arXiv subjects

Haiyan Su

Publications and source records attributed to Haiyan Su.

8 recordsLinked to original sources

arXiv2026

High-order Energy-stable and Charge-conservative Lagrangian FEM for 3D Incompressible Inductionless MHD equations with Variable Density

In this paper, we develop a high-order, energy-stable and charge-conservative Lagrangian finite element method for variable-density incompressible inductionless magnetohydrodynamic (MHD) equations. The method utilizes the moving high-order curved tetrahedral mesh to track the material interface. Second-order Backward Differentiation Formula (BDF2) is used for the temporal discretization of material derivative, together with the second-order Adams--Bashforth method (AB2) for the update of control points of the meshes. High-order isoparametric Taylor-Hood elements with grad-div stabilization are used for the velocity-pressure pair. While, to ensure the discrete charge conservation, high-order parametric $\BH(\Div)$-conforming element is adopted for the current density. In the absence of external force, the unconditional energy-stability of the fully discrete scheme is proven. Finally, 3D numerical experiments are conducted to confirm the expected high-order accuracy for smooth solutions, the energy stability property and the capability of the proposed method.

math.NA↗

arXiv2025

A divergence-free parametric finite element method for 3D Stokes equations on curved domains

The Stokes equations play an important role in the incompressible flow simulation. In this paper, a novel divergence-free parametric mixed finite element method is proposed for solving three-dimensional Stokes equations on domains with piecewise smooth boundaries. The flow velocity and pressure are discretized with high-order parametric Brezzi-Douglas-Marini elements and volume elements, respectively, on curved tetrahedral meshes. Utilizing the interior-penalty discontinuous Galerkin (IPDG) technique, we prove the inf-sup condition for the mixed finite element pair, and high-order optimal error estimates in the energy norm, with the help of the extension and transformation of the true solution to computational domain. Moreover, the discrete velocity is exactly divergence-free, meaning that div uh = 0 holds in the curved computational domain. Numerical experiments are conducted to support the theoretical analyses.

math.NA↗

arXiv2025

Optimal error estimates for a fully discrete, highly efficient decoupled scheme for the 2D/3D diffuse interface two-phase MHD flows

In this paper, we derive optimal L2- and H1-norm error estimates for a fully discrete convex-splitting decoupled finite element method (FEM) for the two-phase diffuse interface magnetohydrodynamics (MHD) system. We use the semi-implicit backward Euler scheme in time and employ the standard inf-sup stable Taylor--Hood or Mini elements to discretize the velocity and pressure. Furthermore, we apply a pressure-correction scheme to decouple the velocity from the pressure. The optimal error estimates are obtained via novel Ritz and Stokes quasi-projection techniques. In addition, the unconditional energy stability of the proposed scheme is ensured. Numerical examples are presented to validate the theoretical analysis.

math.NA↗

arXiv2025

Optimal ${L^2}$ error estimates for 2D/3D incompressible Cahn--Hilliard--magnetohydrodynamic equations

This paper focuses on an optimal error analysis of a fully discrete finite element scheme for the Cahn--Hilliard--magnetohydrodynamic (CH-MHD) system. The method use the standard inf-sup stable Taylor--Hood/MINI elements to solve the Navier--Stokes equations, Lagrange elements to solve the phase field, and particularly, the N\'ed\'elec elements for solving the magnetic induction field. Suffering from the strong coupling and high nonlinearity, the previous works just provide suboptimal error estimates for phase field and velocity field in $L^{2}/\L^2$-norm under the same order elements, and the suboptimal error estimates for magnetic induction field in $\H(\rm curl)$-norm. To this end, we utilize the Ritz, Stokes, and Maxwell quasi-projections to eliminate the low-order pollution of the phase field and magnetic induction field. In addition to the optimal $\L^2$-norm error estimates, we present the optimal convergence rates for magnetic induction field in $\H(\rm curl)$-norm and for velocity field in $\H^1$-norm. Moreover, the unconditional energy stability and mass conservation of the proposed scheme are preserved. Numerical examples are illustrated to validate the theoretical analysis and show the performance of the proposed scheme.

math.NA↗

arXiv2024

The virtual element method for the three dimensional inductionless magnetohydrodynamic model

This paper proposes a novel first-order and a novel second-order fully discrete virtual element schemes based on the scalar auxiliary variable method for the three dimensional inductionless magnetohydrodynamics problem. The backward Eular formula and the backward differential formula are used for the time discretization and two types conservation virtual element formulations are employed for spatial discretization. The main advantages include that the mass conservation in the velocity field and the charge conservation in the current density field are kept by taking characteristics of the virtual element method's discrete complex structures, the nonlinear term is handled explicitly by applying the scalar auxiliary variable method, the current density field is decoupled from the momentum equation, and the velocity field is decoupled from the Ohm's law. The unconditionally stable of the two fully discrete schemes are demonstrated. Finally, we present numerical experiment to verify the valid of the proposed schemes.

math.AP↗

arXiv2023

A full divergence-free of high order virtual finite element method to approximation of stationary inductionless magnetohydrodynamic equations on polygonal meshes

In this present paper we consider a full divergence-free of high order virtual finite element algorithm to approximate the stationary inductionless magnetohydrodynamic model on polygonal meshes. More precisely, we choice appropriate virtual spaces and necessary degrees of freedom for velocity and current density to guarantee that their final discrete formats are both pointwise divergence-free. Moreover, we hope to achieve higher approximation accuracy at higher "polynomial" orders k_{1} \geq 2, k_{2} \geq 1, while the full divergence-free property has always been satisfied. And then we processed rigorous error analysis to show that the proposed method is stable and convergent. Several numerical tests are presented, confirming the theoretical predictions.

math.AP↗

arXiv2023

Error estimates for the highly efficient and energy stable schemes for the 2D/3D two-phase MHD

In this paper, we mainly focus on the rigorous convergence analysis of two fully decoupled, unconditionally energy-stable methods for the diffuse interface two-phase magnetohydrodynamics (MHD) model. The two methods consist of the semi-implicit stabilization method and the invariant energy quadratization (IEQ) method, which are both applied to the phase field system. In addition, the pressure correction method is used for the saddle point system, and appropriate implicit-explicit treatments are employed for the nonlinear coupled terms. We prove the unconditional energy stability of the two schemes. In addition, we mainly establish the error estimates based on the bounds of $\left\|\phi^{k}\right\|_{L^{\infty}}$ and $\left\|\textbf{b}^{k}\right\|_{L^{\infty}}$. Several numerical examples are presented to test the accuracy and stability of the proposed methods.

math.AP↗

arXiv2022

Coupled iterative analysis for the stationary thermally coupled inductionless MHD system based on charge-conservative finite element method

This paper mainly considers three iterations based on charge-conservative finite element approximation in Lipschitz domain for the stationary thermally coupled inductionless MHD equations. Based on the hybrid finite element method, the unknowns of hydrodynamic are discretized by the stable velocity-pressure finite element pair, and the current density along with electric potential are similarly discretized by the comforming finite element pair in $\boldsymbol{H}(\boldsymbol{div}, Ω)\times L^2(Ω)$. And on account of the strong nonlinearity of the equations, we present three coupled iterative methods, namely, the Stokes, Newton and Oseen iteration and the convergence and stability under different uniqueness conditions are analyzed strictly. It is proved especially that the error estimates of velocity, current density, temperature and pressure do not depend on potential. The theoretical analysis is validated by the given numerical results, and for the proposed methods, the applicability and effectiveness are demonstrated.

math.AP↗