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

TL;DR

This paper introduces a finite element method with symmetric stabilization for transient convection-diffusion equations, analyzing stability and error estimates for explicit-implicit time discretizations under CFL conditions.

Contribution

It develops a novel explicit-implicit multistep finite element scheme with stability proof and error bounds, including a new CFL condition for higher-order elements.

Findings

Proves stability of the proposed method.

Derives error estimates of order $ au^2 + h^{p+1/2}$.

Numerical examples confirm theoretical results.

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 $\tau^2 + h^{p+{\frac12}}$ 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. $\tau \leq C h^{4/3}$. The theory is illustrated with some numerical examples.