A charge-conservative finite element method for inductionless MHD equations. Part I: Convergence

A charge-conservative finite element method is proposed to solve the inductionless and incompressible magnetohydrodynamic (MHD) equations in three dimensions. The method yields an exactly divergence-free current density directly. We prove that, as the spatial mesh size h \rightarrow 0, the fully discrete solutions converge to the solutions of the semicontinuous problem weakly in \bfitH ¹(\Omega) \times \bfitH (div, \Omega) upon an extracted subsequence, and as the time step size \tau \rightarrow 0, the semicontinuous solutions converge to the solutions of the continuous problem weakly in \bfitL ²(0, T; \bfitH ¹(\Omega))\times \bfitL ²(0, T; \bfitH (div, \Omega)) upon an extracted subsequence. This yields the existence of the continuous solutions naturally. Three numerical experiments are presented to show the convergence rate of discrete solutions and the charge-conservation of the method.