A parallel fully coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: Drift-diffusion, flow/transport/reaction, resistive MHD

This study considers the performance of a fully coupled algebraic multilevel preconditioner for Newton–Krylov solution methods. The performance of the preconditioner is demonstrated on a set of challenging multiphysics partial differential equation (PDE) applications: a drift‐diffusion approximation for semiconductor devices; a low Mach number formulation for the simulation of coupled flow, transport and non‐equilibrium chemical reactions; and a low Mach number formulation for visco‐resistive magnetohydrodynamics (MHD) systems. These systems contain multiple physical mechanisms that are strongly coupled, highly nonlinear, non‐symmetric and produce solutions with multiple length‐ and time‐scales. In the context of this study the governing PDEs for these systems are discretized in space by a stabilized finite element (FE) method that collocates all unknowns at each node of the FE mesh. The algebraic multilevel preconditioner is based on an aggressive‐coarsening graph‐partitioning of the non‐zero block structure of the Jacobian matrix. The performance of the algebraic multilevel preconditioner is compared with a standard variable overlap additive Schwarz domain decomposition preconditioner. Representative performance and parallel scaling results are presented for a set of direct‐to‐steady‐state and fully implicit transient solutions. The performance studies include parallel weak scaling studies on up to 4096 cores and also includes the solution of systems as large as two billion unknowns carried out on 24000 cores of a Cray XT3/4. In general, the results of this study indicate that on this reasonably diverse set of challenging multiphysics applications the algebraic multilevel preconditioner performs very well. Copyright © 2010 John Wiley & Sons, Ltd.