Convergence rates for Galerkin approximation for magnetohydrodynamic type equations

Abstract

The motion of incompressible electrical conducting fluids can be modeled by magnetohydrodynamics equations, which consider the Navier-Stokes equations coupled with Maxwell’s equations. For the classical Navier- Stokes system, there exists an extensively study of the convergence rate for the Galerkin approximations. Here, we extend the estimates rates of spectral Galerkin approximations for the magnetohydrodynamic equations. We prove optimal error estimates in the 𝐿2 (Ω) and 𝐻1 (Ω)-norms, we obtain a result similar to the Rautmann for the 𝐻2 (Ω)-norm, and we reach basically the same level of knowledge as in the case of the classical Navier-Stokes.