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

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 ${\tau }^2+h^{p+\frac{1}{2}}$ error estimates for the $L^2$-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 $p\ge 1$, a stronger, so-called $4/3$-CFL, i.e. $\tau \le C{h}^{4/3}$. The theory is illustrated with some numerical examples.