Multiscale Petrov-Galerkin Method for High-Frequency Heterogeneous Helmholtz Equations

Abstract

This paper presents a multiscale Petrov-Galerkin finite element method for time-harmonic acoustic scattering problems with heterogeneous coefficients in the high-frequency regime. We show that the method is pollution-free also in the case of heterogeneous media provided that the stability bound of the continuous problem grows at most polynomially with the wave number k. By generalizing classical estimates of Melenk (Ph.D. Thesis, 1995) and Hetmaniuk (Commun. Math. Sci. 5, 2007) for homogeneous medium, we show that this assumption of polynomially wave number growth holds true for a particular class of smooth heterogeneous material coefficients. Further, we present numerical examples to verify our stability estimates and implement an example in the wider class of discontinuous coefficients to show computational applicability beyond our limited class of coefficients.

Appendix: Proof of Stability

1.1 Technical and Auxiliary Lemmas

We will now proceed by recalling and demonstrating a few technical and auxiliary Lemmas used in the proof of Theorem 1. We begin with two critical technical lemmas that remain unchanged from the homogeneous case examined in [11] and are repeated here for completeness.

Lemma 2

Let $m \in W^{1,\infty }(\Omega )^{d}$ and for all $q \in H^{1}(\Omega )$ we have

$$\displaystyle\begin{array}{rcl} \int _{\partial \Omega }\vert q\vert ^{2}m \cdot \nu ds =\int _{ \Omega }\mbox{ div}(m)\vert q\vert ^{2}dx + 2\mathop{\mathrm{Re}}\nolimits \int _{ \Omega }qm \cdot \nabla \bar{q}dx.& &{}\end{array}$$

(38)

Proof

See [11], Lemma 3.1. □ 

Lemma 3

Let $m \in W^{1,\infty }(\Omega )^{d}$ and for all $q \in H_{\Gamma _{D}}^{1}(\Omega ) \cap H^{3/2+\delta },\delta> 0,$ we have

$$\displaystyle\begin{array}{rcl} & & \int _{\partial \Omega \setminus \Gamma _{D}}\vert \nabla q\vert ^{2}m \cdot \nu ds -\int _{ \Gamma _{D}}\vert \partial _{\nu }q\vert ^{2}m \cdot \nu ds \\ & & =\int _{\Omega }\mbox{ div}(m)\vert \nabla q\vert ^{2}dx - 2\mathop{\mathrm{Re}}\nolimits \int _{ \Omega }\nabla q \cdot (\nabla \bar{q}\nabla )mdx \\ & & -2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\Delta q(m \cdot \nabla \bar{q})dx + 2\mathop{\mathrm{Re}}\nolimits \int _{\partial \Omega \setminus \Gamma _{D}}\partial _{\nu }q(m \cdot \nabla \bar{q})ds{}\end{array}$$

(39)

Proof

See [8]. □ 

Here we will present a few auxiliary Lemmas.

Lemma 4

Let $\Omega \subset \mathbb{R}^{d}$ be a bounded connected Lipschitz domain. Let $u \in H^{1}(\Omega )$ be a weak solution of (1) , with $f \in L^{2}(\Omega )$ and $g \in L^{2}(\Gamma _{R})$ . Then, we have for any ε > 0

$$\displaystyle\begin{array}{rcl} k^{2}\left \Vert u\right \Vert _{ L^{2}(\Gamma _{R})}^{2} \leq \frac{1} {\beta _{min}}\left (\frac{1} {\epsilon } \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + k^{2}\epsilon \left \Vert u\right \Vert _{ L^{2}(\Omega )}^{2} + \frac{1} {\beta _{min}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ).& &{}\end{array}$$

(40)

Proof

Taking v = u into the variational form (5) and looking at the imaginary part we have

$$\displaystyle\begin{array}{rcl} \mathfrak{I}(a(u,u)) = -(k\beta (x)u,u) = \mathfrak{I}((g,u)_{L^{2}(\Gamma _{R})} + (\,f,u)_{L^{2}(\Omega )}),& & {}\\ \end{array}$$

and so

$$\displaystyle\begin{array}{rcl} & & k\beta _{min}\left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2} {}\\ & & \quad \leq \left \Vert u\right \Vert _{L^{2}(\Omega )}\left \Vert f\right \Vert _{L^{2}(\Omega )} + \left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})} {}\\ & & \quad \leq \frac{1} {2k\xi _{1}}\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{k\xi _{1}} {2} \left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {2\xi _{2}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} + \frac{\xi _{2}} {2}\left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2}. {}\\ \end{array}$$

Multiplying by k, dividing by β _min , and setting ξ ₂ = β _min k we obtain

$$\displaystyle\begin{array}{rcl} k^{2}\left \Vert u\right \Vert _{ L^{2}(\Gamma _{R})}^{2} \leq \frac{1} {\beta _{min}}\bigg( \frac{1} {2\xi _{1}}\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2}& +& \frac{k^{2}\xi _{ 1}} {2} \left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & +& \frac{1} {2\beta _{min}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} + \frac{k^{2}\beta _{ min}} {2} \left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2}\bigg), {}\\ \end{array}$$

and we obtain

$$\displaystyle\begin{array}{rcl} \frac{k^{2}} {2} \left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2}& \leq & \frac{1} {\beta _{min}}\left ( \frac{1} {2\xi _{1}}\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{k^{2}\xi _{ 1}} {2} \left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {2\beta _{min}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ). {}\\ \end{array}$$

Taking ξ ₁ = ε > 0 we arrive at the estimate. □ 

We will also need the estimate below.

Lemma 5

Let $\Omega \subset \mathbb{R}^{d}$ be a bounded connected Lipschitz domain. Let $u \in H^{1}(\Omega )$ be a weak solution of (1) with $f \in L^{2}(\Omega )$ and $g \in L^{2}(\Gamma _{R})$ . Then, we have

$$\displaystyle{ \begin{array}{ll} &\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \quad \leq \frac{1} {A_{min}}\bigg[k^{2}\left (V _{max}^{2} + \frac{\xi _{4}} {\beta _{min}} + \frac{\xi _{3}} {2}\right )\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \qquad \qquad + \left ( \frac{1} {2k^{2}\xi _{3}} + \frac{1} {\beta _{min}\xi _{4}} \right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}} \right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\bigg]. \end{array} }$$

(41)

for any ξ ₃ ,ξ ₄ > 0.

Proof

Taking v = u into the variational form (5) and looking at the real part we have

$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{Re}}\nolimits (a(u,u))& =& (A(x)\nabla u,\nabla u)_{L^{2}(\Omega )} - (k^{2}V ^{2}(x)u,u)_{ L^{2}(\Omega )} {}\\ & =& \mathop{\mathrm{Re}}\nolimits ((g,u)_{L^{2}(\Gamma _{R})} + (\,f,u)_{L^{2}(\Omega )}), {}\\ \end{array}$$

and so we have

$$\displaystyle\begin{array}{rcl} \left \Vert A^{\frac{1} {2} }\nabla u\right \Vert _{L^{2}(\Omega )}^{2} \leq k^{2}\left \Vert V u\right \Vert _{L^{2}(\Omega )}^{2} + \left \Vert u\right \Vert _{L^{2}(\Omega )}\left \Vert f\right \Vert _{L^{2}(\Omega )} + \left \Vert u\right \Vert _{L^{2}(\Gamma _{ R})}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}.& & {}\\ \end{array}$$

Using the maximal and minimal values we have for any ξ ₃ > 0 that

$$\displaystyle\begin{array}{rcl} A_{min}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2}& \leq & k^{2}\left \Vert V u\right \Vert _{ L^{2}(\Omega )}^{2} + \left \Vert u\right \Vert _{ L^{2}(\Omega )}\left \Vert f\right \Vert _{L^{2}(\Omega )} + \left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})} \\ & \leq & \left (k^{2}V _{ max}^{2} + \frac{k^{2}\xi _{ 3}} {2} \right )\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {2k^{2}\xi _{3}}\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} \\ & +& \frac{1} {4k^{2}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} + k^{2}\left \Vert u\right \Vert _{ L^{2}(\Gamma _{R})}^{2}. {}\end{array}$$

(42)

Using estimate (40) we may write for any ε > 0

$$\displaystyle\begin{array}{rcl} k^{2}\left \Vert u\right \Vert _{ L^{2}(\Gamma _{R})}^{2} \leq \frac{1} {\beta _{min}}\left (k^{2}\epsilon \left \Vert u\right \Vert _{ L^{2}(\Omega )}^{2} + \frac{1} {\epsilon } \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {\beta _{min}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ).& &{}\end{array}$$

(43)

Inserting the above inequality into (42) we obtain

$$\displaystyle\begin{array}{rcl} & & A_{min}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & & \qquad \leq \left (k^{2}V _{ max}^{2} + \frac{k^{2}\xi _{ 3}} {2} \right )\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {2k^{2}\xi _{3}}\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {4k^{2}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} {}\\ & & \qquad + \frac{1} {\beta _{min}}\left (k^{2}\epsilon \left \Vert u\right \Vert _{ L^{2}(\Omega )}^{2} + \frac{1} {\epsilon } \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{1} {\beta _{min}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ). {}\\ \end{array}$$

Taking ε = ξ ₄ the above inequality becomes

$$\displaystyle\begin{array}{rcl} A_{min}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2}& \leq & k^{2}\left (V _{ max}^{2} + \frac{\xi _{4}} {\beta _{min}} + \frac{\xi _{3}} {2}\right )\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & +& \left ( \frac{1} {2k^{2}\xi _{3}} + \frac{1} {\beta _{min}\xi _{4}}\right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}}\right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}. {}\\ \end{array}$$

Thus, we obtained our estimate. □ 

1.2 Proof of Main Stability Result

We are now in a position to prove Theorem 1. The key observation is that the Laplacian may be rewritten using (1) and combined with the technical and auxiliary lemmas. This leads to the conditions on the coefficients (12).

Proof (Proof of Theorem 1)

Using (39) where we write

$$\displaystyle{-\Delta u = \frac{1} {A}(\,f + k^{2}V ^{2}u + \nabla A \cdot \nabla u),}$$

∂ _ν u = 0 on $\Gamma _{N}$, and ∂ _ν u = ik β u + g on $\Gamma _{R}$, we obtain

$$\displaystyle{ \begin{array}{ll} &\int _{\partial \Omega \setminus \Gamma _{D}}\vert \nabla u\vert ^{2}m \cdot \nu ds -\int _{\Gamma _{D}}\vert \partial _{\nu }u\vert ^{2}m \cdot \nu ds \\ &\qquad =\int _{\Omega }\mbox{ div}(m)\vert \nabla u\vert ^{2}dx - 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\nabla u \cdot (\nabla \bar{u}\nabla )mdx \\ &\qquad \qquad + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega } \frac{1} {A}(\,f + k^{2}V ^{2}u + \nabla A \cdot \nabla u)(m \cdot \nabla \bar{u})dx \\ &\qquad \qquad + 2\mathop{\mathrm{Re}}\nolimits \int _{\Gamma _{R}}(ik\beta u + g)(m \cdot \nabla \bar{u})ds. \end{array} }$$

(44)

Using (38) with the transform $m \rightarrow \frac{V ^{2}} {A} m$, we have

$$\displaystyle\begin{array}{rcl} & & k^{2}\int _{ \partial \Omega }\vert u\vert ^{2}\left (\frac{V ^{2}} {A} \right )m \cdot \nu ds {}\\ & & \qquad \qquad = k^{2}\int _{ \Omega }\mbox{ div}\left (\frac{V ^{2}} {A} m\right )\vert u\vert ^{2}dx + 2k^{2}\mathop{ \mathrm{Re}}\nolimits \int _{ \Omega }u\left (\frac{V ^{2}} {A} \right )m \cdot \nabla \bar{u}dx. {}\\ \end{array}$$

Using this to replace the term $\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left (\frac{V ^{2}} {A} \right )u(m \cdot \nabla \bar{u})dx$, we have

$$\displaystyle\begin{array}{rcl} & & \int _{\partial \Omega \setminus \Gamma _{D}}\vert \nabla u\vert ^{2}m \cdot \nu ds -\int _{ \Gamma _{D}}\vert \partial _{\nu }u\vert ^{2}m \cdot \nu ds {}\\ & & \quad =\int _{\Omega }\mbox{ div}(m)\vert \nabla u\vert ^{2}dx - 2\mathop{\mathrm{Re}}\nolimits \int _{ \Omega }\nabla u \cdot (\nabla \bar{u}\nabla )mdx {}\\ & & \qquad + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left ( \frac{f} {A}\right )(m \cdot \nabla \bar{u})dx + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left (\frac{\nabla A} {A} \right ) \cdot \nabla u(m \cdot \nabla \bar{u})dx {}\\ & & \qquad + 2\mathop{\mathrm{Re}}\nolimits \int _{\Gamma _{R}}(ik\beta u + g)(m \cdot \nabla \bar{u})ds {}\\ & & \qquad - k^{2}\int _{ \Omega }\mbox{ div}\left (\frac{V ^{2}} {A} m\right )\vert u\vert ^{2}dx + k^{2}\int _{ \partial \Omega }\vert u\vert ^{2}\left (\frac{V ^{2}} {A} \right )m \cdot \nu ds. {}\\ \end{array}$$

Expanding out the boundary terms in each of the portions we have

$$\displaystyle{ \begin{array}{ll} & -\int _{\Gamma _{D}}\vert \partial _{\nu }u\vert ^{2}m \cdot \nu ds +\int _{\Gamma _{N}}\vert \nabla u\vert ^{2}m \cdot \nu ds \\ &\qquad +\int _{\Gamma _{R}}\vert \nabla u\vert ^{2}m \cdot \nu ds + k^{2}\int _{\Omega }\mbox{ div}\left (\frac{V ^{2}} {A} m\right )\vert u\vert ^{2}dx \\ & =\int _{\Omega }\mbox{ div}(m)\vert \nabla u\vert ^{2}dx - 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\nabla u \cdot (\nabla \bar{u}\nabla )mdx \\ &\qquad + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left ( \frac{f} {A}\right )(m \cdot \nabla \bar{u})dx + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left (\frac{\nabla A} {A} \right ) \cdot \nabla u(m \cdot \nabla \bar{u})dx \\ &\qquad + k^{2}\int _{\Gamma _{N}}\vert u\vert ^{2}\left (\frac{V ^{2}} {A} \right )m \cdot \nu ds + k^{2}\int _{ \Gamma _{R}}\vert u\vert ^{2}\left (\frac{V ^{2}} {A} \right )m \cdot \nu ds \\ &\qquad + 2\mathop{\mathrm{Re}}\nolimits \int _{\Gamma _{R}}(ik\beta u + g)(m \cdot \nabla \bar{u})ds.\end{array} }$$

(45)

Now we suppose we make the geometric assumptions made by Hetmaniuk [11] outlined in (9). Recall, we have for m = x − x ₀, thus we compute

$$\displaystyle\begin{array}{rcl} \mbox{ div}(x - x_{0})& =& d\mbox{ in }\Omega, {}\\ \nabla u \cdot (\nabla \bar{u}\nabla )(x - x_{0})& =& \vert \nabla u\vert ^{2}\mbox{ in }\Omega, {}\\ (x - x_{0})\cdot \nu & \leq & 0\mbox{ on }\Gamma _{D}, {}\\ (x - x_{0})\cdot \nu & =& 0\mbox{ on }\Gamma _{N}, {}\\ (x - x_{0})\cdot \nu & \geq & \eta \mbox{ on }\Gamma _{R}. {}\\ \end{array}$$

Using the above relations in (45) we obtain

$$\displaystyle{ \begin{array}{ll} &\eta \int _{\Gamma _{R}}\vert \nabla u\vert ^{2}ds + k^{2}\int _{\Omega }\mbox{ div}\left (\frac{V ^{2}} {A} (x - x_{0})\right )\vert u\vert ^{2}dx \\ & \leq (d - 2)\int _{\Omega }\vert \nabla u\vert ^{2}dx + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left ( \frac{f} {A}\right )((x - x_{0}) \cdot \nabla \bar{u})dx \\ & + 2\mathop{\mathrm{Re}}\nolimits \int _{\Omega }\left (\frac{\nabla A} {A} \right )\nabla u((x - x_{0}) \cdot \nabla \bar{u})dx \\ & + k^{2}\int _{\Gamma _{R}}\vert u\vert ^{2}\left (\frac{V ^{2}} {A} \right )(x - x_{0}) \cdot \nu ds + 2\mathop{\mathrm{Re}}\nolimits \int _{\Gamma _{R}}(ik\beta u + g)(m \cdot \nabla \bar{u})ds.\end{array} }$$

(46)

Recall, (10), where we define the following function

$$\displaystyle{ \begin{array}{ll} S(x)&:= \mbox{ div}\left (\left (\frac{V ^{2}(x)} {A(x)} \right )(x - x_{0})\right ) \\ & = d\left (\frac{V ^{2}(x)} {A(x)} \right ) + \left (2\frac{V (x)\nabla V (x)} {A(x)} -\frac{V ^{2}(x)\nabla A(x)} {A^{2}(x)} \right ) \cdot (x - x_{0}),\end{array} }$$

(47)

and from (12), we have a minimum for S(x) exists and is positive

$$\displaystyle{S_{min} =\min _{x\in \Omega }S(x)> 0.}$$

Further, from (12), we have C _G to be the minimal constant so that

$$\displaystyle\begin{array}{rcl} 2\left \vert \int _{\Omega }\left (\frac{\nabla A} {A} \right )\nabla u((x - x_{0}) \cdot \nabla \bar{u})dx\right \vert \leq C_{G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2}.& &{}\end{array}$$

(48)

Returning to inequality (46), we obtain

$$\displaystyle{ \begin{array}{ll} &\eta \left \Vert \nabla u\right \Vert _{L^{2}(\Gamma _{R})}^{2} + k^{2}S_{min}\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \leq (d - 2)\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} + C_{G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \quad + C_{1}\left ( \frac{1} {A_{min}}\left \Vert f\right \Vert _{L^{2}(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )} + \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}\left \Vert \nabla u\right \Vert _{L^{2}(\Gamma _{R})}\right ) \\ &\quad + C_{1}\left (k^{2}\left (\frac{V _{max}^{2}} {A_{min}} \right )\left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2} + k\left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}\left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}\left \Vert \nabla u\right \Vert _{L^{2}(\Gamma _{R})}\right ),\end{array} }$$

(49)

where C ₁ is independent of k and the bounds (3). Note that on the right hand side we have for any ξ ₅, ξ ₆ > 0 the terms

$$\displaystyle\begin{array}{rcl} k\left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}\left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}\left \Vert \nabla u\right \Vert _{L^{2}(\Gamma _{R})}& \leq & \frac{k^{2}} {2\xi _{5}} \left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2} + \frac{\xi _{5}} {2}\left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}^{2}\left \Vert \nabla u\right \Vert _{ L^{2}(\Gamma _{R})}^{2} {}\\ \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}\left \Vert \nabla u\right \Vert _{L^{2}(\Gamma _{R})}& \leq & \frac{1} {2\xi _{6}}\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} + \frac{\xi _{6}} {2}\left \Vert \nabla u\right \Vert _{L^{2}(\Gamma _{R})}^{2}. {}\\ \end{array}$$

We choose ξ ₅, ξ ₆ so that

$$\displaystyle{ \frac{\eta } {2} = C_{1} \frac{\xi _{5}} {2}\left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}^{2} = C_{ 1} \frac{\xi _{6}} {2},}$$

and so

$$\displaystyle{\frac{k^{2}} {2\xi _{5}} \leq \frac{C_{1}} {2\eta } \left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}^{2}k^{2}.}$$

We then obtain

$$\displaystyle{ \begin{array}{ll} k^{2}S_{min}\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} & \leq C_{1}\left (\left (\frac{C_{1}} {2\eta } \left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}^{2} + \frac{V _{max}^{2}} {A_{min}} \right )k^{2}\left \Vert u\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ) \\ & + C_{1}\left ( \frac{1} {A_{min}}\left \Vert f\right \Vert _{L^{2}(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )} + \frac{C_{1}} {2\eta } \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ) \\ & + (d - 2)\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} + C_{G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2}. \end{array} }$$

(50)

Taking $C_{2}^{bd} = C_{1}\left (\frac{C_{1}} {2\eta } \left \Vert \beta \right \Vert _{L^{\infty }(\Gamma _{R})}^{2} + \frac{V _{max}^{2}} {A_{min}} \right )$ and letting ε = β _min ξ ₇∕C ₂ ^bd in the inequality (40) we have the relation

$$\displaystyle\begin{array}{rcl} C_{2}^{bd}k^{2}\left \Vert u\right \Vert _{ L^{2}(\Gamma _{R})}^{2} \leq \frac{(C_{2}^{bd})^{2}} {\beta _{min}^{2}\xi _{7}} \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + k^{2}\xi _{ 7}\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} + \frac{C_{2}^{bd}} {\beta _{min}^{2}} \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}.& &{}\end{array}$$

(51)

Applying this above inequality to (50), we obtain

$$\displaystyle{ \begin{array}{ll} &k^{2}(S_{min} -\xi _{7})\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \qquad \leq C_{1}\left ( \frac{1} {A_{min}}\left \Vert f\right \Vert _{L^{2}(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )} + \frac{C_{1}} {2\eta } \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ) \\ &\qquad \qquad + \left ((d - 2) + C_{G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\right )\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \qquad \qquad + \frac{(C_{2}^{bd})^{2}} {\beta _{min}^{2}\xi _{7}} \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{C_{2}^{bd}} {\beta _{min}^{2}} \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}.\end{array} }$$

(52)

Recall the estimate (41), with $C_{3}^{bd} = \left ((d - 2) + C_{G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\right )$, and taking $\xi _{4} = \frac{\xi _{3}} {2} =\xi _{8}$

$$\displaystyle\begin{array}{rcl} & & C_{3}^{bd}\left \Vert \nabla u\right \Vert _{ L^{2}(\Omega )}^{2} {}\\ & & \ \leq \frac{C_{3}^{bd}k^{2}} {A_{min}} \left (V _{max}^{2} + \frac{\xi _{8}} {\beta _{min}} +\xi _{8}\right )\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & & \quad + \frac{C_{3}^{bd}} {A_{min}}\left ( \frac{1} {4k^{2}\xi _{8}} + \frac{1} {\beta _{min}\xi _{8}}\right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{C_{3}^{bd}} {A_{min}}\left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}}\right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}. {}\\ \end{array}$$

and so, using the above estimate (52)we obtain

$$\displaystyle{ \begin{array}{ll} &k^{2}(S_{min} -\xi _{7} - \frac{C_{3}^{bd}} {A_{min}}\left (V _{max}^{2} + \frac{\xi _{8}} {\beta _{min}} +\xi _{8}\right ))\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \leq C_{1}\left ( \frac{1} {A_{min}}\left \Vert f\right \Vert _{L^{2}(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )} + \frac{C_{1}} {2\eta } \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\right ) \\ & + \frac{C_{3}^{bd}} {A_{min}}\left ( \frac{1} {4k^{2}\xi _{8}} + \frac{1} {\beta _{min}\xi _{8}} \right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{C_{3}^{bd}} {A_{min}}\left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}} \right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} \\ & + \frac{(C_{2}^{bd})^{2}} {\beta _{min}^{2}\xi _{7}} \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{C_{2}^{bd}} {\beta _{min}^{2}} \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}.\end{array} }$$

(53)

Finally to deal with the remaining term on the right hand side that contains ∇u, we note using (41), letting $\frac{\xi _{4}} {\beta _{min}} = \frac{\xi _{3}} {2} = \frac{V _{max}^{2}} {2}$, and multiplying by ξ ₉∕(2A _min ), ξ ₉ > 0, we obtain

$$\displaystyle\begin{array}{rcl} & & \frac{\xi _{9}} {2A_{min}}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & & \qquad \leq \frac{\xi _{9}} {2A_{min}^{2}}\bigg[2V _{max}^{2}k^{2}\left \Vert u\right \Vert _{ L^{2}(\Omega )}^{2} + \left ( \frac{2} {\beta _{min}^{2}V _{max}^{2}} + \frac{1} {2k^{2}V _{max}^{2}}\right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & & \qquad \qquad \qquad \qquad + \left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}}\right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}\bigg], {}\\ \end{array}$$

and so

$$\displaystyle\begin{array}{rcl} & & \frac{1} {A_{min}}\left \Vert f\right \Vert _{L^{2}(\Omega )}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )} {}\\ & & \qquad \leq \frac{1} {2\xi _{9}A_{min}}\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{\xi _{9}} {2A_{min}}\left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & & \qquad \leq \frac{\xi _{9}V _{max}^{2}} {A_{min}^{2}} k^{2}\left \Vert u\right \Vert _{ L^{2}(\Omega )}^{2} {}\\ & & \qquad \qquad + \left ( \frac{1} {2A_{min}\xi _{9}} + \frac{\xi _{9}} {2A_{min}^{2}}\left ( \frac{2} {\beta _{min}^{2}V _{max}^{2}} + \frac{1} {2k^{2}V _{max}^{2}}\right )\right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} {}\\ & & \qquad \qquad + \frac{\xi _{9}} {2A_{min}^{2}}\left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}}\right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}. {}\\ \end{array}$$

Applying this into (53), we obtain

$$\displaystyle{ \begin{array}{ll} &k^{2}(S_{min} -\xi _{7} - \frac{C_{3}^{bd}} {A_{min}}\left (V _{max}^{2} + \frac{\xi _{8}} {\beta _{min}} +\xi _{8}\right ) -\frac{C_{1}\xi _{9}V _{max}^{2}} {A_{min}^{2}} )\left \Vert u\right \Vert _{L^{2}(\Omega )}^{2} \\ & \leq C_{1}\left ( \frac{1} {2A_{min}\xi _{9}} + \frac{\xi _{9}} {2A_{min}^{2}} \left ( \frac{2} {\beta _{min}^{2}V _{max}^{2}} + \frac{1} {2k^{2}V _{max}^{2}} \right )\right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} \\ & + C_{1}\left (\frac{C_{1}} {2\eta } + \frac{\xi _{9}} {2A_{min}^{2}} \left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}} \right )\right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} \\ & + \frac{C_{3}^{bd}} {A_{min}}\left ( \frac{1} {4k^{2}\xi _{8}} + \frac{1} {\beta _{min}\xi _{8}} \right )\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} \\ & + \frac{C_{3}^{bd}} {A_{min}}\left ( \frac{1} {\beta _{min}^{2}} + \frac{1} {4k^{2}} \right )\left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2} + \frac{(C_{2}^{bd})^{2}} {\beta _{min}^{2}\xi _{7}} \left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \frac{C_{2}^{bd}} {\beta _{min}^{2}} \left \Vert g\right \Vert _{L^{2}(\Gamma _{R})}^{2}. \end{array} }$$

(54)

Hence, we see that the critical term is $S_{min} -\frac{C_{3}^{bd}V _{ max}^{2}} {A_{min}}.$ Recall,

$$\displaystyle{C_{3}^{bd}:= \left ((d - 2) + C_{ G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\right ),}$$

thus, from (12), we have

$$\displaystyle\begin{array}{rcl} S_{min} -\left ((d - 2) + C_{G}\left \Vert \left (\frac{\nabla A} {A} \right )\right \Vert _{L^{\infty }(\Omega )}\right )\frac{V _{max}^{2}} {A_{min}}> 0.& &{}\end{array}$$

(55)

Since (55) is assumed to hold, we take ξ ₇, ξ ₈, and ξ ₉, so that

$$\displaystyle{\left (S_{min} -\frac{C_{3}^{bd}V _{max}^{2}} {A_{min}} -\xi _{7} -\frac{C_{3}^{bd}\xi _{8}} {A_{min}} \left ( \frac{1} {\beta _{min}} + 1\right ) -\frac{C_{1}\xi _{9}V _{max}^{2}} {A_{min}^{2}} \right )>\delta }$$

for some δ > 0, and taking C ₄ ^bd to be the global constant bound for (54) we obtain

$$\displaystyle\begin{array}{rcl} k^{2}\left \Vert u\right \Vert _{ L^{2}(\Omega )}^{2} \leq \frac{C_{4}^{bd}} {\delta } \left (1 + \frac{1} {k^{2}}\right )\left (\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \left \Vert g\right \Vert _{ L^{2}(\Gamma _{R})}^{2}\right ),& &{}\end{array}$$

(56)

and using (41), and taking C ₅ ^bd to be the global constant bound we obtain

$$\displaystyle\begin{array}{rcl} \left \Vert \nabla u\right \Vert _{L^{2}(\Omega )}^{2} \leq C_{ 5}^{bd}\left (1 + \frac{1} {k^{2}}\right )\left (\left \Vert f\right \Vert _{L^{2}(\Omega )}^{2} + \left \Vert g\right \Vert _{ L^{2}(\Gamma _{R})}^{2}\right ),& &{}\end{array}$$

(57)

as desired. □