Implicit-explicit multistep formulations for finite element discretisations using continuous interior penalty

¹ Department of Mathematics, University College London, London, UK
² Division of Applied Mathematics, Brown University, Providence, RI, USA

^* Corresponding author: johnny_guzman@brown.edu

Received: 7 June 2021
Accepted: 14 December 2021

Abstract

We consider a finite element method with symmetric stabilisation for the discretisation of the transient convection–diffusion equation. For the time-discretisation we consider either the second order backwards differentiation formula or the Crank–Nicolson method. Both the convection term and the associated stabilisation are treated explicitly using an extrapolated approximate solution. We prove stability of the method and the τ² + h^p+½ error estimates for the L²-norm under either the standard hyperbolic CFL condition, when piecewise affine (p = 1) approximation is used, or in the case of finite element approximation of order p ≥ 1, a stronger, so-called 4/3-CFL, i.e. τ ≤ Ch^4/3. The theory is illustrated with some numerical examples.