We are concerned with the three-dimensional time-dependent incompressible magnetohydrodynamic (MHD) equations with magnetic vector potential formulations. Compared with the traditional B formulations, the new MHD system has the advantage that it can ensure a direct exact discrete divergence-free magnetic induction in practical numerical computations. Using a mixed finite element approach, we discretize the velocity and pressure by stable finite elements, and the magnetic vector potential by Nedelec edge elements. We first discuss the existence and uniqueness of the weak solution for the new continuous MHD system. Next, under a reasonable hypothesis of the regularity for the exact solution, the optimal estimates for the velocity and magnetic vector potential are rigorously established. Finally, several numerical experiments have been provided to illustrate the performance and convergence rates of the numerical scheme.

毛士鹏，中国科学院数学与系统科学研究院研究员。2008年博士毕业于中国科学院数学与系统科学研究院计算数学所。曾经在法国的INRIA以及在瑞士苏黎世高工(ETHZ)做博士后和研究助理，主要研究方向为有限元方法及其应用，计算流体力学和磁流体力学等。在 Math. Comp., Numer. Math., SIAM. J. Numer. Math., SIAM J.Sci.Comput., Math. Model Meth. Appl. Sci. (M3AS) 等国际SCI杂志上发表论文60余篇。