A Priori Error Analysis of a Discontinuous Galerkin Scheme for the Magnetic Induction Equation

Abstract

Abstract We perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened 4 / 3 {4/3} -CFL condition ( Δ ⁢ t ∼ h 4 / 3 {\Delta t\sim h^{4/3}} ). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in L 2 {L^{2}} -sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree l ≥ 1 {l\geq 1} under the standard CFL condition.