The magnetohydrodynamics (MHD) equations are generally known to be difficult to solve numerically, due to their highly nonlinear structure and the strong coupling between the electromagnetic and hydrodynamic variables, especially for high Reynolds and coupling numbers. In this work, we present a scalable augmented Lagrangian preconditioner for a finite element discretization of the $B$-$E$ formulation of the incompressible viscoresistive MHD equations. For stationary problems, our solver achieves robust performance with respect to the Reynolds and coupling numbers in two dimensions and good results in three dimensions. We extend our method to fully implicit methods for time-dependent problems which we solve robustly in both two and three dimensions. Our approach relies on specialized parameter-robust multigrid methods for the hydrodynamic and electromagnetic blocks. The scheme ensures exactly divergence-free approximations of both the velocity and the magnetic field up to solver tolerances. We confirm the robustness of our solver by numerical experiments in which we consider fluid and magnetic Reynolds numbers and coupling numbers up to 10,000 for stationary problems and up to 100,000 for transient problems in two and three dimensions.


  1. magnetohydrodynamics (MHD)
  2. multigrid
  3. augmented Lagrangian

MSC codes

  1. 65N55
  2. 65N30
  3. 65F10
  4. 65F08

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: B1018 - B1044
ISSN (online): 1095-7197


Submitted: 30 April 2021
Accepted: 6 June 2022
Published online: 15 August 2022


  1. magnetohydrodynamics (MHD)
  2. multigrid
  3. augmented Lagrangian

MSC codes

  1. 65N55
  2. 65N30
  3. 65F10
  4. 65F08



Funding Information

Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/L015811/1, EP/V001943/1, EP/R029423/1

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