含p-拉普拉斯粘性的三维磁流体动力学有限元方法的稳定性、收敛性与误差分析
Stability, Convergence, and Error Analysis of Finite Element Methods for 3D Magnetohydrodynamics with $p$-Laplacian Viscosity
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中文总结 AI 辅助
本文针对含p-拉普拉斯粘性的三维不可压缩磁流体动力学流动，提出全离散有限元方法，证明其收敛性与无条件误差估计，扩展了非牛顿流体的有限元分析并通过数值模拟验证了方法有效性。
中文摘要 AI 辅助
本文针对具有非线性p-拉普拉斯粘性的三维不可压缩磁流体动力学（MHD）流动，提出了一种全离散有限元方法。该格式将空间有限元与半隐式欧拉时间离散化相结合，通过时间平移紧性论证与Minty单调性方法，证明了其对弱解的收敛性。在额外正则性假设下，本文推导了速度场与磁场的无条件误差估计，无需时间步长与网格尺寸间的耦合限制。该框架将不可压缩MHD系统的有限元分析扩展至非牛顿剪切增稠流体（p>2），同时可退化为经典牛顿流体情形（p=2）。最后，本文提供了数值模拟以验证理论收敛率并展示所提方法的鲁棒性。
英文摘要
This paper develops a fully discrete finite element method for three-dimensional incompressible magnetohydrodynamic (MHD) flows with nonlinear $p$-Laplace viscosity. The scheme combines spatial finite elements with a semi-implicit Euler time discretisation. Convergence to a weak solution is proved using a time-translation compactness argument together with Minty's monotonicity method. Under additional regularity assumptions, we derive unconditional error estimates for both the velocity and magnetic field, with no coupling restriction between the time step and mesh size. The framework extends finite element analysis of incompressible MHD systems to non-Newtonian shear-thickening fluids ($p>2$), while recovering the classical Newtonian case ($p=2$). Finally, numerical simulations are provided to validate the theoretical convergence rates and demonstrate the robustness of the proposed method.
