Kaibo Hu, Young-Ju Lee, Jinchao Xu
We construct finite element methods for the incompressible magnetohydrodynamics (MHD) system that precisely preserve magnetic and cross helicity, the energy law and the magnetic Gauss law at the discrete level. The variables are discretized as discrete differential forms in a de Rham complex. We present numerical tests to show the performance of the algorithm.
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We construct a structure-preserving finite element method and time-stepping scheme for inhomogeneous, incompressible magnetohydrodynamics (MHD). The method preserves energy, cross-helicity (when the fluid density is constant), magnetic helicity, mass, total squared density, pointwise incompressibility, and the constraint $\operatorname{div} B = 0$ to machine precision, both at the spatially and temporally discrete levels.
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 provide a more appropriate description of fully ionized plasmas than the standard MHD equations on length scales close to or smaller than the ion skin depth. We introduce a stationary discrete variational formulation of Hall MHD that enforces the magnetic Gauss's law exactly (up to solver tolerances) and prove the well-posedness and convergence of a Picard linearization. For the transient problem, …
Error estimates for a helicity-preserving finite element discretisation of an incompressible magnetohydrodynamics system
本文推导了不可压缩磁流体动力学模型七场公式的有限元逼近误差估计，证明该方法在离散层面保持系统能量、磁螺旋度与交叉螺旋度。
Error estimates for structure-preserving discretization of the incompressible MHD system
本文基于能量估计和物理结构，分析了不可压缩磁流体动力学系统结构保持离散化的误差，证明了在CFL条件下最优收敛阶，并通过数值测试验证理论结果。
We introduce a mimetic dual-field discretization which conserves mass, kinetic energy and helicity for three-dimensional incompressible Navier-Stokes equations. The discretization makes use of a conservative dual-field mixed weak formulation where two evolution equations of velocity are employed and dual representations of the solution are sought for each variable. A temporal discretization, which staggers the evolution equations and handles the nonlinearity such that the resulting discrete algebraic systems are linear and decoupled, is constructed. The spatial discretization is mimetic in the…
We present a scalable block preconditioning strategy for the trace system coming from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block (the upper block), for which a system algebraic multigrid (AMG) is used for the approx…
Analysis of an HDG Method for Linearized Incompressible Resistive MHD Equations
本文提出了一种用于静止线性化不可压缩磁流体动力学（MHD）方程的混合不连续配方法，介绍了MHD方程的对偶鞍点形式的HDG通量，实现了不连续配方法的混合化，并给出了在二维和三维简单网格上的先验误差估计，证明了流体速度和磁变量的最优收敛性以及其余变量的近似最优收敛性，通过数值实验验证了理论结果。
Decoupled structure-preserving discretization of incompressible MHD equations with general boundary conditions
该研究针对带一般边界条件的不可压缩MHD方程，在混合有限元框架下提出蛙跳式解耦保结构格式，实现流体与麦克斯韦部分完全解耦和部分线性化，验证了其精度及守恒耗散特性，并应用于经典基准算例。
A dual-field structure-preserving mixed finite element discretization for incompressible Hall MHD equations
本文提出了一种新的双场结构保持的混合有限元离散化方法，用于不可压缩霍尔磁流体动力学方程，该方法保持质量、磁高斯定律和电流密度的守恒，并遵循一个离散能量定律，精确捕捉耗散情况下的能量耗散，并在理想情况下还原为能量守恒。数值实验展示了时间与空间精度以及结构保持性质。
We construct a structure-preserving finite element method and time-stepping scheme for compressible barotropic magnetohydrodynamics (MHD) both in the ideal and resistive cases, and in the presence of viscosity. The method is deduced from the geometric variational formulation of the equations. It preserves the balance laws governing the evolution of total energy and magnetic helicity, and preserves mass and the constraint $ \operatorname{div}B = 0$ to machine precision, both at the spatially and temporally discrete levels. In particular, conservation of energy and magnetic helicity hold at the …
Numerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow
研究不可压缩电阻磁流体动力学自由表面流，采用热力学一致的扩散界面方法建模，通过匹配渐近论证建立收敛性，设计完全解耦线性有限元格式，经数值研究验证了模型和算法的可靠性、鲁棒性并进行定量比较。
In this paper we study finite element method for three-dimensional incompressible resistive magnetohydrodynamic equations, in which the velocity, the current density, and the magnetic induction are divergence-free. It is desirable that the discrete solutions should also satisfy divergence-free conditions exactly especially for the momentum equations. Inspired by constrained transport method,we devise a new stable mixed finite element method that can achieve the goal. We also prove the well-posedness of the discrete solutions. To solve the resulting linear algebraic equations, we propose a GMRE…
An energy- and helicity-conserving enriched galerkin method for the incompressible Navier-Stokes equations
本文提出了一种能量和螺旋度守恒的增强型伽辽金方法，用于求解不可压缩纳维-斯托克斯方程，通过两种数值方案实现了动能和螺旋度的精确守恒，并验证了方法的稳定性和误差估计。
Parametric charge-conservative mixed finite element method for 3D incompressible inductionless MHD equations on curved domains
本文提出了一种参数保守混合有限元方法，用于求解三维曲域上的不可压缩无感应磁流体动力学方程，通过等参数泰勒-霍德元素和参数Brezzi-Douglas-Marini元素实现最优收敛性并保证电流密度无散性。
A robust solver for the finite element approximation of stationary incompressible MHD equations in 3D
本文提出了一种稳健的求解器，用于三维不可压缩磁流体动力学方程的有限元离散问题，采用混合有限元方法，利用H1符合有限元近似速度和压力，H(curl)符合边元近似磁场，并提出高效预处理器加速GMRES方法求解线性化MHD问题。
Robust globally divergence-free HDG finite element method for steady thermally coupled incompressible MHD flow
本文针对稳态热耦合不可压缩MHD流提出任意阶HDG有限元方法，实现速度与磁场的全局无散逼近，并证明离散格式的存在唯一性及最优误差估计。
Robust Preconditioners for Incompressible MHD Models
本文开发了两类鲁棒预条件子，用于不可压缩磁流体动力学系统的结构保持离散化。通过分析离散系统的适定性，设计了块预条件子并严格分析其性能，证明其对大部分物理和离散参数具有鲁棒性，并改进了鞍点问题的块三角预条件子估计。
Robust H(curl)-based finite element methods for the incompressible MHD system
本文提出并分析了基于H(curl)符合离散化的有限元方法，用于不可压缩磁流体动力学系统，研究了稳定化机制对非凸多边形域适用性、压力鲁棒性及雷诺数鲁棒性的影响。
Magnetohydrodynamics (MHD) is a continuum level model for conducting fluids subject to external magnetic fields, e.g. plasmas and liquid metals. The efficient and robust solution of the MHD system poses many challenges due to it's nonlinear, non self-adjoint, and highly coupled nature. In this paper, we develop a robust and accurate a posteriori error estimate for the numerical solution of the MHD equations based on the exact penalty method. The error estimate also isolates particular contributions of error in a quantity of interest (QoI) to inform discretization choices to arrive at accurate …
A Divergence-Free and $H(div)$-Conforming Embedded-Hybridized DG Method for the Incompressible Resistive MHD equations
提出一种无散度且$H(div)$协调的E-HDG方法求解定常不可压缩电阻性MHD方程，利用连续与间断面未知量减少自由度，通过Picard迭代处理非线性问题，并保持速度场和磁场的散度为机器零且具压力鲁棒性。
High-order Energy-stable and Charge-conservative Lagrangian FEM for 3D Incompressible Inductionless MHD equations with Variable Density
针对带移动界面的变密度不可压无感应MHD方程，提出拉格朗日框架下高阶能量稳定且电荷守恒的混合有限元方法，通过数值实验验证了其高阶精度与无条件能量稳定性。
A fully discretization, unconditionally energy stable finite element method solving the thermodynamically consistent diffuse interface model for incompressible two-phase MHD flows with large density ratios
针对现有扩散界面两相磁流体动力学模型仅适用于密度匹配场景的局限，研究基于Onsager变分原理与守恒律推导大密度比下的热力学一致模型，提出全离散有限元求解格式并证明其理论性质，通过数值实验验证了格式的有效性。
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