Optimal error estimate of the penalty method for the 2D/3D time-dependent MHD equations

In this article, we mainly consider a first-order decoupling penalty method for the 2D/3D time-dependent incompressible magnetohydrodynamic (MHD) equations in a convex domain. This method applies a penalty term to the constraint “divu = 0,” which allows us to transform the saddle point problem into two small problems to solve. The time discretization is based on the backward Euler scheme. Moreover, we derive the optimal error estimate for the penalty method under semi-discretization with the relationship 𝜖 = O(Δt). Finally, we give abundant of numerical tests to verify the theoretical result and the spatial discretization is based on Lagrange finite element.