I'm studying a paper (see citation below) on numerical analysis, and came across this estimate. I am unable to figure out what was done in the final step.

Preliminary information: $\Delta{t}$ is the time step, $n$ is the iteration count from $0$ to final iteration $(N+1)$ such that $T=N\Delta{t}$ is the final time at $n=N+1$, $\vec{u}$ is a solution vector, $\vec{f}$ is a vector of forcing functions, and $E_k$ is the energy at iteration $n = k$. We also have various positive constants $C_1, C_2, C_3, C_a$.

Here is the estimate in the text:

$E_n + \frac{C_a\Delta{t}^2(C_1+C_2)}{1+(C_1+C_2)\Delta{t}}\|\nabla\vec{u}^{n+1}\|^2+\frac{C_a\Delta{t}^2(C_1+C_2)}{3(1+(C_1+C_2)\Delta{t})}\|\nabla\vec{u}^n\|^2$

$\le C_3\|\vec{f}^{n+1}\|^2\Delta{t} + (1 + (C_1 + C_2)\Delta{t})E_{n-1}$

$\le e^{(C_1+C_2)T}E_0 + \frac{C_3}{C_1+C_2}e^{(C_1+C_2)T} \max\limits_{n}\|\vec{f}^{n+1}\|^2$

Here is my attempt at what happened at the last step, before the final inequality:

For clarity, take $X_n = \frac{C_a\Delta{t}^2(C_1+C_2)}{1+(C_1+C_2)\Delta{t}}\|\nabla\vec{u}^{n+1}\|^2+\frac{C_a\Delta{t}^2(C_1+C_2)}{3(1+(C_1+C_2)\Delta{t})}\|\nabla\vec{u}^n\|^2$

Since $X_n \ge 0$, then $E_n \le E_n + X_n$ so

$E_n \le C_3\|\vec{f}^{n+1}\|^2\Delta{t} + (1 + (C_1 + C_2)\Delta{t})E_{n-1}$

Problem: Ignoring the first term on the right, I figured the second term can be bounded as:

$E_n \le \left(1 + (C_1 + C_2)\Delta{t}\right)E_{n-1}$

$\quad\: \le \left(1 + (C_1 + C_2)\Delta{t}\right)^{n-1}E_0$

$\quad\: \le e^{(1 + (C_1 + C_2)T}E_0 \qquad$ since $e^{nx} \ge (1+x)^n\ \forall\ n,x\in\mathbb{R}_+$

The parts I can't figure out:

• what happened to the first term i.e. $C_3\|\vec{f}^{n+1}\|^2\Delta{t}$.
• How to re-combine the results with the omitted $X_n$.

Thanks.

Paper: Chen, Wenbin; Gunzburger, Max; Sun, Dong; Wang, Xiaoming, Efficient and long-time accurate second-order methods for the Stokes-Darcy system, SIAM J. Numer. Anal. 51, No. 5, 2563-2584 (2013). ZBL1282.76094.