Block Preconditioners for Stable Mixed Nodal and Edge finite element Representations of Incompressible Resistive MHD

Abstract

The scalable iterative solution of strongly coupled three-dimensional incompressible resistive magnetohydrodynamics (MHD) equations is very challenging because disparate time scales arise from the electromagnetics, the hydrodynamics, as well as the coupling between these systems. This study considers a mixed finite element discretization of a dual saddle point formulation of the incompressible resistive MHD equations using a stable nodal (Q2/Q1) discretization for the hydrodynamics and a stable edge-node discretization of a reduced form of the Maxwell equations. This paper presents new approximate block factorization preconditioners for this system which reduce the system to approximate Schur complement systems that can be solved using algebraic multilevel methods. These preconditioners include a new augmentation-based approximation for the magnetic induction saddle point system as well as efficient approximations of the Schur complements that arise from the complex coupling between the Navier-Stokes equations and the Maxwell equations.

Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics

“…The temporal scales give rise to very stiff modes that make explicit time integration methods intractable due to stability limitations. To overcome this, unconditionally stable implicit time integration methods are often used [15,34,42,44,45,54,55,62]. These time integration approaches lead to large sparse linear systems whose solution requires effective parallel preconditioners.…”

A face‐based monolithic approach for the incompressible magnetohydrodynamics equations

“…In recent years, the motion of an electrically conducting fluid within a three‐dimensional lid‐driven cavity has been extensively investigated in the literature . In here, the calculations are carried out for a lid‐driven cubic cavity with a horizontally applied magnetic field, B =(1,0,0) ⊤ corresponding to the work of Li and Zheng due to its relatively high Hartmann number.…”

A semi-implicit energy conserving finite element method for the dynamical incompressible magnetohydrodynamics equations

“…This example was tested in [21], where a divergence free approach was used. Some similar and simplified two-dimensional model were tested by several authors with different methods, e.g., see [25,29,34]. It should be noted that due to the different nondimensionalization procedure, the unknown B in the above MHD equations (4.18)-(4.21) is a little bit different with the one used in (1.1)-(1.4).…”

“…where V h (Ω) denotes a certain FE space. Let us remark that this approach and its generalizations has been widely used, see [3,11,14,17,18,20,29,30,31,35]. Finally, it should be noted that the regularity of B on non-convex and non-smooth domains is lower than H 1 (Ω).…”

“…Sermane and Temam proved the existence and uniqueness of local strong solution on regular domains [33]. There have been numerous works on numerical methods for the incompressible MHD equations, see [3,14,17,21,22,25,29,31,35]. Due to the nonlinear coupling of the unknowns, the divergence free constraints and the low regularity of the exact solutions, it is a challenging task to design efficient numerical schemes for the dynamical incompressible MHD equations.…”