physicists guidelines as to appropriate linear solvers for their problems via detailed performance analysis, in terms of both the scalability and the constants in the solution times. We show, as expected, almost perfect scalability of multilevel methods and quantify the solution costs on a common computational platform - the IBM SP Power3.