A projection method for Navier–Stokes equations with a boundary condition including the total pressure

We consider a projection method for time-dependent incompressible Navier–Stokes equations with a total pressure boundary condition. The projection method is one of the numerical calculation methods for incompressible viscous fluids often used in engineering. In general, the projection method needs additional boundary conditions to solve a pressure-Poisson equation, which does not appear in the original Navier–Stokes problem. On the other hand, many mechanisms generate flow by creating a pressure difference, such as water distribution systems and blood circulation. We propose a new additional boundary condition for the projection method with a Dirichlet-type pressure boundary condition and no tangent flow. We demonstrate stability for the scheme and establish error estimates for the velocity and pressure under suitable norms. A numerical experiment verifies the theoretical convergence results. Furthermore, the existence of a weak solution to the original Navier–Stokes problem is proved by using the stability.

1. If \(d=2\), then we define

   $$\begin{aligned} v\times w:=v_x w_y-v_y w_x\in {\mathbb R}\qquad \text{ for } \text{ all } v=(v_x,v_y),w=(w_x,w_y)\in {\mathbb R}^2. \end{aligned}$$

2. If \(d=2\), then \(\nabla \times v\) and \((\nabla \times v)\times w\) denote the scalar and vector functions, respectively, defined as follows: for all \(v=(v_x,v_y),w=(w_x,w_y)\in {\mathbb R}^2\),

   $$\begin{aligned} \nabla \times v:=\partial _x v_y-\partial _y v_x,\quad (\nabla \times v)\times w:=(w_y(\partial _y v_x-\partial _x v_y), w_x(\partial _x v_y-\partial _y v_x)). \end{aligned}$$

   .

3. If one replaces the bilinear form a with \(a_\gamma (u,v):=(\nabla \times u,\nabla \times v) + \gamma (\text{ div }\,u, \text{ div }\,v)\) for a fixed \(0 < \gamma \ne 1\), then the corresponding strong form is different from (1.3), but Theorems 2.13, 2.15, and 2.20 hold since we have for all \(\varphi \in H\), \(a_\gamma (\varphi ,\varphi ) \ge (\min \{1,\gamma \}/c_{a})\Vert \varphi \Vert _1^2\) by Lemma 2.2 (cf. [8, Lemma 2.11]). If we choose a smaller value as \(\gamma \), the coefficients of Theorems 2.13, 2.15, and  2.20 are larger (see (3.4)).

4. Here, it holds that for all \(i,j=1,\ldots ,d\) and \(k=1,2,\ldots ,N\),

   $$\begin{aligned} (g_k)_i:=\sum _{l=1}^d\frac{\partial (u^*_k)_l}{\partial x_i}(u^*_{k-1})_l -(u^*_{k-1})_i\text{ div }\,u^*_k,\qquad (h_k)_{ij}:= -(u^*_k)_i(u^*_{k-1})_j. \end{aligned}$$

5. Since \(p_2=2+\varepsilon \) and \(p_3=3\), we have \(p_2/{\tilde{q}}_2 = \varepsilon /2\) and \(p_3/{\tilde{q}}_3=1/2\).

Authors and Affiliations

1. Department of Science and Technology, Faculty of Science and Technology, Seikei University, 3-3-1 Kichijoji-Kitamachi, Musashino-shi, Tokyo, 180-8633, Japan

   Kazunori Matsui

Corresponding author

Correspondence to Kazunori Matsui.

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This work was supported by JSPS KAKENHI Grant Number 19J20514.

Proofs of Lemmas 2.4, 2.6, 3.2, 3.3, and Corollary 2.22

The purpose of this appendix is to provide the Proofs of Lemmas 2.4, 2.6, 3.2, 3.3, and Corollary 2.22. The continuity of the operator T follows from Lemmas 2.1 and 2.2 [14, Corollary 4.1]. By using Lemma 2.2 again, we prove the second inequality of Lemma 2.4.

Proof of Lemma 2.4

By Lemmas 2.1 and 2.2, there exists a unique solution \((w,r)\in H\times {L^2(\varOmega )}\) to (2.1) for all \(e\in {L^2(\varOmega )}^d\) and T is a continuous operator;

for a constant \(c_1>0\) independent of e [14, Corollary 4.1]. It is easy to check that T is a linear operator.

Next, we show the second inequality of Lemma 2.4. By the first equation of (2.1), it holds that for all \(\varphi \in V\),

By Lemma 2.2, we have

which implies that \(\Vert w\Vert _1\le c_a\Vert e\Vert _{V^*}\). On the other hand, by Lemma 2.2, it holds that for all \(e\in {L^2(\varOmega )}^d\),

\(\square \)

In order to prove Lemma 2.6 and Corollary 2.22, we define \({\tilde{p}}_d, {\tilde{q}}_d\) as

Here, since \(p_2=2+\varepsilon \) and \(p_3=3\), we have \(({\tilde{p}}_2, {\tilde{q}}_2) = (\frac{4+2\varepsilon }{4+\varepsilon }, 2+\frac{4}{\varepsilon })\) and \(({\tilde{p}}_2, {\tilde{q}}_2) = (\frac{6}{5}, 6)\). By the Sobolev embeddings [10, Theorem III.2.33], it holds that \({H^1(\varOmega )}\subset L^{{\tilde{q}}_d}(\varOmega )\), \({H^1(\varOmega )}\subset L^{p_d}(\varOmega )\), \(H^2(\varOmega ) \subset L^\infty (\varOmega )\) and the embeddings are continuous.

Proof of Lemma 2.6

(i) For all \(u \in L^{p_d}(\varOmega )^d, v, w \in H\), we have

for two constants \(c_1,{\tilde{c}}_1>0\), which implies the third inequality of Lemma 2.6.

(ii) For all \(u \in L^{p_d}(\varOmega )^d, v \in H, w \in H \cap H^2(\varOmega )^d\), we have

for two constants \(c_2,{\tilde{c}}_2>0\).

(iii) For all \(u \in {H^1(\varOmega )}^d, v \in H \cap H^2(\varOmega )^d, w \in H\), we have

for two constants \(c_3,{\tilde{c}}_3>0\).

(ix) For all \(u \in H^2(\varOmega )^d, v, w\in H\), we have

for two constants \(c_4,{\tilde{c}}_4>0\). \(\square \)

Next, we prove Lemmas 3.2 and 3.3.

Proof of Lemma 3.2

It holds that for all \(\varphi \in H\) and \(k=1,2,\ldots ,N\),

which implies that

Next, we show the second inequality of the conclusion. For all \(\varphi \in H\) and \(k=2,3,\ldots ,N\), we have

where we have used the coordinate transformation \((s_1,s_2)\mapsto ({\tilde{s}}_1,{\tilde{s}}_2):=(s_1+s_2,-s_1+s_2)\). Therefore, we obtain

\(\square \)

Proof of Lemma 3.3

It holds that for all \(k = 1,2,\ldots ,N\) and \(t \in [t_{k-1}, t_k]\),

which implies that \(\Vert x - x_\tau \Vert _{L^\infty (E)} \le \sqrt{\tau } \left\| \frac{\partial x}{\partial t}\right\| _{L^2(E)}\) and

On the other hand, we have

\(\square \)

We prove Corollary 2.22 by using the boundedness from Theorem 2.13 and the Aubin–Lions compactness lemma.

Proof of Corollary 2.22

By the first and third equations of (PM), it holds that for all \(v\in V\) and \(k=1,2,\ldots ,N\),

where \(g_k\) and \(h_k\) are defined^{Footnote 4} by

which implies that for all \(v\in V\) and \(\theta \in C^\infty _0(0,T)\),

Here, \({\bar{f}}_\tau \rightarrow f\) strongly in \(L^2(H^*)\) and \({\bar{p}}^b_\tau \rightarrow p^b\) strongly in \(L^2({H^1(\varOmega )})\) as \(\tau \rightarrow 0\). By Theorem 2.13 and Lemma 3.1, there exists a constant \(c_1>0\) such that

In particular, it holds that

which implies that \(\Vert u^*_1-u_0\Vert _0 \le \Vert u^*_1-u_1\Vert _0 + \Vert u_1-u_0\Vert _0 \le 2\sqrt{c_1}\). Furthermore, by the first equation of (PM) and Lemmas 2.2, 2.6, we have

where \(c_2:=\sqrt{c_1}(c_a + c_d\Vert u_0\Vert _{L^{p_d}}) + \Vert f\Vert _{L^2(H^*)}\). Let \(u^\circ _0:=u^*_1\), \(u^\circ _k:=u^*_k\) for all \(k=1,2,\ldots ,N\) and let \({\hat{u}}^\circ _\tau \) be the piecewise linear interpolant of \((u^\circ _k)_{k=0}^N \subset H\).

From the uniform estimates (A.2), one can show that there exist a sequence \((\tau _k)_{k\in {\mathbb N}}\) and three functions \(u\in L^2(H)\cap L^\infty ({L^2(\varOmega )}^d)\cap W^{1,4/p_d}(V^*)\) (in particular, \(u\in C([0,T];V^*)\)), \(g\in L^{4/p_d}(L^{{\tilde{p}}_d}(\varOmega )^d)\) and \(h\in L^{4/p_d}({L^2(\varOmega )}^{d\times d})\) such that \(\tau _k\rightarrow 0\) and

as \(k\rightarrow \infty \). Here, we note that \({\bar{u}}^*_{\tau _k}\), \({\hat{u}}^\circ _{\tau _k}\) and \({\hat{u}}_{\tau _k}\) possess a common limit function. Indeed, the weak convergence (A.5) of \({\bar{u}}^*_\tau \) immediately follows from the uniform estimates (A.2). Since we have \(1/{\tilde{p}}_d = 1/2 + 1/p_d\), \(p_d/4 = 1/2 + p_d/(2{\tilde{q}}_d)\), and

for a constant \(c_3>0\) (cf. [10, Theorem II.5.5]),^{Footnote 5} it holds that

for constants \(c_4\) and \(c_5\). Hence, by (A.2), the weak convergences (A.11) and (A.12) hold. Moreover, since there exists a constant \(c_6>0\) such that \(|(g_k,v)| \le \Vert g_k\Vert _{L^{{\tilde{p}}_d}}\Vert v\Vert _{L^{{\tilde{q}}_d}} \le c_6\Vert g_k\Vert _{L^{{\tilde{p}}_d}}\Vert v\Vert _1\) for all \(k=1,2,\ldots ,N\) and \(v\in {H^1(\varOmega )}^d\), we have

and \(\Vert \frac{\partial {\hat{u}}_\tau }{\partial t}\Vert _{L^{4/{p_d}}(V^*)}\) and \(\Vert \frac{\partial {\hat{u}}^\circ _\tau }{\partial t}\Vert _{L^{4/{p_d}}(V^*)}\) are also bounded. Hence, (A.10) holds. Furthermore, \(\Vert {\hat{u}}^\circ _\tau \Vert _{L^2(H^1)}\) is bounded: by (A.2) and (A.3),

which implies the strong convergence (A.7) of \({\hat{u}}^\circ _\tau \) in \(L^2({L^2(\varOmega )}^d)\) from the Aubin–Lions lemma [10, Theorem II.5.16 (i)]. Since we have for all \(t\in (t_{k-1},t_k), k=1,2,\ldots ,N\),

the functions \({\bar{u}}^*_{\tau _k}\), \({\hat{u}}^\circ _{\tau _k}\) and \({\hat{u}}_{\tau _k}\) possess a common limit function u, and the strong convergences (A.6) and (A.9) hold: by (A.2) and (A.3),

It also holds that

Since \(\Vert {\hat{u}}^\circ _\tau \Vert _{L^\infty (L^2)}\) and \(\Vert \frac{\partial {\hat{u}}^\circ _\tau }{\partial t}\Vert _{L^{4/p_d}(V^*)}\) are bounded, we obtain the strong convergence (A.8) of \({\hat{u}}^\circ _\tau \) in \(C([0,T];V^*)\) [10, Theorem II.5.16 (ii)]. In particular, \({\hat{u}}^\circ _\tau (0)\) converges to u(0) in \(V^*\). On the other hand, by (A.4), \({\hat{u}}^\circ _\tau (0)=u^*_1\) converges to \(u_0\) in \(V^*\). Through the uniqueness of the limit in \(V^*\), we have indeed obtained that \(u(0)=u_0\).

From (A.1) with \(\varepsilon :=\varepsilon _k\), taking \(k\rightarrow \infty \), it holds that for all \(v\in V\) and \(\theta \in C^\infty _0(0,T)\),

Next, we show that

We set \({\bar{v}}_\tau (t):=u^*_{k-1}\) for \(t\in (t_{k-1},t_k],k=1,2,\ldots ,N\). Then it holds that

and hence it follows from (A.6) that \({\bar{v}}_{\tau _k}\rightarrow u\) strongly in \(L^2({L^2(\varOmega )}^d)\) as \(k\rightarrow \infty \). Since \(\nabla {\bar{u}}^*_\tau \rightharpoonup \nabla u\) weakly in \(L^2({L^2(\varOmega )}^{d\times d})\) and \(\text{ div }\,{\bar{u}}^*_\tau \rightharpoonup \text{ div }\,u\) weakly in \(L^2({L^2(\varOmega )})\) as \(k\rightarrow \infty \), we have

as \(k\rightarrow \infty \) (cf. [10, Proposition II.2.12]). On the other hand, we also know (A.11) and (A.12). The convergence in these spaces imply the convergence in the distributions sense, therefore (A.13) holds by the uniqueness of the limit in \({\mathcal {D}}'((0,T)\times \varOmega )\). Hence, it holds that for all \(v\in V\) and \(\theta \in C^\infty _0(0,T)\),

which is equivalent to the following

\(\square \)

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