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.
翻译:我们考虑的是具有对称稳定法的有限元素方法,用于对瞬时对流对流反扩散方程式进行分解。 在时间分解时,我们考虑的是第二顺序向后偏差公式或Crank-Nicolson法。对流期和相关的稳定化都使用外推近似解决方案进行明确处理。我们证明这种方法的稳定性,以及在标准双曲式CFL条件下对$L2美元(p=1美元)的错误估计值($\tau2 + h ⁇ p ⁇ p ⁇ frac12+$),或者在使用整形折线折线($p=1美元)近似值时,或者在定点元素近似值($p\ge1美元,一种更强的、所谓的4/3美元CFL,即$\tau\leq Ch ⁇ 4/3美元)的情况下,该方法的稳定性和$$leq\leq Ch*4/3美元。 理论用一些数字例子加以说明。