论文标题
曲柄的最佳误差估计 - Nicolson的有限元投影方法
Optimal error estimates of a Crank--Nicolson finite element projection method for magnetohydrodynamic equations
论文作者
论文摘要
在本文中，我们提出和分析了磁性水力动力学（MHD）方程的完全离散的有限元投影方法。修改后的曲柄 - Nicolson方法和Galerkin有限元方法分别在时间和空间中分别离散模型，并将适当的半显微处理处理应用于流体对流项和两个耦合项。这些半平近近似值导致线性系统具有可变系数，可以从理论上证明其唯一的溶解度。此外，我们在Stokes solver中使用Van Kan Type \ Cite {Vankan1986}的二阶解耦投影方法，该方法基于前一个时间级的压力梯度计算中间速度字段，并通过Helmholtz Decomportions Intermiped Velocitie velocitie velocitie。理论上证明了该方案的能量稳定性，其中需要详细分析脱钩的Stokes求解器。错误估计已在提议的解耦有限元投影方案的离散$ l^\ infty（0，t; l^2）$ norm中证明。提供数值示例以说明理论结果。
In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete $L^\infty(0,T;L^2)$ norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results.
