A new conservative finite element solver for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations is presented.The solver utilizes magnetic vector potential and current density as solution variables, which are discretized by H(curl)-conforming edge-element and H(div)-conforming face element respectively. As a result, the divergence-free constraints of discrete current density and magnetic induction are both satisfied. Moreover the solutions also preserve the total magnetic helicity. The generated linear algebraic equation is a typical dual saddle-point problem that is ill-conditioned and indefinite. To efficiently solve it, we develop a block preconditioner based on constraint preconditioning framework and devise a preconditioned FGMRES solver. Numerical experiments verify the conservative properties, the convergence rate of the discrete solutions and the robustness of the preconditioner.
翻译:演示了三维稳定磁流动力动动动方程式的一个新的保守的有限元素求解器。 解答器将磁矢量潜能值和当前密度作为溶解变量,分别由H(cur)相容边缘元素和H(div)相容面元素分离。结果,离异当前密度和磁感应的无差异限制得到了满足。此外,解决方案还保留了总磁热度。生成的线性代数方程式是一个典型的双峰点问题,其条件不完善且不固定。为了有效解决这一问题,我们开发了一个基于制约前提框架的区块前置装置,并设计了一个具有先决条件的硬化器。 数字实验核查了保守性、离散溶剂的趋同率和前置器的坚固性。