A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation

Abstract

In this paper we devise a first-order-in-time, second-order-in-space, convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. The unconditional unique solvability, energy stability and $\ell ^\infty (0, T; \ell ^4)$ stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the $\ell ^\infty (0, T; \ell ^2)$ and $\ell ^\infty (0, T; \ell ^\infty )$ norms. The proofs of these estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The $\ell ^2$ convergence proof requires no refinement path constraint, while the one involving the $\ell ^\infty$ norm requires only a mild linear refinement constraint, $s \le C h$. The key estimates for the error analyses take full advantage of the unconditional $\ell ^\infty (0, T; \ell ^4)$ stability of the numerical solution and an interpolation estimate of the form $\left\| \phi \right\| _4 \le C \left\| \phi \right\| _2^\alpha \left\| \nabla _h\phi \right\| _2^{1-\alpha },\alpha = \frac{4-D}{4},D=1,2,3$, which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.

Appendices

Appendix A: Finite difference discretization of space

Our primary goal in this appendix is to define some finite-difference operators and provide some summation-by-parts formulae in one and two space dimensions that are used to derive and analyze the numerical schemes. Everything extends straightforwardly to 3D. We make extensive use of the notation and results for cell-centered functions from [39, 40]. The reader is directed to those references for more complete details.

In 1D we will work on the interval $\Omega = (0,L)$, with $L = m\cdot h$, and in 2D, we work with the rectangle $\Omega = (0,L_1)\times (0,L_2)$, with $L_1 = m\cdot h$ and $L_2 = n\cdot h$, where $m$ and $n$ are positive integers and $h>0$ is the spatial step size. Define $p_r := (r-\frac{1}{2})\cdot h$, where $r$ takes on integer and half-integer values. For any positive integer $\ell$, define $E_\ell = \left\{ p_r \ |\ r=\frac{1}{2},\ldots , \ell +\frac{1}{2}\right\} ,C_\ell = \left\{ p_r \ |\ r=1,\ldots , \ell \right\} ,C_{\overline{\ell }} = \left\{ p_r\cdot h\ |\ r=0,\ldots , \ell +1\right\}$. We need the 1D grid function spaces

$$\begin{aligned} {\mathcal C}_m = \left\{ \phi : C_m\rightarrow \mathbb {R} \right\} , \ \ {\mathcal E}_m = \left\{ u: E_m \rightarrow \mathbb {R} \right\} , \end{aligned}$$

and the 2D grid function spaces

$$\begin{aligned}&{\mathcal C}_{m\times n} = \left\{ \phi : C_m\times C_n \rightarrow \mathbb {R} \right\} ,\quad {\mathcal V}_{m\times n} = \left\{ f: E_m\times E_n \rightarrow \mathbb {R} \right\} ,\\&{\mathcal E}^\mathrm{ew}_{m\times n} = \left\{ u: E_m\times C_n \rightarrow \mathbb {R} \right\} ,\quad \, {\mathcal E}^\mathrm{ns}_{m\times n} = \left\{ v: C_m\times E_n \rightarrow \mathbb {R} \right\} , \end{aligned}$$

We use the notation $\phi _{i,j} := \phi \left( p_i,p_j\right)$ for cell-centered functions, i.e., those in the space ${\mathcal C}_{m\times n}$. In component form east-west edge-centered functions, i.e., those in the space ${\mathcal E}^\mathrm{ew}_{m\times n}$, are identified via $u_{i+\frac{1}{2},j}:=u(p_{i+\frac{1}{2}},p_j)$. In component form north-south edge-centered functions, i.e., those in the space ${\mathcal E}^\mathrm{ns}_{m\times n}$, are identified via $u_{i+\frac{1}{2},j}:=u(p_{i+\frac{1}{2}},p_j)$. The functions of ${\mathcal V}_{m\times n}$ are called vertex-centered functions. In component form vertex-centered functions are identified via $f_{i+\frac{1}{2},j+\frac{1}{2}}:=f(p_{i+\frac{1}{2}},p_{j+\frac{1}{2}})$. The 1D cell-centered and edge-centered functions are easier to express.

We will need the 1D grid inner-products $\left( \, \cdot \, | \, \cdot \, \right)$ and $\left[ \, \cdot \, \Vert \, \cdot \, \right]$ and the 2D grid inner-products $\left( \, \cdot \, \Vert \, \cdot \, \right)$, $\left[ \, \cdot \, \Vert \, \cdot \, \right] _\mathrm{ew}$, $\left[ \, \cdot \, \Vert \, \cdot \, \right] _\mathrm{ns}$ that are defined in [39, 40].

We shall say the cell-centered function $\phi \in {\mathcal C}_{m\times n}$ is periodic if and only if, for all $p,\, q\in \mathbb {Z}$,

$$\begin{aligned} \phi _{i+ p\cdot m, j+ q\cdot n} = \phi _{i, j} \quad i = 1,\ldots ,m, \quad j = 1,\ldots , n. \end{aligned}$$

(6.1)

Here we have abused notation a bit, since $\phi$ is not explicitly defined on an infinite grid. Of course, $\phi$ can be extended as a periodic function in a perfectly natural way, which is the context in which we view the last definition. Similar definitions are implied for periodic edge-centered and vertex-centered grid functions. The 1D and 3D cases are analogous and are suppressed.

The reader is referred to [39, 40] for the precise definitions of the edge-to-center difference operators $d_x : {\mathcal E}_{m\times n}^\mathrm{ew}\rightarrow {\mathcal C}_{m\times n}$ and $d_y : {\mathcal E}_{m\times n}^\mathrm{ns}\rightarrow {\mathcal C}_{m\times n}$; the $x$—dimension center-to-edge average and difference operators, respectively, $A_x,\, D_x: {\mathcal C}_{m\times n}\rightarrow {\mathcal E}_{m\times n}^\mathrm{ew}$; the $y$—dimension center-to-edge average and difference operators, respectively, $A_y,\, D_y: {\mathcal C}_{m\times n}\rightarrow {\mathcal E}_{m\times n}^\mathrm{ns}$; and the standard 2D discrete Laplacian, $\Delta _h: {\mathcal C}_{m \times n}\rightarrow {\mathcal C}_{m\times n}$.These operators have analogs in 1D and 3D that should be clear to the reader.

We will use the grid function norms defined in [39, 40]. The reader is referred to those references for the precise definitions of $\left\| \, \cdot \, \right\| _2,\left\| \, \cdot \, \right\| _\infty , \left\| \, \cdot \, \right\| _p (1\le p < \infty ), \left\| \, \cdot \, \right\| _{0,2}, \left\| \, \cdot \, \right\| _{1,2}$, and $\left\| \phi \right\| _{2,2}$. We will specifically use the following inverse inequality in 2D: for any $\phi \in {\mathcal C}_{m\times n}$ and all $1\le p < \infty$

$$\begin{aligned} \left\| \phi \right\| _{\infty }\le h^{-\frac{2}{p}}\left\| \phi \right\| _p. \end{aligned}$$

(6.2)

Again, the analogous norms in 1D and 3D should be clear.

Using the definitions given in this appendix and in [39, 40], we obtain the following summation-by-parts formulae whose proofs are simple:

Proposition 6.1

If $\phi \in {\mathcal C}_{m\times n}$ and $f\in {\mathcal E}_{m\times n}^\mathrm{ew}$ are periodic then

$$\begin{aligned} h^2\, \left[ D_x \phi \Vert f \right] _\mathrm{ew} = -h^2\, \left( \phi \Vert d_x f \right) , \end{aligned}$$

(6.3)

and if $\phi \in {\mathcal C}_{m\times n}$ and $f\in {\mathcal E}_{m\times n}^\mathrm{ns}$ are periodic then

$$\begin{aligned} h^2\, \left[ D_y\phi \Vert f \right] _\mathrm{ns} = -h^2\, \left( \phi \Vert d_y f \right) . \end{aligned}$$

(6.4)

Proposition 6.2

Let $\phi ,\, \psi \in {\mathcal C}_{\overline{m}\times \overline{n}}$ be periodic grid functions. Then

$$\begin{aligned} h^2\, \left[ D_x\phi \Vert D_x\psi \right] _\mathrm{ew} +h^2\, \left[ D_y\phi \Vert D_y\psi \right] _\mathrm{ns} = -h^2\, \left( \phi \Vert \Delta _h\psi \right) . \end{aligned}$$

(6.5)

Proposition 6.3

Let $\phi ,\, \psi \in {\mathcal C}_{m \times n}$ be periodic grid functions. Then

$$\begin{aligned} h^2\, \left( \phi \Vert \Delta _h\psi \right) = h^2\, \left( \Delta _h\phi \Vert \psi \right) . \end{aligned}$$

(6.6)

Analogs in 1D and 3D of the summation-by-parts formulae above are straightforward.

Appendix B: Proof of Lemma 2.2

Our goal in this section is to prove Lemma 2.2. We will need the following well-known result, whose proof we omit for the sake of brevity.

Lemma 7.1

Suppose that $\phi \in {\mathcal C}_{m\times n}$ is periodic and of mean zero, i.e., $\left( \phi \Vert \mathbf{1} \right) = 0$. Then, $\phi$ satisfies a discrete Poincare-type inequality of the form

$$\begin{aligned} \left\| \phi \right\| _2 \le C\left\| \nabla _h\phi \right\| _2 , \end{aligned}$$

(7.1)

for some $C>0$ that is independent of $h$ and $\phi$.

Analogous results also hold for 1D and 3D grid functions. Now, we state and prove and estimate bounding the discrete 4-norm.

Lemma 7.2

Suppose that $\left\| \phi \right\| \in {\mathcal C}_{m\times n}$ is periodic and of mean zero, i.e., $\left( \phi \Vert \mathbf{1} \right) = 0$. Then,

$$\begin{aligned} \left\| \phi \right\| _4 \le C_{10} \left\| \phi \right\| _2^{\frac{1}{2}}\left\| \nabla _h\phi \right\| _2^{\frac{1}{2}} , \end{aligned}$$

(7.2)

for some constant $C_{10}>0$ that is independent of $h$.

Proof

Appealing to Lemma 6 of [40], we have

$$\begin{aligned} \left| \phi _{i,j}\right| ^2&\le \frac{h}{L_1}\sum _{i'=1}^m \left( \phi _{i',j}\right) ^2 + 2h\sum _{i'=1}^m D_x\phi _{i'+\frac{1}{2},j} A_x\phi _{i'+\frac{1}{2},j} =: Q_j ,\end{aligned}$$

(7.3)

$$\begin{aligned} \left| \phi _{i,j}\right| ^2&\le \frac{h}{L_2}\sum _{j'=1}^n \left( \phi _{i,j'}\right) ^2 + 2h\sum _{j'=1}^n D_y\phi _{i,j'+\frac{1}{2},} A_y\phi _{i,j'+\frac{1}{2}} =: R_i\ , \end{aligned}$$

(7.4)

for any $1\le i\le m$ and any $1\le j\le n$. Then

$$\begin{aligned} \left\| \phi \right\| _4^4&\le h^2 \sum _{i = 1}^m\sum _{j = 1}^n R_i Q_j\nonumber \\&= \left( \frac{1}{L_2} \left\| \phi \right\| _2^2 \!+\!2 h^2 \left[ D_y\phi \Vert A_y\phi \right] _\mathrm{ns} \right) \left( \frac{1}{L_1} \left\| \phi \right\| _2^2 \!+\!2 h^2 \left[ D_x\phi \Vert A_x\phi \right] _\mathrm{ew} \right) .\quad \quad \quad \end{aligned}$$

(7.5)

Since $\phi$ is periodic and of mean zero, it satisfies the discrete Poincare-type inequality (7.1). Using this and the Cauchy-Schwartz inequality, we have

$$\begin{aligned} \left\| \phi \right\| _4^4&\le C\left( \left\| \phi \right\| _2\left\| \nabla _h\phi \right\| _2 +2 h^2 \left[ D_y\phi \Vert A_y\phi \right] _\mathrm{ns} \right) \left( \left\| \phi \right\| _2\left\| \nabla _h\phi \right\| _2 \!+\!2 h^2 \left[ D_x\phi \Vert A_x\phi \right] _\mathrm{ew} \right) \nonumber \\&\le C\left( \left\| \phi \right\| _2\left\| \nabla _h\phi \right\| _2 +\sqrt{h^2\left[ D_y\phi \Vert D_y\phi \right] _\mathrm{ns}}\left\| \phi \right\| _2 \right) \nonumber \\&\times \left( \left\| \phi \right\| _2\left\| \nabla _h\phi \right\| _2 +\sqrt{h^2\left[ D_x\phi \Vert D_x\phi \right] _\mathrm{ew}}\left\| \phi \right\| _2 \right) \nonumber \\&\le C\left( \left\| \phi \right\| _2\left\| \nabla _h\phi \right\| _2 +\left\| \nabla _h\phi \right\| _2\left\| \phi \right\| _2 \right) ^2. \end{aligned}$$

(7.6)

The result follows. $\square$

Remark 7.3

It is easy to conclude a 1D version of the previous Lemma as a corollary: if $\phi \in {\mathcal C}_m$ is periodic and of mean zero, i.e., $\left( \phi | \mathbf{1} \right) = 0$, then,

$$\begin{aligned} \left\| \phi \right\| _4 \le C_{10} \left\| \phi \right\| _2^{\frac{1}{2}}\left\| \nabla _h\phi \right\| _2^{\frac{1}{2}}. \end{aligned}$$

(7.7)

This result is not optimal in 1D. We claim that for 1D and 3D cases, we can derive the following, dimension-dependent estimates: if $\phi$ is mean zero and periodic, then

$$\begin{aligned} \left\| \phi \right\| _4&\le C \left\| \phi \right\| _2^{\frac{3}{4}} \left\| \nabla _h \phi \right\| _2^{\frac{1}{4}} , \quad (\hbox {in 1D}) ,\end{aligned}$$

(7.8)

$$\begin{aligned} \left\| \phi \right\| _4&\le C \left\| \phi \right\| _2^{\frac{1}{4}} \left\| \nabla _h \phi \right\| _2^{\frac{3}{4}} , \quad (\hbox {in 3D}). \end{aligned}$$

(7.9)

The continuous space versions of estimates of these type can be found in the book by Temam [35].

Now we prove a 1D version of Lemma 2.2. The extension to 2D and 3D will follow by similar arguments, working dimension-by-dimension.

Lemma 7.4

Suppose $\phi ,\, \psi \in \mathcal {C}_{n}$ are periodic with equal means, i.e., $\left( \phi -\psi | \mathbf{1} \right) =0$. Suppose that $\phi$ and $\psi$ are in the class of grid functions satisfying

$$\begin{aligned} \left\| \phi \right\| _4 + \left\| \phi \right\| _{\infty } + \left\| D\phi \right\| _{\infty } \le C_{11}, \quad \left\| \psi \right\| _4 \le C_{11}, \end{aligned}$$

(7.10)

where $C_{11}$ is an $h$-independent positive constant. Then there exists a positive constant $C_{12}$ that depends only on $C_{10}$ and $C_{11}$ such that

$$\begin{aligned} 2h\left( \phi ^3 - \psi ^3 | \Delta _h \left( \phi -\psi \right) \right) \!\le \! \frac{C_{12}}{\alpha ^3} \left\| \phi -\psi \right\| _2^2 \!+\! \alpha \left\| D\left( \phi -\psi \right) \right\| _2^2 , \quad \! \forall \alpha > 0.\quad \quad \end{aligned}$$

(7.11)

Proof

Using summation-by-parts,

$$\begin{aligned} h\left( \phi ^3 - \psi ^3 | \Delta _h \left( \phi -\psi \right) \right) = -h \left[ D\left( \phi ^3 - \psi ^3\right) \Vert D\left( \phi -\psi \right) \right] ,\nonumber \end{aligned}$$

where $\Delta _h$ is the 1D discrete lapacian and $D$ is the center-to-edge difference [39, 40]. Denote

$$\begin{aligned} \mathbb {A}\phi _{i+\frac{1}{2}} := \frac{1}{2}\phi _{i+1}^2+\frac{1}{2}\phi _{i}^2+\frac{1}{2}\left( \phi _{i+1}+\phi _{i}\right) ^2. \end{aligned}$$

(7.12)

Then $D \phi ^3_{i+\frac{1}{2}} = \mathbb {A}\phi _{i+\frac{1}{2}}D\phi _{i+\frac{1}{2}}$ and $D \psi ^3_{i+\frac{1}{2}} = \mathbb {A}\psi _{i+\frac{1}{2}}D\psi _{i+\frac{1}{2}}$, and, furthermore,

$$\begin{aligned} -h \left[ D\left( \phi ^3 - \psi ^3\right) \Vert D\left( \phi -\psi \right) \right] = -h\left[ \mathbb {A}\phi D\phi -\mathbb {A}\psi D\psi \Vert D\left( \phi -\psi \right) \right] .\quad \quad \quad \end{aligned}$$

(7.13)

By adding and subtracting $\mathbb {A}\psi D\phi$, we have

$$\begin{aligned}&-h \left[ D\left( \phi ^3 - \psi ^3\right) \Vert D\left( \phi -\psi \right) \right] \nonumber \\&\quad = -h\left[ \mathbb {A}\phi D\phi +\mathbb {A}\psi D\phi -\mathbb {A}\psi D\phi -\mathbb {A}\psi D\psi \Vert D\left( \phi -\psi \right) \right] . \end{aligned}$$

(7.14)

Observing that $\mathbb {A}\psi \ge 0$ and

$$\begin{aligned} \left[ \mathbb {A}\psi D\phi -\mathbb {A}\psi D\psi \Vert D\left( \phi -\psi \right) \right] = \left[ \mathbb {A}\psi \Vert \left( D\left( \phi -\psi \right) \right) ^2 \right] \end{aligned}$$

(7.15)

yields

$$\begin{aligned} -h \left[ D\left( \phi ^3 - \psi ^3\right) \Vert D\left( \phi -\psi \right) \right]&\le -h\left[ \mathbb {A}\phi D\phi -\mathbb {A}\psi D\phi \Vert D\left( \phi -\psi \right) \right] \nonumber \\&\le h\Big |\left[ \mathbb {A}\phi D\phi -\mathbb {A}\psi D\phi \Vert D\left( \phi -\psi \right) \right] \Big | \nonumber \\&\le h\Big [ \ \big |\mathbb {A}\phi -\mathbb {A}\psi \big | \ \big | D\phi \big | \ \Big | \ \big |D\left( \phi -\psi \right) \big | \ \Big ] \nonumber \\&\le C_{11} h\Big [ \ \big |\mathbb {A}\phi -\mathbb {A}\psi \big | \ \Big | \ \big |D\left( \phi -\psi \right) \big | \ \Big ] \nonumber \\&\le C_{11} h\left[ \mathbb {A}\phi -\mathbb {A}\psi \Vert \mathbb {A}\phi -\mathbb {A}\psi \right] ^\frac{1}{2}\nonumber \\&\times \left[ D\left( \phi -\psi \right) \Vert D\left( \phi -\psi \right) \right] ^\frac{1}{2}. \end{aligned}$$

(7.16)

Using the definitions, and Cauchy’s inequality, one can show

$$\begin{aligned} \left( \mathbb {A}\phi -\mathbb {A}\psi \right) ^2_{i+\frac{1}{2}}&\le \frac{15}{4}\left( \left( \phi _{i+1}+\psi _{i+1}\right) ^2+\left( \phi _{i}+\psi _{i}\right) ^2\right) \nonumber \\&\times \left( \left( \phi _{i+1}-\psi _{i+1}\right) ^2 +\left( \phi _{i}-\psi _{i}\right) ^2\right) . \end{aligned}$$

(7.17)

Now, define

$$\begin{aligned} \mathbb {H}_{i+\frac{1}{2}} :=&\frac{1}{2}\left( \left( \phi _{i+1}+\psi _{i+1}\right) ^2+\left( \phi _{i}+\psi _{i}\right) ^2\right) ,\end{aligned}$$

(7.18)

$$\begin{aligned} \mathbb {G}_{i+\frac{1}{2}} :=&\frac{1}{2}\left( \left( \phi _{i+1}-\psi _{i+1}\right) ^2 +\left( \phi _{i}-\psi _{i}\right) ^2\right) , \end{aligned}$$

(7.19)

so that, applying the Cauchy-Schwartz inequality, we have

$$\begin{aligned} \left[ \mathbb {A}\phi -\mathbb {A}\psi \Vert \mathbb {A}\phi -\mathbb {A}\psi \right]&\le 15 \left[ \mathbb {H} \Vert \mathbb {G} \right] \nonumber \\&\le 15 \left[ \mathbb {H} \Vert \mathbb {H} \right] ^{\frac{1}{2}}\left[ \mathbb {G} \Vert \mathbb {G} \right] ^{\frac{1}{2}} . \end{aligned}$$

(7.20)

The definition of $\mathbb {H}_{i+\frac{1}{2}}$ shows that

$$\begin{aligned} \mathbb {H}^2_{i+\frac{1}{2}}&= \frac{1}{4}\left( \left( \phi _{i+1}+\psi _{i+1}\right) ^2+\left( \phi _{i}+\psi _{i}\right) ^2\right) ^2 \nonumber \\&\le 4\left( \left( \phi _{i+1}\right) ^4+\left( \psi _{i+1}\right) ^4+\left( \phi _{i}\right) ^4+\left( \psi _{i}\right) ^4\right) . \end{aligned}$$

(7.21)

Thus by the definition of $\left[ \cdot \Vert \cdot \right]$ and using periodicity, we get

$$\begin{aligned} \left[ \mathbb {H} \Vert \mathbb {H} \right] \le \frac{8}{h}\left\| \phi \right\| ^4_{4}+\frac{8}{h}\left\| \psi \right\| ^4_{4}. \end{aligned}$$

(7.22)

Similarly,

$$\begin{aligned} \left[ \mathbb {G} \Vert \mathbb {G} \right] \le \frac{1}{h}\left\| \phi -\psi \right\| ^4_{4}, \end{aligned}$$

(7.23)

where we are using the 1D norms: $\left\| \phi \right\| ^p_p = h\sum _{i=1}^m |\phi _i|^p$. Thus

$$\begin{aligned} \left[ \mathbb {A}\phi -\mathbb {A}\psi \Vert \mathbb {A}\phi -\mathbb {A}\psi \right] \le \frac{1}{h}\left[ 1{,}800\left( \left\| \phi \right\| ^4_{4}+\left\| \psi \right\| ^4_{4}\right) \right] ^{\frac{1}{2}}\left\| \phi -\psi \right\| ^2_4. \end{aligned}$$

(7.24)

Combining the above results, we have

$$\begin{aligned}&- 2h \left[ D \left( \phi ^3-\psi ^3\right) \Vert D \left( \phi -\psi \right) \right] \nonumber \\&\quad \le 2C_{11} \left[ 1{,}800\left( \left\| \phi \right\| ^4_{4}+\left\| \psi \right\| ^4_{4}\right) \right] ^{\frac{1}{4}}\left\| \phi -\psi \right\| _4 \left\| D \left( \phi -\psi \right) \right\| _{2} \nonumber \\&\quad \le 2^{\frac{5}{4}}\, 1{,}800\, C^2_{11}\, \left\| \phi -\psi \right\| _4 \left\| D \left( \phi -\psi \right) \right\| _{2}. \end{aligned}$$

(7.25)

We now recall Lemma 7.2, specifically the (non-optimal) 1D estimate (7.7). Under the condition $\left( \phi -\psi | 1 \right) =0$, we have

$$\begin{aligned} \left\| \phi -\psi \right\| _4 \le C_{10} \left\| \phi -\psi \right\| _2^{\frac{1}{2}} \left\| D\left( \phi -\psi \right) \right\| _2^{\frac{1}{2}}. \end{aligned}$$

(7.26)

Therefore,

$$\begin{aligned} 2h\left( \phi ^3 - \psi ^3 | \Delta _h \left( \phi -\psi \right) \right) \le M \left\| \phi -\psi \right\| _2^{\frac{1}{2}} \left\| D\left( \phi -\psi \right) \right\| _2^{\frac{3}{2}}. \end{aligned}$$

(7.27)

Observe that $M$ is independent of $h$ and depends upon $C_{10}$ and $C_{11}$.

To finish up, we use the Young inequality

$$\begin{aligned} a \cdot b \le \frac{a^p}{p} + \frac{b^q}{q} , \quad \forall \ a, b > 0 , \quad \frac{1}{p} + \frac{1}{q} = 1 , \end{aligned}$$

(7.28)

with the choices $p\!=\!4,q\!=\!\frac{4}{3},a \!=\! \left( \frac{3}{4} \alpha ^{-1} \right) ^{\frac{3}{4}} M \left\| \phi \!-\!\psi \right\| _2^{\frac{1}{2}}, b \!=\! \left( \frac{4}{3} \alpha \right) ^{\frac{3}{4}} \left\| D\left( \phi \!-\!\psi \right) \right\| _2^{\frac{3}{2}}$. We obtain

$$\begin{aligned} M \left\| \phi -\psi \right\| _2^{\frac{1}{2}} \cdot \left\| D\left( \phi -\psi \right) \right\| _2^{\frac{3}{2}}&= a \cdot b \le \frac{1}{4} a^4 + \frac{3}{4} b^{\frac{4}{3}}\nonumber \\&= \frac{1}{4} M^4 \cdot \frac{\left( \frac{3}{4} \right) ^3}{\alpha ^3} \left\| \phi -\psi \right\| _2^2 + \alpha \left\| D\left( \phi -\psi \right) \right\| _2^2.\nonumber \\ \end{aligned}$$

(7.29)

The result is proven by taking $C_{12} = \frac{1}{4} M^4 \cdot \left( \frac{3}{4} \right) ^3$. $\square$

Remark 7.5

The form of the derived inequality (7.11) is valid for both of the 1D and 2D cases. For the 3D case, a combination of (7.9) and the estimates like those derived above leads to the following result

$$\begin{aligned} 2h^3 \left( \phi ^3 - \psi ^3 | \! | \! | \Delta _h \left( \phi -\psi \right) \right) \le \frac{C_{12}}{\alpha ^7} \left\| \phi -\psi \right\| _2^2 + \alpha \left\| \nabla _h \left( \phi -\psi \right) \right\| _2^2 , \quad \forall \alpha > 0 ,\nonumber \\ \end{aligned}$$

(7.30)

the only changes being the $\alpha ^7$ replaces $\alpha ^3$ and we use the triple summation $\left( \, \cdot \, | \! | \! | \, \cdot \, \right)$. As a result, an unconditional $\ell ^\infty (0,T; \ell ^2)$ convergence in 3D can be derived in the same manner. The details are omitted in this paper for the sake of brevity.

Appendix C: Proof of Lemma 2.3

Our goal in this appendix is to prove Lemma 2.3. We will do this by proving the 1D version and leaving the 2D argument to the reader. Suppose $\psi \in {\mathcal C}_m$ and $f \in {\mathcal E}_m$ are periodic grid functions. Then $\left[ f \star \psi \right] : {\mathcal E}_m \times {\mathcal C}_m \rightarrow {\mathcal C}_m$is defined component-wise as follows

$$\begin{aligned} \left[ f \star \psi \right] _i := h\sum ^{m}_{k=1}f_{k+\frac{1}{2}}\psi _{i-k}. \end{aligned}$$

(8.1)

Note very carefully that the order is important in the definition of the discrete convolution $\left[ \, \cdot \, \star \, \cdot \, \right]$.

Lemma 8.1

Suppose $\phi , \psi \in \mathcal {C}_{n}$ are periodic. Assume that $\mathsf {f} \in C_\mathrm{per}^\infty (0,L)$ is even and set $f_{i+\frac{1}{2}} := \mathsf {f}\left( p_{i+\frac{1}{2}}\right)$, so that $f\in \mathcal {E}_m$. Then, for any $\alpha >0$,

$$\begin{aligned} - 2h\left( \left[ f \star \psi \right] | \Delta _h \phi \right) \le \frac{C}{\alpha } \left\| \psi \right\| ^2_2 + \alpha \left\| D \phi \right\| ^2_2 , \end{aligned}$$

(8.2)

for some $C>0$ that depends upon $\mathsf {f}$, but is independent of $h$.

Proof

Using summation-by-parts and the periodic boundary conditions,

$$\begin{aligned} - h\left( \left[ f \star \psi \right] | \Delta _h\phi \right) = h\left[ D \left[ f \star \psi \right] \Vert D \phi \right] . \end{aligned}$$

(8.3)

By definition, and using periodicity,

$$\begin{aligned} h D \left[ f \star \psi \right] _{i+\frac{1}{2}} = h\sum _{k=1}^m f_{k+\frac{1}{2}}D\psi _{i-k+\frac{1}{2}} = h\sum _{k=1}^m d f_{i+1-k}\psi _k, \end{aligned}$$

(8.4)

where $D$ is the center-to-edge difference and $d$ is the edge-to-center difference [39, 40]. Then, using periodicity, Cauchy’s inequality, and summation shifts, we have

$$\begin{aligned} \Big | h\left[ D \left[ f \star \psi \right] \Vert D\phi \right] \Big |&= \left| h\sum _{i=1}^m D \left[ f \star \psi \right] _{i+\frac{1}{2}} D\phi _{i+\frac{1}{2}}\right| \nonumber \\&\le h^2\sum _{i=1}^m \sum _{k=1}^m \left| d f_{i+1-k}\right| \cdot \left| \psi _k D\phi _{i+\frac{1}{2}}\right| \nonumber \\&\le h^2\sum _{i=1}^m \sum _{k=1}^m \left| d f_{i+1-k}\right| \left( \frac{1}{2\beta } \left( \psi _k\right) ^2 +\frac{\beta }{2} \left( D\phi _{i+\frac{1}{2}}\right) ^2\right) \nonumber \\&= \frac{1}{2\beta } h^2\sum _{i=1}^m \sum _{k=1}^m \left| d f_{i+1-k}\right| \left( \psi _k\right) ^2\\&+\frac{\beta }{2} h^2\sum _{i=1}^m \sum _{k=1}^m \left| d f_{i+1-k}\right| \left( D\phi _{i+\frac{1}{2}}\right) ^2 \nonumber \\&= \frac{1}{2\beta } \left\| d f \right\| _1 \left\| \psi \right\| _2^2 +\frac{\beta }{2} \left\| d f \right\| _1 \left\| D\phi \right\| _2^2 \nonumber \\&= \frac{L}{2\beta } \left\| d f \right\| _2 \left\| \psi \right\| _2^2 +\frac{L\beta }{2} \left\| d f \right\| _2 \left\| D\phi \right\| _2^2 \nonumber \\&= \frac{LC}{2\beta } \left\| \psi \right\| _2^2 +\frac{LC\beta }{2} \left\| D\phi \right\| _2^2 , \nonumber \end{aligned}$$

for any $\beta >0$. Here we have used the fact that

$$\begin{aligned} \left\| df \right\| _2 \le \left\| \frac{d \mathsf {f}}{dx} \right\| _{L^2} +C \le C , \end{aligned}$$

(8.5)

which follows by a consistency argument. The result is completed by taking $\beta = \frac{\alpha }{LC}$. $\square$