# An augmented Lagrangian preconditioner for the magnetohydrodynamics equations at high Reynolds and coupling numbers

@article{Laakmann2021AnAL, title={An augmented Lagrangian preconditioner for the magnetohydrodynamics equations at high Reynolds and coupling numbers}, author={Fabian Laakmann and Patrick E. Farrell and Lawrence Mitchell}, journal={ArXiv}, year={2021}, volume={abs/2104.14855} }

. 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…

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## References

SHOWING 1-10 OF 57 REFERENCES

Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics

- Computer ScienceSIAM J. Sci. Comput.
- 2016

This paper compares the two relaxation procedures within a multigrid-preconditioned GMRES method employed within Newton's method, and uses structured grids, inf-sup stable finite elements, and geometric interpolation to isolate the effects of the different relaxation methods.

A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D

- Computer ScienceJ. Comput. Phys.
- 2017

A Block Preconditioner for an Exact Penalty Formulation for Stationary MHD

- Computer ScienceSIAM J. Sci. Comput.
- 2014

A finite element discretization of an exact penalty formulation for the stationary MHD equations posed in two-dimensional domains has the benefit of implicitly enforcing the divergence-free condition on the magnetic field without requiring a Lagrange multiplier.

Preconditioners for Mixed Finite Element Discretizations of Incompressible MHD Equations

- Computer ScienceSIAM J. Sci. Comput.
- 2017

This work proposes a preconditioner that exploits the block structure of the underlying linear system, utilizing and combining effective solvers for the mixed Maxwell and the Navier--Stokes subproblems.

Scalable implicit incompressible resistive MHD with stabilized FE and fully-coupled Newton–Krylov-AMG

- Computer Science
- 2016

Robust preconditioners for incompressible MHD models

- Computer ScienceJ. Comput. Phys.
- 2016

A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations

- Computer Science
- 2021

An augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes achieves both Reynolds-Robust performance and Reynolds-robust error estimates.

A New Approximate Block Factorization Preconditioner for Two-Dimensional Incompressible (Reduced) Resistive MHD

- Computer ScienceSIAM J. Sci. Comput.
- 2013

This paper proposes and explores the performance of several candidate block preconditioners for the one-fluid visco-resistive MHD model and proposes an operator-split approximation that reduces the system into two $2\times2$ operators.

On the Divergence-free Condition and Conservation Laws in Numerical Simulations for Supersonic Magnetohydrodynamical Flows

- Computer Science
- 1998

An approach to maintain exactly the eight conservation laws and the divergence-free condition of magnetic fields is proposed for numerical simulations of multidimensional magnetohdyrodynamic (MHD)…

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

- Computer ScienceSIAM J. Sci. Comput.
- 2016

New approximate block factorization preconditioners for this system are presented which reduce the system to approximate Schur complement systems that can be solved using algebraic multilevel methods.