A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

Abstract

Abstract. The magnetohydrodynamics (MHD) equations are used to model the flow of electrically conducting fluids in such applications as liquid metals and plasmas. This system of non-self adjoint, nonlinear PDEs couples the Navier-Stokes equations for fluids and Maxwell's equations for electromagnetics. There has been recent interest in fully coupled solvers for the MHD system because they allow for fast steady-state solutions that do not require pseudo-time stepping. When the fully coupled system is discretized, the strong coupling can make the resulting algebraic systems difficult to solve, requiring effective preconditioning of iterative methods for efficiency. In this work, we consider a finite element discretization of an exact penalty formulation for the stationary MHD equations. This formulation has the benefit of implicitly enforcing the divergence free condition on the magnetic field without requiring a Lagrange multiplier. We consider extending block preconditioning techniques developed for the Navier-Stokes equations to the full MHD system. We analyze operators arising in block decompositions from a continuous perspective and apply arguments based on the existence of approximate commutators to develop new preconditioners that account for the physical coupling. This results in a family of parameterized block preconditioners for both Picard and Newton linearizations. We develop an automated method for choosing the relevant parameters and demonstrate the robustness of these preconditioners for a range of the physical non-dimensional parameters and with respect to mesh refinement. Key words. magnetohydrodynamics, iterative methods, preconditionersAMS subject classifications. 76W05, 65F08, 65M221. Introduction. The magnetohydrodynamics (MHD) model describes the flow of electrically conducting fluids in the presence of magnetic fields. A principal application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics [13]. MHD is also used to model the flow of liquid metals, for instance in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis [19]. The model consists of a non-self-adjoint, nonlinear system of partial differential equations (PDEs) that couple the Navier-Stokes equations for fluid flow to a reduced set of Maxwell's equations for electromagnetics. Because multiple physical processes are represented in the model, the PDEs can span over a range of length-and time-scales, making the equations difficult to solve and requiring a robust, accurate means of approximating the solution. Decoupled solution methods which solve the fluid and magnetic systems separately and possibly couple the systems by an outer iteration have been commonly employed as solvers for the transient and steady MHD systems. In the context of transient systems these methods are commonly used in operator splitting techniques, for steady state solves a fixed point iteration serves to couple the system (see e.g. [2], and th...

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“…e.g. [13,14,16,18,21,24,[26][27][28]31] and the references therein). In [18], Gunzburger et al studied well-posedness and the finite element method for the stationary incompressible MHD equations.…”

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“…The performance studies include the solution of systems as large as two billion unknowns using 24 000 cores . Phillips et al extended the block preconditioning techniques developed for the incompressible Navier‐Stokes equations to the stationary MHD equations in two dimensions. Cyr et al proposed and investigated the performance of several candidate block preconditioners for the incompressible resistive MHD equations and their result showed that the split approximate block preconditioner is scalable and competitive with other preconditioners, including a fully coupled algebraic multigrid method.…”

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