AI 中文总结
该研究针对带旋度型噪声的随机电阻霍尔磁流体动力学系统，开发了保结构混合有限元方法并证明其解收敛，得到该系统的构造性存在结果，数值实验验证了方法有效性。
AI 中文摘要
我们在有界凸多面体区域上研究随机电阻霍尔磁流体动力学（Hall--MHD）系统，该系统满足非线性完全导电边界条件。该系统由动量方程中的乘性高斯强迫以及广义欧姆定律中电阻项和霍尔项的随机扰动所产生的感应方程中的结构化旋度型噪声驱动。我们基于兼容的离散de Rham复形开发了一种全离散、线性隐式、保结构的混合有限元方法，该方法能精确保持磁高斯定律。我们证明了离散解的子序列收敛到有限能量弱鞅解，从而得到了带旋度型噪声的随机Hall--MHD系统的构造性存在性结果。数值实验展示了该方法的随机动力学、精确散度保持性及收敛性。
英文摘要
We study the stochastic resistive Hall--magnetohydrodynamic (Hall--MHD) system on bounded convex polyhedral domains, subject to nonlinear perfectly conducting boundary conditions. The system is driven by multiplicative Gaussian forcing in the momentum equation together with structured curl-type noise in the induction equation arising from stochastic perturbations of the resistive and Hall parts in the generalised Ohm law. We develop a fully discrete, linearly implicit, structure-preserving mixed finite element method based on a compatible discrete de Rham complex, which preserves the magnetic Gauss law exactly. We prove subsequential convergence of the discrete solutions to a finite-energy weak martingale solution, thereby obtaining a constructive existence result for the stochastic Hall--MHD system with curl-type noise. Numerical experiments illustrate the stochastic dynamics, exact divergence preservation, and convergence of the method.
