Analysis of the Fourier series Dirichlet-to-Neumann boundary condition of the Helmholtz equation and its application to finite element methods

Abstract

It is well known that the Fourier series Dirichlet-to-Neumann (DtN) boundary condition can be used to solve the Helmholtz equation in unbounded domains. In this work, applying such DtN boundary condition and using the finite element method, we analyze and solve a two dimensional transmission problem describing elastic waves inside a bounded and closed elastic obstacle and acoustic waves outside it. We are mainly interested in analyzing the DtN boundary condition of the Helmholtz equation in order to establish the well-posedness results of the approximated variational equation, and further derive a priori error estimates involving effects of both the finite element discretization and the truncation of DtN map. Finally, some numerical results are presented to illustrate the accuracy of the numerical scheme.

Proof of Theorem 6

Before proving Theorem 6, we present some preliminary results.

Definition 1

Let $\langle \cdot ,\cdot \rangle _1$ and $\langle \cdot ,\cdot \rangle _2$ be inner products on $(H^1(\varOmega ))^2\times (H^1(\varOmega ))^2$ and $H^1(\varOmega _R)\times H^1(\varOmega _R)$ defined by, for $\forall \; u,v\in (H^1(\varOmega ))^2$ and $\forall \; p,q\in H^1(\varOmega _R)$,

$$\begin{aligned} \langle u,v\rangle _1= & {} \int _{\varOmega }{ \left[ \lambda (\nabla \cdot u)(\nabla \cdot \overline{v})+\frac{\mu }{2}\left( \nabla {u}+(\nabla { u})^T\right) :\left( \nabla \overline{v}+(\nabla \overline{v})^T\right) +{\widetilde{\alpha }}{u} \cdot \overline{v}\right] dx}, \\ \langle p,q\rangle _2= & {} \int _{\varOmega _R}{\left( \nabla p\cdot \nabla \overline{q}+p \overline{q}\right) dx}+b_1(p,q), \end{aligned}$$

which further induce the norms $|||\cdot |||_1$ on $(H^1(\varOmega ))^2$ and $|||\cdot |||_2$ on $H^1(\varOmega _R)$, respectively, i.e., $\forall \,u\in (H^1(\varOmega ))^2, p\in H^1(\varOmega _R)$,

$$\begin{aligned} |||u|||_1^2=\langle {u},{u}\rangle _1, \quad |||p|||_2^2=\langle p,p\rangle _2. \end{aligned}$$

From the above definition together with (16) and (18), we conclude that there exist constants $\alpha _0>0,\beta _0>0,\gamma _0>0$ such that

$$\begin{aligned} \alpha _0\Vert u\Vert _{\left( H^1(\varOmega )\right) ^2}^2\le & {} |||u|||_1^2 \le \beta _0\Vert u\Vert _{\left( H^1(\varOmega )\right) ^2}^2, \end{aligned}$$

(50)

$$\begin{aligned} \Vert p\Vert _{H^1(\varOmega _R)}^2\le & {} |||p|||_2^2 \le \gamma _0\Vert p\Vert _{H^1(\varOmega _R)}^2. \end{aligned}$$

(51)

Now, let $\{{\widetilde{\varPhi }}_i,{\widetilde{\lambda }}_i\}$ and $\{{\widehat{\varphi }}_i,{\widehat{\lambda }}_i\}$ be eigenpairs satisfying

$$\begin{aligned} \langle {\widetilde{\varPhi }}_i,\varTheta \rangle _1= & {} {\widetilde{\lambda }}_i \left( {\widetilde{\varPhi }}_i,\varTheta \right) _{\left( L^2(\varOmega )\right) ^2} , \quad \forall \; \varTheta \in \left( H^1(\varOmega )\right) ^2, \end{aligned}$$

(52)

$$\begin{aligned} \langle \widehat{\varphi _i},\theta \rangle _2= & {} {\widehat{\lambda }}_i\left( {\widehat{\varphi }}_i,\theta \right) _{L^2(\varOmega _R)}, \quad \forall \; \theta \in H^1(\varOmega _R), \end{aligned}$$

(53)

where $\left( \cdot ,\cdot \right) _{H}$ is the classical $L^2$ inner product on H. Without loss of generalities, we assume that $0<{\widetilde{\lambda }}_1\le {\widetilde{\lambda }}_2\le \ldots$, $0<{\widehat{\lambda }}_1\le {\widehat{\lambda }}_2\le \ldots$, and $({{\widetilde{\varPhi }}}_i,{{\widetilde{\varPhi }}}_j)_{(L^2(\varOmega ))^2} =({{\widehat{\varphi }}}_i,{{\widehat{\varphi }}}_j)_{L^2(\varOmega _R)} =\delta _{ij}$.

Lemma 4

Let $u=\sum _{n=1}^{\infty }{{\widetilde{c}}_n{\widetilde{\varPhi }}_n}$, $p=\sum _{n=1}^{\infty }{{\widehat{c}}_n{\widehat{\varphi }}_n}$ and $U=(u,p)$.

• (1) If $U\in {\mathcal {H}}^0$,

  $$\begin{aligned} \Vert U\Vert _{{\mathcal {H}}^0}^2=\sum _{n=1}^{\infty }{(|\widetilde{c}_n|^2 +|{\widehat{c}}_n|^2)}. \end{aligned}$$

  (54)

• (2) If $U\in {\mathcal {H}}^1$,

  $$\begin{aligned} |||u|||_1^2 =\sum _{n=1}^{\infty }{{\widetilde{\lambda }}_i|\widetilde{c}_n|^2},\quad |||p|||_2^2=\sum _{n=1}^{\infty }{{\widehat{\lambda }}_i|{\widehat{c}}_n|^2}. \end{aligned}$$

  (55)

• (3) Let $H_{M_1}=\underset{{{\widetilde{\lambda }}}_i\le M_1}{span}\{{{\widetilde{\varPhi }}}_i\},H_{M_2}=\underset{{{\widehat{\lambda }}}_i\le M_2}{span}\{{{\widehat{\varphi }}}_i\}$ and we define

  $$\begin{aligned} H_{M_1}^{\perp }= & {} \{{v}\in (H^1(\varOmega ))^2:\langle {v},\varTheta \rangle _1=0, \quad \forall \; \varTheta \in H_{M_1}\},\\ H_{M_2}^{\perp }= & {} \{q\in H^1(\varOmega _R): \langle q,\theta \rangle _1=0, \quad \forall \; \theta \in H_{M_2}\}. \end{aligned}$$

  Then we have

  $$\begin{aligned} |||u|||_1^2\le M_1\Vert u\Vert _{(H^0(\varOmega ))^2}^2,\quad |||p|||_2^2\le M_2\Vert p\Vert _{H^0(\varOmega _R)}^2,\quad \forall \; U\in H_{M_1}\times H_{M_2},\nonumber \\ \end{aligned}$$

  (56)

  and

  $$\begin{aligned} \Vert u\Vert _{(H^0(\varOmega ))^2}^2\le \frac{1}{M_1}|||u|||_1^2,\quad \Vert p\Vert _{H^0(\varOmega _R)}^2\le \frac{1}{M_2}|||p|||_2^2,\quad \forall \; U\in H_{M_1}^{\perp }\times H_{M_2}^{\perp }.\nonumber \\ \end{aligned}$$

  (57)

Lemma 5

Suppose that $U=(u,p)\in {\mathcal {H}}^1$ satisfies

$$\begin{aligned} \langle {u},\varTheta \rangle _1= & {} (\widetilde{f}_1,\varTheta )_{(L^2(\varOmega ))^2},\quad \forall \; \varTheta \in (H^1(\varOmega ))^2, \\ \langle p,\theta \rangle _2= & {} ({{\widehat{f}}}_2,\theta )_{L^2(\varOmega _R)},\quad \forall \; \theta \in H^1(\varOmega _R) \end{aligned}$$

with $(\widetilde{f}_1,{{\widehat{f}}}_2)\in {\mathcal {H}}^0$. Then there exists a positive constant c such that

$$\begin{aligned} \Vert u\Vert _{(H^2(\varOmega ))^2}\le c\Vert \widetilde{f}_1\Vert _{(H^0(\varOmega ))^2},\quad \Vert p\Vert _{H^2(\varOmega _R)}\le c\Vert {{\widehat{f}}}_2\Vert _{H^0(\varOmega _R)}. \end{aligned}$$

(58)

Proof

The first assertion for u follows from the classical regularity estimates [11]. We now prove the regularity results for p. It follows that $b_1(\cdot ,\cdot )$ is the corresponding sesquilinear form of the DtN map for the solution of homogeneous Laplace equation outside $\varOmega$. Now we define

$$\begin{aligned} \widetilde{p}(x)= {\left\{ \begin{array}{ll} p(x),\quad &{}|x|\le R ,\\ \sum _{n\ge 0}\left( \frac{|x|}{R}\right) ^n(a_n\cos n\theta +b_n\sin n\theta ),\quad &{}|x|>R, \end{array}\right. } \end{aligned}$$

where $p(R,\theta )=\sum _{n\ge 0}(a_n\cos n\theta +b_n\sin n\theta )$. Note that for $\varphi \in C_0^\infty ({{\mathbb {R}}}^2)$,

$$\begin{aligned} \int _{{{\mathbb {R}}}^2\backslash \overline{\varOmega }}\nabla \widetilde{p}\cdot \nabla \overline{\varphi }dx +\int _{\varOmega _R}\widetilde{p}\,\overline{\varphi }\,dx= & {} \langle \widetilde{p},\varphi \rangle _2-\int _{|x|\ge R}\varDelta \widetilde{p}\,\overline{\varphi }\,dx\\= & {} \langle p,\varphi \rangle _2\\= & {} ({{\widehat{f}}}_2,\varphi ). \end{aligned}$$

Then we conclude from interior regularity estimates that $\widetilde{p}=p\in H^2(\varOmega _R)$ and

$$\begin{aligned} \Vert p\Vert _{H^2(\varOmega _R)}\le & {} c\{\Vert \widetilde{p}\Vert _{H^1(\varOmega _{R+1})} +\Vert {\widehat{f}}_2\Vert _{H^0(\varOmega _R)}\}\\\le & {} c\Vert {\widehat{f}}_2\Vert _{H^0(\varOmega _R)}. \end{aligned}$$

$\square$

Corollary 1

If $U=(u,p)\in H_{M_1}\times H_{M_2}$,

$$\begin{aligned} \Vert u\Vert _{(H^2(\varOmega ))^2}\le c M_1^{1/2}|||u|||_1,\quad \Vert p\Vert _{H^2(\varOmega _R)}\le cM_2^{1/2}|||p|||_2. \end{aligned}$$

(59)

Proof

For $U=(u,p)\in H_{M_1}\times H_{M_2}$, we can see that

$$\begin{aligned} u=\sum _{{{\widetilde{\lambda }}}_n\le M_1}{\widetilde{c}_n{\widetilde{\varPhi }}_n}, \quad p=\sum _{{{\widehat{\lambda }}}_n\le M_2}{{\widehat{c}}_n{\widehat{\varphi }}_n}. \end{aligned}$$

Then (52) and (53) imply that

$$\begin{aligned} \langle {u},\varTheta \rangle _1= & {} \left( \sum _{{{\widetilde{\lambda }}}_n\le M_1}{{{\widetilde{\lambda }}}_n\widetilde{c}_n{\widetilde{\varPhi }}_n},\varTheta \right) , \quad \forall \; \varTheta \in (H^1(\varOmega ))^2,\\ \langle p,\theta \rangle _2= & {} \left( \sum _{{{\widehat{\lambda }}}_n\le M_2}{{{\widehat{\lambda }}}_n{\widehat{c}}_n{\widehat{\varphi }}_n},\theta \right) , \quad \forall \; \theta \in H^1(\varOmega _R). \end{aligned}$$

Therefore, (59) follows from Lemma 5 and (55). $\square$

Corollary 2

If $U=(u,p)\in H_{M_1}^{\perp }\times H_{M_2}^{\perp }$, there exist positive constants $M_0,C_1$ such that for $M_1,M_2\ge M_0$,

$$\begin{aligned} \Vert U\Vert _{{\mathcal {H}}^1}^2\le C_1 \text{ Re }\{A^N(U,U)\}. \end{aligned}$$

(60)

Proof

From (50) and (51), we know

$$\begin{aligned} \alpha _0\Vert U\Vert _{{\mathcal {H}}^1}^2 \le |||{u}|||_1^2+\alpha _0|||p|||_2^2 \le \max \{\alpha _0,1\}(|||{u}|||_1^2 +|||p|||_2^2). \end{aligned}$$

According to Definition 1 and $b_1^N(p, p)\ge 0$, we have

$$\begin{aligned} \alpha _0\Vert U\Vert _{{\mathcal {H}}^1}^2\le & {} \max \{\alpha _0,1\}\text{ Re }\{A^N({U},{U})\} +\max \{\alpha _0,1\} (2\mu +\rho \omega ^2)\Vert {u}\Vert _{(H^0(\varOmega ))^2}^2\nonumber \\&\quad +\, \max \{\alpha _0,1\}(k^2+1)\Vert p\Vert _{H^0(\varOmega _R)}^2 \nonumber \\&\quad +\, \max \{\alpha _0,1\}\text{ Re }\{b_2^N(p,p)-a_3(u,p)-a_4(p,{u})\}. \end{aligned}$$

(61)

The Sobolev embedding theorem and the interpolation inequality for $0<\theta =1/2+\varepsilon < 1$, $0<\varepsilon <1/2$ give

$$\begin{aligned} |b_2^N(p,p)-a_3({u},p)-a_4(p,{u})|\le & {} c(\Vert {u}\Vert _{(H^0(\varGamma ))^2}^2 +\Vert p\Vert _{H^0(\varGamma _R)}^2)\nonumber \\\le & {} c(\Vert { u}\Vert _{(H^{1/2+\varepsilon }(\varOmega ))^2}^2+\Vert p\Vert _{H^{1/2+\varepsilon } (\varOmega _R )}^2)\nonumber \\\le & {} c(\Vert {u}\Vert _{(H^0(\varOmega ))^2}^{1-2\varepsilon }\Vert {u}\Vert _{(H^1(\varOmega ))^2}^{1+2\varepsilon } +\Vert p\Vert _{H^0(\varOmega _R )}^{1-2\varepsilon }\Vert p\Vert _{H^1(\varOmega _R )}^{1+2\varepsilon }).\nonumber \\ \end{aligned}$$

(62)

A combination of (50), (51) and (57) leads to

$$\begin{aligned} \Vert {u}\Vert _{(H^0(\varOmega ))^2}^2\le \frac{\beta _0}{M_1}\Vert {u}\Vert _{(H^1(\varOmega ))^2}^2,\Vert p\Vert _{H^0(\varOmega _R )}^2\le \frac{\gamma _0}{M_2}\Vert p\Vert _{H^1(\varOmega _R )}^2. \end{aligned}$$

(63)

Thus, we conclude from (61)–(63) that

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}^2\le & {} C_\alpha \text{ Re }\{A^N({U},{U})\}+C_\alpha (2\mu +\rho \omega ^2)\Vert { u} \Vert _{(H^0(\varOmega ))^2}^2+C_\alpha (k^2+1)\Vert p\Vert _{H^0(\varOmega _R)}^2 \nonumber \\&\quad +\, C_\alpha |b_2^N(p,p)-a_3({u},p)-a_4(p,{u})|\nonumber \\\le & {} C_\alpha \text{ Re }\{A^N({U},{U})\}\nonumber \\&\quad +\, C_\alpha \frac{\beta _0^2(2\mu +\rho \omega ^2)}{M_1^2} \Vert {u}\Vert _{(H^1(\varOmega ))^2}^2 + C_\alpha \frac{\gamma _0^2(1+k^2)}{M_2^2}\Vert p\Vert _{H^1(\varOmega _R )}^2 \nonumber \\&\quad +\, C_\alpha \frac{c\beta _0^{1/2-\varepsilon }}{M_1^{1/2-\varepsilon }}\Vert { u}\Vert _{(H^1(\varOmega ))^2}^2+C_\alpha \frac{c\gamma _0^{1/2-\varepsilon }}{M_2^{1/2-\varepsilon }}\Vert p\Vert _{H^1(\varOmega _R)}^2, \end{aligned}$$

(64)

where $C_\alpha =\max \{1,1/\alpha _0\}$. Let $M_0>0$ be such that

$$\begin{aligned} 1-C_\alpha \frac{\beta _0^2(2\mu +\rho \omega ^2)}{M_0^2}-C_\alpha \frac{c\beta _0^{1/2 -\varepsilon }}{M_0^{1/2-\varepsilon }} >\frac{1}{2}, \end{aligned}$$

(65)

$$\begin{aligned} 1-C_\alpha \frac{\gamma _0^2(1+k^2)}{M_0^2}-C_\alpha \frac{c\gamma _0^{1/2-\varepsilon }}{M_0^{1/2-\varepsilon }} >\frac{1}{2}. \end{aligned}$$

(66)

Then for $M_1,M_2\ge M_0$, there exists a constant $C_1=2C_\alpha$ such that (60) holds. $\square$

Corollary 3

If ${U}=({u},p)\in H_{M_1}\times H_{M_2}$ and ${V}=(v,q)\in H_{M_1}^{\perp }\times H_{M_2}^{\perp }$, we have

$$\begin{aligned} |A({U},{V})|\le c\gamma _0^{1/4-\varepsilon /2} M_2^{\varepsilon /2-1/4} \Vert p\Vert _{H^1(\varOmega _R)}\Vert q\Vert _{H^1(\varOmega _R)}. \end{aligned}$$

(67)

where $c>0$ and $0<\varepsilon <1/2$ are constants independent of U and V.

Proof

It easily follows that

$$\begin{aligned} |A({U},{V})|= & {} |\langle {u},{v}\rangle _1-({\widetilde{\alpha }}+\rho \omega ^2)({u},{ v})_{(H^0(\varOmega ))^2}\\&+\,\langle p,q\rangle _2-(k^2+1)(p,q)_{H^0(\varOmega _R)}-b_2(p,q)|\\= & {} |b_2(p,q)|\\\le & {} c\Vert p\Vert _{H^{1/2+\varepsilon }(\varOmega _R)}\Vert q\Vert _{H^{1/2+\varepsilon }(\varOmega _R)}, \end{aligned}$$

where $c>0$ is a constant. Then the interpolation inequality and (63) lead to (67). $\square$

Corollary 4

If ${U}=({u},p)\in H_{M_1}\times H_{M_2}$, there exist ${V}=({v},q)\in H_{M_1}\times H_{M_2}$ and a positive constant c such that

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}\le c\frac{\text{ Re }\{A({U},{V})\}}{\Vert { V}\Vert _{{\mathcal {H}}^1}}. \end{aligned}$$

(68)

Proof

Firstly, we observe that for ${W}=({w},\varpi )\in {\mathcal {H}}^1$, we can rewrite it as

$$\begin{aligned} {W}={W}_M+{W}_M^{\perp }=({w}_{M_1}+{ w}_{M_1}^{\perp },\varpi _{M_2}+\varpi _{M_2}^{\perp }), \end{aligned}$$

(69)

where ${w}_{M_1}\in H_{M_1}$, ${w}_{M_1}^{\perp }\in H_{M_1}^{\perp }$, $\varpi _{M_2}\in H_{M_2}$ and $\varpi _{M_2}^{\perp }\in H_{M_2}^{\perp }$. Then from (50) and (51) we know that

$$\begin{aligned} \Vert {W}_M\Vert _{{\mathcal {H}}^1}^2= & {} \Vert {w}_{M_1}\Vert _{(H^1(\varOmega ))^2}^2 +\Vert \varpi _{M_2}\Vert _{H^1(\varOmega _R)}^2\nonumber \\\le & {} \frac{1}{\alpha _0}|||{w}_{M_1}|||_1 +|||\varpi _{M_2}|||_2\nonumber \\\le & {} \frac{1}{\alpha _0}|||{w}|||_1 +|||\varpi |||_2\nonumber \\\le & {} \frac{\beta _0}{\alpha _0}\Vert { w}\Vert _{(H^1(\varOmega ))^2}^2 +\gamma _0\Vert \varpi \Vert _{H^1(\varOmega _R)}^2 \nonumber \\\le & {} \max \left\{ \frac{\beta _0}{\alpha _0},\gamma _0\right\} \Vert { W}\Vert _{{\mathcal {H}}^1}^2. \end{aligned}$$

(70)

Similarly, we have

$$\begin{aligned} \Vert {W}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2 \le \max \left\{ \frac{\beta _0}{\alpha _0},\gamma _0\right\} \Vert {W}\Vert _{{\mathcal {H}}^1}^2. \end{aligned}$$

(71)

From the stability of (15), we know that there exists a ${W}=({ w},\varpi )\in {\mathcal {H}}^1$ such that

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}\le c\frac{\text{ Re }\{A({U},{W})\}}{\Vert { W}\Vert _{{\mathcal {H}}^1}}, \end{aligned}$$

where $c>0$ is a constant. Let ${V}={W}_M$. Therefore, Corollary 3, Eqs. (70) and (71) yield

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}\le & {} c\frac{\text{ Re }\{A({U},{W})\}}{\Vert { W}\Vert _{{\mathcal {H}}^1}} \\= & {} c\frac{\text{ Re }\{A({U},{V})+A({U},{W}-{V})\}}{\Vert {W}\Vert _{{\mathcal {H}}^1}} \\\le & {} c \max \left\{ \frac{\beta _0}{\alpha _0},\gamma _0\right\} \frac{\text{ Re }\{A({ U},{V})\}}{\Vert {V}\Vert _{{\mathcal {H}}^1}}+ c\gamma _0^{1/4-\varepsilon /2} M_2^{\varepsilon /2-1/4}\max \left\{ \frac{\beta _0}{\alpha _0},\gamma _0\right\} \Vert {U}\Vert _{{\mathcal {H}}^1}. \end{aligned}$$

Let $M_0>0$ be such that

$$\begin{aligned} 1-c\gamma _0^{1/4-\varepsilon /2} M_0^{\varepsilon /2-1/4}\max \left\{ \frac{\beta _0}{\alpha _0},\gamma _0\right\} >\frac{1}{2}. \end{aligned}$$

(72)

Thus, (68) holds for $M_2\ge M_0$. $\square$

Proof of Theorem 6

Here we prove that for all ${U}=({u},p)\in {\mathcal {H}}^1$, there exists a constant $N_{0}=N_0\ge 0$ such that

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1} \le c \sup _{(\mathbf{0},0)\ne {V}\in {\mathcal {H}}_{h}}\frac{|A^N({U},{V})|}{\Vert {V}\Vert _{{\mathcal {H}}^1}}, \end{aligned}$$

(73)

where $c>0$ is a constant. Then the uniqueness follows immediately and then Theorem 6 follows from the Fredholm Alternative theorem. We use the same form as (69) to represent U as

$$\begin{aligned} {U}={U}_M+{U}_M^{\perp }=({u}_{M_1}+{ u}_{M_1}^{\perp },p_{M_2}+p_{M_2}^{\perp }), \end{aligned}$$

where ${u}_{M_1}\in H_{M_1}$, ${u}_{M_1}^{\perp }\in H_{M_1}^{\perp }$, $p_{M_2}\in H_{M_2}$ and $p_{M_2}^{\perp }\in H_{M_2}^{\perp }$.

If ${U}=0$, then (73) holds trivially.

If ${U}\ne 0$, we can obtain from Corollary 4 that there exist ${ V}_M=({v}_{M_1},q_{M_2})\in H_{M_1}\times H_{M_2}$ and a positive constant c such that

$$\begin{aligned} \Vert {U}_M\Vert _{{\mathcal {H}}^1}\le c\frac{\text{ Re }\{A({U}_M,{V}_M)\}}{\Vert \mathbf{V}_M\Vert _{{\mathcal {H}}^1}} \end{aligned}$$

(74)

with $\Vert {U}_M\Vert _{{\mathcal {H}}^1}=\Vert {V}_M\Vert _{{\mathcal {H}}^1}$ after scaling. Now we define ${V}={V}_M+{U}_M^{\perp }$. Then (74) together with Corollary 2 give

$$\begin{aligned} \Vert {U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert {U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2\le & {} c(\text{ Re }\{A({U}_M,{V}_M)\}+\text{ Re }\{A^N({U}_M^{\perp },{U}_M^{\perp })\}) \nonumber \\= & {} c|A^N({U},{V})|+c|A^N({U}_M,{U}_M^{\perp }|+c|A^N({U}_M^{\perp },{V}_M)| \nonumber \\&\quad +\, c|b(p_{M_2},q_{M_2})-b^N(p_{M_2},q_{M_2})|. \end{aligned}$$

(75)

According to Theorem 3, we obtain

$$\begin{aligned} |b(p_{M_2},q_{M_2})-b^N(p_{M_2},q_{M_2})|\le & {} \Vert (S-S^N)p_{M_2}\Vert _{H^{-3/2}(\varGamma _R)}\Vert q_{M_2}\Vert _{H^{3/2}(\varGamma _R)} \nonumber \\\le & {} cN^{-2}\Vert p_{M_2}\Vert _{H^{3/2}(\varGamma _R)} \Vert q_{M_2}\Vert _{H^{3/2}(\varGamma _R)}\nonumber \\\le & {} cN^{-2}\Vert p_{M_2}\Vert _{H^2(\varOmega _R)}\Vert q_{M_2}\Vert _{H^2(\varOmega _R)} \nonumber \\\le & {} c_1N^{-2}M_2\gamma _0\Vert {U}_M\Vert _{{\mathcal {H}}^1}^2. \end{aligned}$$

(76)

Additionally, we have that

$$\begin{aligned} |A^N({U}_M,{U}_M^{\perp })|\le & {} |A({U}_M,{U}_M^{\perp })|+ |b(p_{M_2},p_{M_2}^{\perp })-b^N(p_{M_2},q_{M_2}^{\perp })|. \end{aligned}$$

(77)

Then Theorem 3 gives

$$\begin{aligned} |b(p_{M_2},p_{M_2}^{\perp })-b^N(p_{M_2},q_{M_2}^{\perp })|\le & {} \Vert (S-S^N)p_{M_2}\Vert _{H^{-1/2}(\varGamma _R)}\Vert q_{M_2}^{\perp }\Vert _{H^{1/2}(\varGamma _R)} \nonumber \\\le & {} cN^{-1}\Vert p_{M_2}\Vert _{H^{3/2}(\varGamma _R)} \Vert q_{M_2}^{\perp }\Vert _{H^{1/2}(\varGamma _R)}\nonumber \\\le & {} cN^{-1}\Vert p_{M_2}\Vert _{H^2(\varOmega _R)}\Vert q_{M_2}^{\perp }\Vert _{H^1(\varOmega _R)} \nonumber \\\le & {} c_2N^{-1}M_2^{1/2}\gamma _0^{1/2}\Vert {U}_M\Vert _{{\mathcal {H}}^1} \Vert { U}_M^{\perp }\Vert _{{\mathcal {H}}^1}. \end{aligned}$$

(78)

Therefore, Corollary, (77), (78) and the arithmetic-geometric mean inequality lead to

$$\begin{aligned} |A^N({U}_M,{U}_M^{\perp })|\le & {} (c_3\gamma _0^{1/4-\varepsilon /2} M_2^{\varepsilon /2-1/4}+c_4N^{-1}M_2^{1/2}\gamma _0^{1/2})(\Vert { U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert {U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2).\nonumber \\ \end{aligned}$$

(79)

Similarly, we have

$$\begin{aligned} |A^N({U}_M^{\perp },{V}_M)|\le & {} (c_5\gamma _0^{1/4-\varepsilon /2} M_2^{\varepsilon /2-1/4}+c_6N^{-1}M_2^{1/2}\gamma _0^{1/2})(\Vert { U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert {U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2).\nonumber \\ \end{aligned}$$

(80)

Here, we need that $M_1,M_2\ge M_0$ where $M_0>0$ satisfies (65), (66) and (72). In additionally, we suppose that $M_0>0$ and $N_0\ge 0$ are large enough such that

$$\begin{aligned} 1-c_1N_0^{-2}M_0\gamma _0-(c_3+c_5)\gamma _0^{1/4-\varepsilon /2} M_0^{\varepsilon /2-1/4}-(c_5+c_6)N_0^{-1}M_0^{1/2}\gamma _0^{1/2}>\frac{1}{2},\nonumber \\ \end{aligned}$$

(81)

which further implies that there is a positive constant c such that

$$\begin{aligned} \Vert {U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert {U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2 \le c|A^N({ U},{V})|. \end{aligned}$$

(82)

Now, since

$$\begin{aligned}&\alpha _0\Vert {u}\Vert _{(H^1{\varOmega })^2}^2+\Vert p\Vert _{H^1(\varOmega _R)}^2 \\&\quad \le |||{ u}|||_1^2+|||p|||_2^2 \\&\quad = |||{u}_{M_1}|||_1^2+|||{ u}_{M_1}^{\perp }|||_1^2+|||p_{M_2}|||_2^2 +|||p_{M_2}^{\perp }|||_2^2 \\&\quad \le \beta _0(\Vert {u}_{M_1}\Vert _{(H^1{\varOmega })^2}^2+\Vert { u}_{M_1}^{\perp }\Vert _{(H^1{\varOmega })^2}^2) + \gamma _0(\Vert p_{M_2}\Vert _{H^1(\varOmega _R)}^2+\Vert p_{M_2}^{\perp }\Vert _{H^1(\varOmega _R)}^2) \\&\quad \le \max \{\beta _0,\gamma _0\}(\Vert {U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert { U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2), \end{aligned}$$

we have

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}^2 \le \frac{\max \{\beta _0,\gamma _0\}}{\min \{1,\alpha _0\}}(\Vert { U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert {U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2). \end{aligned}$$

(83)

Thus,

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}^2 \le c|A^N({U},{V})| \end{aligned}$$

(84)

for some constant $c>0$. Finally, we obtain from (83) similarly that

$$\begin{aligned} \Vert {V}\Vert _{{\mathcal {H}}^1}\le & {} \sqrt{\frac{\max \{\beta _0,\gamma _0\}}{\min \{1,\alpha _0\}}\left( \Vert { V}_M\Vert _{{\mathcal {H}}^1}^2+\Vert { U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2\right) }\nonumber \\= & {} \sqrt{\frac{\max \{\beta _0,\gamma _0\}}{\min \{1,\alpha _0\}} \left( \Vert { U}_M\Vert _{{\mathcal {H}}^1}^2+\Vert { U}_M^{\perp }\Vert _{{\mathcal {H}}^1}^2 \right) }\nonumber \\\le & {} \sqrt{\frac{\max \{\beta _0,\gamma _0\}}{\min \{1,\alpha _0\}} \left( \frac{1}{\alpha _0}|||{u}|||_1^2+|||p|||_2^2\right) }\nonumber \\\le & {} \sqrt{\frac{\max \{\beta _0,\gamma _0\}}{\min \{1,\alpha _0\}} \max \left\{ \frac{\beta _0}{\alpha _0},\gamma _0\right\} }\Vert {U}\Vert _{{\mathcal {H}}^1}. \end{aligned}$$

(85)

Therefore, (84) and (85) give

$$\begin{aligned} \Vert {U}\Vert _{{\mathcal {H}}^1}\le & {} c\frac{|A^N({U},{V})|}{\Vert {V}\Vert _{{\mathcal {H}}^1}} \\\le & {} c \sup _{(0,0)\ne {V}\in {\mathcal {H}}_{h}}\frac{|A^N({U},{V})|}{\Vert { V}\Vert _{{\mathcal {H}}^1}}, \end{aligned}$$

where $c>0$ is a constant. This completes the proof. $\square$