This article investigates the fully discrete finite element approximation and error analysis for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equations, which are nonlinearly coupled through convection, stresses, and Lorentz forces. To address this highly nonlinear and multi-physics system, we propose a fully discrete energy stable scheme with the finite element projection method for spatial discretization, in which the velocity and pressure are decoupled. The temporal discretization is a combination of the first-order Euler semi-implicit scheme and a convex splitting energy strategy. We show that the proposed scheme is mass-conservative, charge-conservative and unconditional energy stable. The error estimates for the phase variable, chemical potential, velocity, pressure, current density and electric potential are rigorously established. Finally, several three-dimensional numerical experiments are performed to illustrate the features of the proposed numerical method and verify the theoretical conclusions.

中文翻译：

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Cahn-Hilliard 无感应 MHD 问题的完全离散投影方法的误差分析


本文研究了两相不可压缩无感磁流体动力学问题的扩散界面模型的完全离散有限元近似和误差分析。该模型由 Cahn-Hilliard 方程、Navier-Stokes 方程和 Poisson 方程组成，它们通过对流、应力和洛伦兹力进行非线性耦合。为了解决这个高度非线性和多物理系统，我们提出了一种完全离散的能量稳定方案，采用有限元投影方法进行空间离散，其中速度和压力是解耦的。时间离散化是一阶欧拉半隐式格式和凸分裂能量策略的组合。我们证明了所提出的方案是质量守恒、电荷守恒和无条件能量稳定的。严格建立了相位变量、化学势、速度、压力、电流密度和电势的误差估计。最后，进行了几个三维数值实验来说明所提出的数值方法的特点并验证了理论结论。