Resolving an inequality in the final step of derivation of an *a priori* energy estimate

Question

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, and I am not certain if this was just a typo in the paper.

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:

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

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

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

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.

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

EDIT: Most recent attempt, just as the answer below came in:

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

I will use following prelimary estimates $\forall\ x,n\in\mathbb{R}_+$. The second employs the first. $(1+x)^n \le e^{nx}$ and $\sum_{k=1}^n(1+x)^k \le n(1+x)^n \le n e^{nx}$

So the estimate was $$E_n + X_n \le C_3\Delta{t}\|\vec{f}^{n+1}\|^2+(1 +\ (C_1+C_2)\Delta{t})E_{n-1}$$ $$\le C_3\Delta{t}\left(\frac{1+(C_1+C_2)\Delta{t}}{(C_1+C_2)\Delta{t}}\right)\|\vec{f}^{n+1}\|^2+(1 +\ (C_1+C_2)\Delta{t})E_{n-1}$$ i.e. $$E_n + X_n \le \left(1+(C_1+C_2)\Delta{t}\right)\left(\frac{C_3}{C_1+C_2}\|\vec{f}^{n+1}\|^2+E_{n-1}\right) \tag{1}$$ Since $X_n \ge 0$ we also have $$E_n \le \left(1+(C_1+C_2)\Delta{t}\right)\left(\frac{C_3}{C_1+C_2}\|\vec{f}^{n+1}\|^2+E_{n-1}\right) \tag{2}$$ Using (2) in (1), $$E_n + X_n \le \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)$$ $$\cdot\left(\frac{C_3}{C_1+C_2}\|\vec{f}^{n+1}\|^2 + \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)\left(\frac{C_3}{C_1+C_2}\|\vec{f}^{n}\|^2+E_{n-2}\right)\right)$$ Repeating the process $n-1$ times, $$E_n + X_n \le \frac{C_3}{C_1+C_2}\sum_{k=1}^n \|\vec{f}^{k+1}\|^2 \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)^k+ \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)^n E_0$$ i.e. $$E_n + X_n \le \frac{C_3}{C_1+C_2}\max_n\|\vec{f}^{n+1}\|^2\sum_{k=1}^n \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)^k+ \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)^n E_0$$ Using the first preliminary estimate, $$E_n + X_n \le \frac{C_3}{C_1+C_2}\max_n\|\vec{f}^{n+1}\|^2 n \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)^n+ \Bigl(1+(C_1+C_2)\Delta{t}\Bigr)^n E_0$$ And using the second preliminary estimate, $$E_n + X_n \le \frac{nC_3}{C_1+C_2}\max_n\|\vec{f}^{n+1}\|^2 e^{(C_1+C_2)T}+ e^{(C_1+C_2)T} E_0$$

There is a certain $n$ multiplying the first term, which is not present in the paper.

Answer 1

It looks to me like a discrete version of Gronwall's inequality.

If you have a sequence of numbers satisfying

$$ E_n \leq k \Delta t + (1 + \ell \Delta t) E_{n-1} $$

You can rewrite

$$ A_n = (1 + \ell \Delta t)^{-n} E_n $$

to get

$$ A_n \leq \frac{k \Delta t}{(1 + \ell \Delta t)^n} + A_{n-1} $$

which implies

$$ A_n \leq A_0 + k\Delta t \sum_{m = 1}^n \frac{1}{(1 + \ell \Delta t)^m} $$

The sum is a geometric series bounded by $\dfrac{1}{\ell \Delta t}$ (see comment below for the computation)

So you conclude

$$ A_n \leq A_0 + \frac{k}{\ell} $$

This gives

$$ E_n \leq (1 + \ell \Delta t)^n ( E_0 + \frac{k}{\ell} ) $$

which is exactly what is claimed, using that in the above derivation you can use

$$ k = C_3 \max \| f^{n+1} \|^2 $$

and

$$ \ell = C_1 + C_2 $$