Conservative mixed Discontinuous Galerkin finite element method with full decoupling strategy for incompressible MHD problems

In this research, we propose a mixed Discontinuous Galerkin (DG) finite element method (FEM) for the numerical solution of incompressible magnetohydrodynamics (MHD) problems, ensuring the divergence-free condition for the velocity field. Our proposed methodology is characterized by its energy stability. Subsequently, we construct a novel finite element discretization scheme, for which we establish the theoretical framework of energy stability. Furthermore, we incorporate a preconditioning strategy that facilitates the development of an alternative positive definite, fully decoupled linearization approach. A significant advantage of this scheme is its ability to circumvent the complexities associated with solving saddle point problems at each time step. We demonstrate the uniqueness and convergence of the solutions within this fully decoupled, block-structured linear system. This theoretical validation is complemented by a series of numerical experiments that substantiate the effectiveness and accuracy of our proposed method.