• Corpus ID: 233476644

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… 
Robust multigrid techniques for augmented Lagrangian preconditioning of incompressible Stokes equations with extreme viscosity variations
TLDR
To cope with the near-singularity of the (1,1)-block, a multigrid scheme with a discretization-dependent smoother and transfer operators from triangular/tetrahedral to the quadrilateral/hexahedral finite element discretizations is extended.
Structure-preserving and helicity-conserving finite element approximations and preconditioning for the Hall MHD equations
We develop structure-preserving finite element methods for the incompressible, resistive Hall magnetohydrodynamics (MHD) equations. These equations incorporate the Hall current term in Ohm’s law and
Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow
TLDR
This paper extends the classical Vanka relaxation scheme to implicit RK discretizations of saddle point problems, and presents numerical results for the incompressible Stokes, Navier–Stokes, and resistive magnetohydrodynamics equations, confirming that these relaxation schemes lead to robust and scalable monolithic multigrid methods for a challenging range of incompressable fluid-flow models.
Preconditioners for computing multiple solutions in three-dimensional fluid topology optimization
TLDR
This work develops a nested block preconditioning approach which reduces the linear systems to solving two symmetric positivedefinite matrices and an augmented momentum block, and presents multiple solutions in three-dimensional examples computed using the proposed iterative solver.
Transformations for Piola-mapped elements
TLDR
The novel transformation theory recently developed by Kirby is applied to devise the correct map for transforming the basis on a reference cell to a generic physical triangle, enabling the use of the Arnold–Winther elements in the widelyused Firedrake finite element software, composing with its advanced symbolic code generation and geometric multigrid functionality.

References

SHOWING 1-10 OF 57 REFERENCES
Monolithic Multigrid Methods for Two-Dimensional Resistive Magnetohydrodynamics
TLDR
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 Block Preconditioner for an Exact Penalty Formulation for Stationary MHD
TLDR
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
TLDR
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.
Robust preconditioners for incompressible MHD models
A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations
TLDR
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
TLDR
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
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
TLDR
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.
...
...