Error Estimates of a Stabilized Lagrange–Galerkin Scheme of Second-Order in Time for the Navier–Stokes Equations

Abstract

Error estimates with optimal convergence orders are proved for a stabilized Lagrange–Galerkin scheme of second-order in time for the Navier–Stokes equations. The scheme is a combination of Lagrange–Galerkin method and Brezzi–Pitkäranta’s stabilization method. It maintains the advantages of both methods; (i) It is robust for convection-dominated problems and the system of linear equations to be solved is symmetric. (ii) Since the P1 finite element is employed for both velocity and pressure, the number of degrees of freedom is much smaller than that of other typical elements for the equations, e.g., P2/P1. Therefore, the scheme is efficient especially for three-dimensional problems. The second-order accuracy in time is realized by Adams-Bashforth’s (two-step) method for the discretization of the material derivative along the trajectory of fluid particles. The theoretical convergence orders are recognized by two- and three-dimensional numerical results.

Appendix

1.1 Proof of Lemma 5

From (18.22), there exists a non-negative sequence $\{\tilde{z}_n\}_{n\geqslant 2}$ such that

$$\begin{aligned} \frac{1}{\varDelta t}\Bigl ( \frac{3}{2} x_n - 2 x_{n-1} + \frac{1}{2} x_{n-2} + y_n - y_{n-1} \Bigr ) + \tilde{z}_n = a_0 (x_{n-1}+x_{n-2}) + b_n,\quad \forall n\geqslant 2, \end{aligned}$$

where $\tilde{z}_n$ satisfies

$$\begin{aligned} z_n \leqslant \tilde{z}_n, \quad \forall n\geqslant 2. \end{aligned}$$

(18.48)

Let p and $q\in \mathbb {R}~(q<p)$ be the roots of the quadratic equation $(3/2)x^2 - 2x + 1/2 = a_0\varDelta t (x+1)$ and $\lambda \equiv 2/3$. We note that p and q satisfy

$$\begin{aligned} 0 \leqslant q < 1 \leqslant p \end{aligned}$$

(18.49)

from $\varDelta t\in (0, 1/(2a_0)]$. Let any $n\geqslant 2$ be fixed. We have

$$\begin{aligned} x_n - p x_{n-1} + \lambda (y_n - y_{n-1} ) + \lambda \varDelta t\tilde{z}_n = q (x_{n-1} - p x_{n-2}) + \lambda \varDelta t b_n,\\ x_n - q x_{n-1} + \lambda (y_n - y_{n-1} ) + \lambda \varDelta t\tilde{z}_n = p (x_{n-1} - q x_{n-2}) + \lambda \varDelta t b_n, \end{aligned}$$

which lead to

$$\begin{aligned} x_n - p x_{n-1} + \lambda \Bigl \{ \sum _{i=2}^n q^{n-i} y_i&- \sum _{i=1}^{n-1} q^{n-1-i} y_i \Bigl \} + \lambda \varDelta t \sum _{i=2}^n q^{n-i} \tilde{z}_i \nonumber \\&= q^{n-1} (x_1 - p x_0) + \lambda \varDelta t \sum _{i=2}^n q^{n-i} b_i, \end{aligned}$$

(18.50a)

$$\begin{aligned} x_n - q x_{n-1} + \lambda \Bigl \{ \sum _{i=2}^n p^{n-i} y_i&- \sum _{i=1}^{n-1} p^{n-1-i} y_i \Bigl \} + \lambda \varDelta t \sum _{i=2}^n p^{n-i} \tilde{z}_i \nonumber \\&= p^{n-1} (x_1 - q x_0) + \lambda \varDelta t \sum _{i=2}^n p^{n-i} b_i. \end{aligned}$$

(18.50b)

Multiplying (18.50a) by q and (18.50b) by p and subtracting the first equation from the second, we get

$$\begin{aligned} (p-q) x_n&+ \lambda \Bigl \{ \sum _{i=2}^n (p^{n+1-i}-q^{n+1-i}) y_i - \sum _{i=1}^{n-1} (p^{n-i}-q^{n-i}) y_i \Bigl \} \nonumber \\&+ \lambda \varDelta t \sum _{i=2}^n (p^{n+1-i}-q^{n+1-i}) \tilde{z}_i \nonumber \\ =&\ (p^n-q^n)x_1-pq(p^{n-1}-q^{n-1})x_0 + \lambda \varDelta t \sum _{i=2}^n (p^{n+1-i}-q^{n+1-i}) b_i. \end{aligned}$$

(18.51)

The definition of p and q and (18.49) imply that

$$\begin{aligned} (p-q) y_n&- (p^n - q^n) y_1 \leqslant \sum _{i=2}^n (p^{n+1-i}-q^{n+1-i}) y_i - \sum _{i=1}^{n-1} (p^{n-i}-q^{n-i}) y_i,\end{aligned}$$

(18.52a)

$$\begin{aligned} \frac{2}{3} \leqslant p-q&\leqslant p^{n+1-i} - q^{n+1-i} \leqslant p^{n-1} - q^{n-1} \leqslant p^n - q^n,\quad i\in \{ 2,\ldots ,n\}, \end{aligned}$$

(18.52b)

$$\begin{aligned} p^n - q^n&\leqslant p^n = \Bigl \{ \frac{1}{3} \Bigl ( 2 + a_0\varDelta t + \sqrt{1+10a_0\varDelta t + a_0^2\varDelta t^2} \Bigr ) \Bigr \}^n \nonumber \\&\leqslant \{ 1+2a_0\varDelta t \}^n \leqslant \exp (2a_0n\varDelta t). \end{aligned}$$

(18.52c)

Combining (18.51) with (18.52), we have

$$\begin{aligned}&(p-q) \Bigl ( x_n + \lambda y_n + \lambda \varDelta t \sum _{i=2}^n \tilde{z}_i \Bigr ) \nonumber \\&\leqslant (p^n-q^n)x_1 - pq(p^{n-1}-q^{n-1})x_0 + \lambda (p^n - q^n) y_1 + \lambda \varDelta t (p^n-q^n) \sum _{i=2}^n b_i \nonumber \\&\leqslant (p^n-q^n) \Bigl (x_1 + \lambda y_1 + \lambda \varDelta t \sum _{i=2}^n b_i \Bigr ) \nonumber \\&\leqslant \exp (2a_0n\varDelta t) \Bigl (x_1 + \lambda y_1 + \lambda \varDelta t \sum _{i=2}^n b_i \Bigr ), \end{aligned}$$

(18.53)

and obtain the desired result as follows:

$$\begin{aligned} \text {LHS of}~(18.23)&\leqslant x_n + \lambda y_n + \lambda \varDelta t \sum _{i=2}^n \tilde{z}_i \quad \text {(by}~(18.48)) \\&\leqslant \frac{3}{2} \exp (2a_0n\varDelta t) \Bigl (x_1 + \lambda y_1 + \lambda \varDelta t \sum _{i=2}^n b_i \Bigr ) \quad \text {(by}~(18.53), (18.52b)). \quad \end{aligned}$$

   $\square$

1.2 Proof of Lemma 6

Let $t(s)\equiv t^{n-1} +s\varDelta t~(s\in [0,1])$. We prove (18.29a). Let $y(x, s)\equiv x-(1-s)u^{(n-1)*}(x)\varDelta t$. Using the identities

$$\begin{aligned} g^\prime (1)-\Bigl \{ \frac{3}{2}g(1)-2g(0)+\frac{1}{2}g(-1) \Bigr \}&= 2\int _0^1sds\int _{2s-1}^sg^{\prime \prime \prime }(s_1)ds_1,\\ \tilde{g}(1)-2\tilde{g}(0)+\tilde{g}(-1)&= \int _0^1ds\int _{s-1}^s \tilde{g}^{\prime \prime }(s_1)ds_1, \end{aligned}$$

for $g(s) = u(y(\cdot ,s),t(s))$ and $\tilde{g}(s) = u(\cdot ,t(s))$, we have

$$\begin{aligned} R_{h1}^n(x)&= \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u^n(x)\cdot \nabla \Bigr ) u \Bigr \} (x, t^n) \nonumber \\&\quad - \frac{1}{2\varDelta t} \bigl \{ 3u^n - 4u^{n-1} \circ X_1(u^{(n-1)*},\varDelta t) + u^{n-2} \circ X_1(u^{(n-1)*},2\varDelta t) \bigr \}(x) \nonumber \\&= \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u^{(n-1)*}(x)\cdot \nabla \Bigr ) u \Bigr \} (x, t^n) \nonumber \\&\quad - \frac{1}{2\varDelta t} \bigl \{ 3u^n - 4u^{n-1} \circ X_1(u^{(n-1)*},\varDelta t) + u^{n-2} \circ X_1(u^{(n-1)*},2\varDelta t) \bigr \}(x) \nonumber \\&\quad + \bigl \{ \bigl ( (u^n - u^{(n-1)*})(x)\cdot \nabla \bigr ) u^n \bigr \} (x) \nonumber \\&= 2\varDelta t^2 \int _0^1sds \int _{2s-1}^{s} \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u^{(n-1)*}(x)\cdot \nabla \Bigr )^3 u \Bigr \} \bigl ( y(x,s_1), t(s_1) \bigr ) ds_1 \nonumber \\&\quad + \varDelta t^2 \int _0^1ds\int _{s-1}^s \Bigl \{ \Bigl ( \frac{\partial {^2u}}{\partial {t^2}} \bigl ( x, t(s_1) \bigr )\cdot \nabla \Bigr ) u^n \Bigr \} (x) ds_1, \end{aligned}$$

and

$$\begin{aligned} \Vert R_{h1}^n \Vert _0&\leqslant 2\varDelta t^2 \int _0^1sds \int _{2s-1}^{s} \Bigl \Vert \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u^{(n-1)*}(\cdot )\cdot \nabla \Bigr )^3 u \Bigr \} \bigl ( y(\cdot , s_1), t(s_1) \bigr )\Bigr \Vert _0 ds_1 \nonumber \\&\quad + \varDelta t^2 \int _0^1ds\int _{s-1}^s \Bigl \Vert \Bigl ( \frac{\partial {^2u}}{\partial {t^2}} \bigl ( \cdot , t(s_1) \bigr )\cdot \nabla \Bigr ) u^n \Bigr \Vert _0 ds_1 \nonumber \\&\leqslant c_u \varDelta t^2 \int _{-1}^{1} \Bigl ( \Bigl \Vert \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + \nabla \Bigr )^3 u \Bigr \} \bigl ( \cdot , t(s_1) \bigr )\Bigr \Vert _0 + \Bigl \Vert \frac{\partial {^2u}}{\partial {t^2}} \bigl ( \cdot , t(s_1) \bigr ) \Bigr \Vert _0 \Bigr ) ds_1 \quad \text {(by}~(18.10)) \nonumber \\&\leqslant c_u^\prime \varDelta t^{3/2} \bigl ( \Vert u \Vert _{Z^3(t^{n-2},t^n)}+\Vert u \Vert _{H^2(t^{n-2},t^n; L^2)} \bigr ) \leqslant 2 c_u^\prime \varDelta t^{3/2} \Vert u \Vert _{Z^3(t^{n-2},t^n)}, \end{aligned}$$

which implies (18.29a).

Inequality (18.29b) is obtained as follows:

$$\begin{aligned} \Vert R_{h2}^n \Vert _0&\leqslant \alpha _{42} \Vert u^{(n-1)*} - u_h^{(n-1)*}\Vert _0 (2 \Vert u^{n-1}\Vert _{1,\infty } + \Vert u^{n-2}\Vert _{1,\infty }) \nonumber \\&\leqslant 3 \alpha _{42} \Vert u\Vert _{C(W^{1,\infty })} \Vert 2(\eta ^{n-1}-e_h^{n-1})-(\eta ^{n-2}-e_h^{n-2})\Vert _0 \nonumber \\&\leqslant c_u ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + \Vert \eta ^{n-1}\Vert _0 + \Vert \eta ^{n-2}\Vert _0 ) \\&\leqslant c_u \{ \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + \alpha _{31} h (\Vert (u,p)^{n-1}\Vert _{H^2\times H^1} + \Vert (u,p)^{n-2}\Vert _{H^2\times H^1} ) \} \nonumber \\&\leqslant c_{(u,p)} ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + h ). \nonumber \end{aligned}$$

(18.54)

We prove (18.29c). Let $y(x,s) \equiv x - (1-s) u_h^{(n-1)*}(x) \varDelta t$. Since we have

$$\begin{aligned} R_{h3}^n&= \frac{1}{\varDelta t} \Bigl \{ \frac{3}{2}\Bigl [ \eta \bigl ( y(\cdot ,s), t(s) \bigr ) \Bigr ]_{s=0}^1 - \frac{1}{2}\Bigl [ \eta \bigl ( y(\cdot ,s), t(s) \bigr ) \Bigr ]_{s=-1}^0 \Bigr \} \\&= \frac{3}{2} \int _0^1 \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u_h^{(n-1)*}(\cdot )\cdot \nabla \Bigr ) \eta \Bigr \} \bigl ( y(\cdot ,s), t(s) \bigr ) ds \\&\quad - \frac{1}{2} \int _{-1}^0 \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u_h^{(n-1)*}(\cdot )\cdot \nabla \Bigr ) \eta \Bigr \} \bigl ( y(\cdot ,s), t(s) \bigr ) ds, \end{aligned}$$

(18.29c) is obtained as follows:

$$\begin{aligned} \Vert R_{h3}^n \Vert _0&\leqslant \frac{3}{2}\int _0^1 \Bigl \Vert \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u_h^{(n-1)*}(\cdot )\cdot \nabla \Bigr ) \eta \Bigr \} \bigl ( y(\cdot , s), t(s) \bigr ) \Bigr \Vert _0 ds \\&\quad + \frac{1}{2}\int _{-1}^0 \Bigl \Vert \Bigl \{ \Bigl ( \frac{\partial {}}{\partial {t}} + u_h^{(n-1)*}(\cdot )\cdot \nabla \Bigr ) \eta \Bigr \} \bigl ( y(\cdot , s), t(s) \bigr ) \Bigr \Vert _0 ds \\&\leqslant \frac{3}{2} \int _{-1}^1 \Bigl ( \Bigl \Vert \frac{\partial {\eta }}{\partial {t}} \bigl ( y(\cdot , s), t(s) \bigr ) \Bigr \Vert _0 + \Vert u_h^{(n-1)*}\Vert _{0,\infty } \bigl \Vert \nabla \eta \bigl ( y(\cdot , s), t(s) \bigr ) \bigr \Vert _0 \Bigr ) ds \\&\leqslant \frac{3}{\sqrt{2}} \int _{-1}^1 \Bigl \{ \Bigl \Vert \frac{\partial {\eta }}{\partial {t}} \bigl ( \cdot , t(s) \bigr ) \Bigr \Vert _0 + \Vert u_h^{n-1}\Vert _{0,\infty } \bigl \Vert \nabla \eta \bigl ( \cdot , t(s) \bigr ) \bigr \Vert _0 \Bigr \} ds \quad \text {(by}~(18.10)) \\&\leqslant \frac{3}{\sqrt{2\varDelta t}} \Bigl ( \Bigl \Vert \frac{\partial {\eta }}{\partial {t}} \Bigr \Vert _{L^2(t^{n-2},t^n; L^2)} + \Vert u_h^{(n-1)*}\Vert _{0,\infty } \bigl \Vert \nabla \eta \bigr \Vert _{L^2(t^{n-2},t^n; L^2)} \Bigr ) \\&\leqslant \frac{3\alpha _{31}h}{\sqrt{2\varDelta t}} (\Vert u_h^{(n-1)*}\Vert _{0,\infty } + 1) \Vert (u,p) \Vert _{H^1(t^{n-2},t^n; H^2\times H^1)} \\&\leqslant \frac{ch}{\sqrt{\varDelta t}} (\Vert u_h^{(n-1)*}\Vert _{0,\infty } + 1) \Vert (u,p) \Vert _{H^1(t^{n-2},t^n; H^2\times H^1)}. \end{aligned}$$

We get (18.29d) from the estimate

$$\begin{aligned} \Vert R_{h4}^n \Vert _0&= \frac{1}{2\varDelta t} \bigl \Vert -4 \bigl \{ e_h^{n-1} - e_h^{n-1} \circ X_1(u_h^{(n-1)*},\varDelta t) \bigr \} + \bigl \{ e_h^{n-2} - e_h^{n-2} \circ X_1(u_h^{(n-1)*},2\varDelta t) \bigr \} \bigr \Vert _0 \\&\leqslant c\alpha _{40} \Vert u_h^{(n-1)*}\Vert _{0,\infty } (\Vert e_h^{n-1}\Vert _1 + \Vert e_h^{n-2}\Vert _1). \end{aligned}$$

   $\square$

1.3 Proof of Lemma 7

Inequality (18.44a) is obtained by combining (18.20b) with (18.54). For (18.44b) we divide $R_{h3}^n$ into three terms,

$$\begin{aligned} R_{h3}^n&= \overline{D}_{\varDelta t}^{(2)} \eta ^n + \frac{1}{2\varDelta t}\bigl [ 4 \bigl \{ \eta ^{n-1} - \eta ^{n-1} \circ X_1(u^{(n-1)*},\varDelta t) \bigr \} \\&\qquad \qquad \qquad - \bigl \{ \eta ^{n-2} - \eta ^{n-2} \circ X_1(u^{(n-1)*},2\varDelta t) \bigr \} \bigr ] \\&\quad + \frac{1}{2\varDelta t}\bigl [ 4\bigl \{ \eta ^{n-1}\circ X_1(u^{(n-1)*},\varDelta t) - \eta ^{n-1} \circ X_1(u_h^{(n-1)*},\varDelta t) \bigr \} \\&\qquad \qquad - \bigl \{ \eta ^{n-2}\circ X_1(u^{(n-1)*},2\varDelta t) - \eta ^{n-2} \circ X_1(u_h^{(n-1)*},2\varDelta t) \bigr \} \bigr ] \\&\equiv R_{h31}^n + R_{h32}^n + R_{h33}^n. \end{aligned}$$

We have, by virtue of (18.20b),

$$\begin{aligned} \Vert R_{h31}^n \Vert _{V_h^\prime }&\leqslant \Vert \overline{D}_{\varDelta t}^{(2)}\eta ^n\Vert _0 \leqslant \frac{3}{2} \Vert \overline{D}_{\varDelta t}\eta ^n \Vert _0 + \frac{1}{2} \Vert \overline{D}_{\varDelta t}\eta ^{n-1} \Vert _0 \nonumber \\&\leqslant \frac{3}{2\sqrt{\varDelta t}} \Bigl \Vert \frac{\partial {\eta }}{\partial {t}} \Bigr \Vert _{L^2(t^{n-1},t^n; L^2)} + \frac{1}{2\sqrt{\varDelta t}} \Bigl \Vert \frac{\partial {\eta }}{\partial {t}} \Bigr \Vert _{L^2(t^{n-2},t^{n-1}; L^2)} \nonumber \\&\leqslant \frac{c\alpha _{32} h^2}{\sqrt{\varDelta t}} \Vert (u,p) \Vert _{H^1(t^{n-2},t^n; H^2\times H^1)} \leqslant \frac{c h^2}{\sqrt{\varDelta t}} \Vert (u,p) \Vert _{H^1(t^{n-2},t^n; H^2\times H^1)}, \end{aligned}$$

(18.55a)

$$\begin{aligned} \Vert R_{h32}^n \Vert _{V_h^\prime }&\leqslant \alpha _{41} \Vert u^{(n-1)*}\Vert _{1,\infty } (2 \Vert \eta ^{n-1} \Vert _0 + \Vert \eta ^{n-2} \Vert _0) \nonumber \\&\leqslant \alpha _{41} \Vert u^{(n-1)*}\Vert _{1,\infty } 3 \alpha _{32} h^2 \Vert (u,p) \Vert _{C(H^2\times H^1)} \leqslant c_{(u,p)} h^2, \end{aligned}$$

(18.55b)

$$\begin{aligned} \Vert R_{h33}^n \Vert _{V_h^\prime }&= \sup _{v_h \in V_h} \frac{1}{\Vert v_h\Vert _1} \frac{1}{2\varDelta t}\Bigl \{ 4\Bigl ( \eta ^{n-1}\circ X_1(u^{(n-1)*}_h,\varDelta t) - \eta ^{n-1} \circ X_1(u^{(n-1)*},\varDelta t), v_h \Bigr ) \nonumber \\&\qquad \qquad \qquad - \Bigl ( \eta ^{n-2}\circ X_1(u^{(n-1)*}_h,2\varDelta t) - \eta ^{n-2} \circ X_1(u^{(n-1)*},2\varDelta t), v_h \Bigr ) \Bigr \} \nonumber \\&\leqslant \sup _{v_h \in V_h} \frac{1}{\Vert v_h\Vert _1} \frac{1}{2\varDelta t} \bigl ( 4 \Vert \eta ^{n-1}\circ X_1(u^{(n-1)*}_h,\varDelta t) - \eta ^{n-1} \circ X_1(u^{(n-1)*},\varDelta t) \Vert _{0,1} \nonumber \\&\qquad \quad + \Vert \eta ^{n-2}\circ X_1(u^{(n-1)*}_h,2\varDelta t) - \eta ^{n-2} \circ X_1(u^{(n-1)*},2\varDelta t) \Vert _{0,1} \bigr ) \Vert v_h\Vert _{0,\infty } \nonumber \\&\leqslant 2 \alpha _{43} \Vert u_h^{(n-1)*} - u^{(n-1)*} \Vert _0 (\Vert \eta ^{n-1} \Vert _1+\Vert \eta ^{n-2} \Vert _1) \alpha _{21} h^{-d/6} \end{aligned}$$

(18.55c)

$$\begin{aligned}&\leqslant c h^{-d/6} ( \Vert \eta ^{n-1} \Vert _1 + \Vert \eta ^{n-2} \Vert _1 ) ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + \Vert \eta ^{n-1}\Vert _0 + \Vert \eta ^{n-2}\Vert _0 ) \nonumber \\&\leqslant c^\prime \alpha _{32} h^{1-d/6} \Vert (u,p) \Vert _{C(H^2\times H^1)} \bigl ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + \alpha _{32} h^2 \Vert (u,p)\Vert _{C(H^2\times H^1)} \bigr ) \nonumber \\&\leqslant c_{(u,p)} \bigl ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + h^2 \bigr ). \end{aligned}$$

(18.55d)

From (18.55a), (18.55b) and (18.55d) we obtain (18.44b).

For (18.44c) we use the bound on $R_{h3}^n$. $R_{h4}^n$ is obtained by replacing $\eta ^{n-1}$ with $-e_h^{n-1}$ in $R_{h32}^n + R_{h33}^n$. Hence, from (18.55b) and (18.55c) we have

$$\begin{aligned} \Vert R_{h4}^n \Vert _{V_h^\prime }&\leqslant \alpha _{41} \Vert u^{(n-1)*}\Vert _{1,\infty } (2 \Vert e_h^{n-1} \Vert _0 + \Vert e_h^{n-2} \Vert _0) \\&\quad + 2 \alpha _{21} \alpha _{43} h^{-d/6} \Vert u_h^{(n-1)*} - u^{(n-1)*} \Vert _0 (\Vert e_h^{n-1} \Vert _1+\Vert e_h^{n-2} \Vert _1) \\&\leqslant c\bigl \{ \Vert u^{(n-1)*}\Vert _{1,\infty } (\Vert e_h^{n-1}\Vert _0+\Vert e_h^{n-2}\Vert _0) + h^{-d/6} (\Vert e_h^{n-1} \Vert _1+\Vert e_h^{n-2} \Vert _1) \\&\qquad \times ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + \alpha _{32}h^2 \Vert (u,p)\Vert _{C(H^2\times H^1)} ) \bigr \} \\&\leqslant c_{(u,p)} \bigl \{ 1+h^{-d/6} ( \Vert e_h^{n-1} \Vert _1 + \Vert e_h^{n-2} \Vert _1 ) \bigr \} \bigl ( \Vert e_h^{n-1}\Vert _0 + \Vert e_h^{n-2}\Vert _0 + h^2 \bigr ), \end{aligned}$$

which implies (18.44c).   $\square$