Preprint 2024-20
Ana Alonso-Rodriguez, Jessika Camaño, Ricardo Oyarzúa:
Analysis of a FEM with exactly divergence-free magnetic field for the stationary MHD problem
Abstract:
In this work we analyze a mixed finite element method for the stationary incompressible magneto-hydrodynamic problem providing an exactly divergence-free approximation of the magnetic field and a direct approximation of the electric field. The method is based on the introduction of the electric field as a further unknown leading to a mixed formulation where the primary magnetic variables consist of the electric and the magnetic fields, and a Lagrange multiplier included to enforce the divergence-free constraint of the magnetic field, whereas the hydrodynamic unknowns are the velocity and pressure. Then the associated Galerkin scheme can be defined by employing Nédélec and Raviart–Thomas elements of lowest order for the electric and magnetic fields, respectively, discontinuous piecewise constants for the Lagrange multiplier and any inf-sup stable pair of elements for the velocity and pressure, such as the Mini-element. The analysis of the continuous and discrete problems are carried out by means of the Banach–Nečas–Babuška theorem and the Banach fixed-point theorem, under a sufficiently small data assumption and quasi-uniformity of the mesh, the latter for the discrete scheme. Finally, we derive the corresponding Cea’s estimate and provide the theoretical rate of convergence.
