The virtual element method for a 2D incompressible MHD system

We present a novel discretization for the two-dimensional incompressible Magnetohydrodynamics (MHD) system coupling an electromagnetic model and a fluid flow model. Our approach follows the framework of the Virtual Element Method and offers two main advantages. The method can be implemented on unstructured meshes making it highly versatile and capable of handling a broad set of problems involving interfaces, free-boundaries, or adaptive refinements of the mesh. The second advantage concerns the divergence of the magnetic flux field and the fluid velocity. Our approach guarantees that the numerical approximation of the magnetic flux field and the fluid velocity are divergence free if their initial states are divergence free. Importantly, the divergence-free condition for the fluid velocity is satisfied in a pointwise sense. We include a theoretical proof of the condition on the magnetic flux field, energy estimates and a well-posedness study. Numerical testing confirms robustness of the method and its convergence properties on a variety of meshes.