Robust Preconditioners for Incompressible MHD Models

Abstract

In this paper, we develop two classes of robust preconditioners for the structure-preserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is not only applicable to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners.

Cited by in corpus (15)

• A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D
• Helicity-conservative finite element discretization for incompressible MHD systems
• A constrained transport divergence-free finite element method for Incompressible MHD equations
• An augmented Lagrangian preconditioner for the magnetohydrodynamics equations at high Reynolds and coupling numbers
• A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume/finite element scheme for the incompressible MHD equations
• On fixed-point, Krylov, and $2\times 2$ block preconditioners for nonsymmetric problems
• An adaptive scalable fully implicit algorithm based on stabilized finite element for reduced visco-resistive MHD
• A Multigrid Preconditioner for Spatially Adaptive High-order Meshless Method on Fluid-solid Interaction Problems
• Monolithic Multigrid for Magnetohydrodynamics
• Reduced Krylov Basis Methods for Parametric Partial Differential Equations
• Modeling, well-posedness and discretization for a class of models for mixed-dimensional problems with high dimensional gap
• A novel divergence-free Finite Element Method for the MHD Kinematics equations using Vector-potential
• Stability and convergence analysis of A linear, fully decoupled and unconditionally energy stable scheme for magneto-hydrodynamic equations
• Convergence of a $B$-$E$ based finite element method for MHD models on Lipschitz domains
• A Stabilized Hybrid Mixed Finite Element Method for Poroelasticity