Convergence Analysis of the Continuous and Discrete Non-overlapping Double Sweep Domain Decomposition Method Based on PMLs for the Helmholtz Equation

Abstract

In this paper we will analyze the convergence of the non-overlapping double sweep domain decomposition method (DDM) with transmission conditions based on PMLs for the Helmholtz equation. The main goal is to establish the convergence of the double sweep DDM of both the continuous level problem and the corresponding finite element problem. We show that the double sweep process can be viewed as a contraction mapping of boundary data used for local subdomain problems not only in the continuous level and but also in the discrete level. It turns out that the contraction factor of the contraction mapping of the continuous level problem is given by an exponentially small factor determined by PML strength and PML width, whereas the counterpart of the discrete level problem is governed by the dominant term between the contraction factor similar to that of the continuous level problem and the maximal discrete reflection coefficient resulting from fast decaying evanescent modes. Based on this analysis we prove the convergence of approximate solutions in the $H^1$-norm. We also analyze how the discrete double sweep DDM depends on the number of subdomains and the PML parameters as the finite element discretization resolves sufficiently the Helmholtz and PML equations. Our theoretical results suggest that the contraction factor for the propagating modes depends linearly on the number of subdomains. To ensure the convergence, it is sufficient to have the PML width growing logarithmically with the number of subdomains. In the end, numerical experiments illustrating the convergence will be presented as well.

Appendix

In this section, we provide the proof of Lemma 4.9.

Proof of Lemma 4.9

We will estimate

$$\begin{aligned} \chi (w)^{2N_p} = \left( (1+iw+{\mathcal {R}}_{\chi }(w))^{\frac{1}{iw+{\mathcal {R}}_{\chi }(w)}}\right) ^{\frac{2\beta }{h}(iw+{\mathcal {R}}_{\chi }(w))}:={\mathfrak {a}}^{\mathfrak {w}} \end{aligned}$$

where

$$\begin{aligned} {\mathfrak {a}}=(1+iw+{\mathcal {R}}_{\chi }(w))^{\frac{1}{iw+{\mathcal {R}}_{\chi }(w)}} \ \text { and }\ {\mathfrak {w}}=\frac{2\beta }{h}\left( iw+{\mathcal {R}}_{\chi }(w)\right) =2\mu _{h,\ell }\sigma _0\beta \left( i+\frac{{\mathcal {R}}_\chi (w)}{w}\right) . \end{aligned}$$

Since ${\mathfrak {a}}\rightarrow e$ and ${\mathcal {R}}_{\chi }(w)/w\rightarrow 0$ as $w\rightarrow 0$, for any $\epsilon >0$ we can take a positive constant $\delta$ small enough so that

$$\begin{aligned} \sigma _r\frac{|\text {arg}({\mathfrak {a}})|}{\ln |{\mathfrak {a}}|}<\sigma _i\epsilon \ \text { and }\ \left| \sigma _0\frac{{\mathcal {R}}_{\chi }(w)}{w}\right| < \epsilon \min \{\sigma _r,\sigma _i\}. \end{aligned}$$

(7.1)

for $|w|<\delta$.

Noting $|{\mathfrak {a}}^{\mathfrak {w}}| = e^{\mathfrak {R}({\mathfrak {w}})\ln (|{\mathfrak {a}}|)-\mathfrak {I}({\mathfrak {w}})\text {arg}({\mathfrak {a}})},$ we need to estimate $\mathfrak {R}({\mathfrak {w}})$ and $\mathfrak {I}({\mathfrak {w}})$. First, for $\mu _{h,\ell }^2>0$ by using the second inequality of (7.1) we have that

$$\begin{aligned} \mathfrak {R}({\mathfrak {w}})< -2\mu _{h,\ell }\sigma _i\beta (1-\epsilon ) \ \ \text { and }\ \ |\mathfrak {I}({\mathfrak {w}})|< 2\mu _{h,\ell }\sigma _r\beta (1+\epsilon ). \end{aligned}$$

The first inequality of (7.1) with the above estimates leads us to

$$\begin{aligned} \mathfrak {R}({\mathfrak {w}})\ln (|{\mathfrak {a}}|)-\mathfrak {I}({\mathfrak {w}})\text {arg}({\mathfrak {a}})&< -2\mu _{h,\ell }\sigma _i\beta \ln (|{\mathfrak {a}}|)((1-\epsilon )-\epsilon (1+\epsilon ))\\&< -2\mu _{h,\ell }\sigma _i\beta \ln (|{\mathfrak {a}}|)(1-3\epsilon ). \end{aligned}$$

Due to the convergence of finite element eigenvalue approximations, there is $0<\epsilon _h<\epsilon _*$ for each $0<h\le h_0$ such that $\mu _{h,\ell }>\mu _{\min }-\epsilon _h$ for all $\ell$ and $\epsilon _h\rightarrow 0$ as $h\rightarrow 0$. Thus we can further show that for $0<h\le h_0$

$$\begin{aligned} \mathfrak {R}({\mathfrak {w}})\ln (|{\mathfrak {a}}|)-\mathfrak {I}({\mathfrak {w}})\text {arg}({\mathfrak {a}}) <-2(1-3\epsilon )\beta \ln (|{\mathfrak {a}}|)(\sigma _\mu -\epsilon _h\sigma _i) \end{aligned}$$

In the same way, one can show that for $\mu _{h,\ell }^2<0$

$$\begin{aligned} \mathfrak {R}({\mathfrak {w}})\ln (|{\mathfrak {a}}|)-\mathfrak {I}({\mathfrak {w}})\text {arg}({\mathfrak {a}}) <-2(1-3\epsilon )\beta \ln (|{\mathfrak {a}}|)(\sigma _\mu - \epsilon _h\sigma _r). \end{aligned}$$

Thus, we have

$$\begin{aligned} \lim _{w\rightarrow 0}|\chi (w)|^{2N_p} \le e^{2\epsilon _h(1-3\epsilon )\sigma _{M}\beta } e^{-2(1-3\epsilon )\sigma _\mu \beta } \end{aligned}$$

where $\sigma _M=\max \{\sigma _r,\sigma _i\}$. Since $\epsilon$ can be arbitrarily small, it holds that

$$\begin{aligned} \lim _{w\rightarrow 0}|\chi (w)|^{2N_p} \le e^{2\epsilon _h\sigma _M\beta }e^{-2\sigma _\mu \beta }. \end{aligned}$$

From the fact that $e^{\epsilon _h\sigma _M\beta }\rightarrow 1$ as $h\rightarrow 0$, it then follows that for any $\epsilon >0$ there exist $0<\delta _0$ and $0<{{\hat{h}}}_1\le h_0$ such that if $|w|\le \delta _0$ and $0<h\le {{\hat{h}}}_1$, then

$$\begin{aligned} |\chi (w)|^{2N_p} \le (1+\epsilon )e^{-2\sigma _\mu \beta }. \end{aligned}$$

(7.2)

Next, we will prove that (7.2) still holds for $|w|>\delta _0$ and for sufficiently small h. To this end, we write $w=h\mu _{h,\ell }\sigma _0=re^{i\theta }$, where $r=|w|$ and $\theta =\arg (w)$ with $0<\theta <\pi$. Let ${{\widetilde{\chi }}}_\theta (r)=\chi (w)$ as a function of r. Noting that $|{{\widetilde{\chi }}}_\theta (r)|^2=1-2r\sin (\theta )+O(r^2)$ resulting from the asymptotic behavior of $\chi$ in (4.42), it is revealed that $|{{\widetilde{\chi }}}_\theta (r)|$ is a decreasing function near the origin. By taking into account the fact $\lim _{r\rightarrow \infty }|{{\widetilde{\chi }}}_\theta (r)|=2-\sqrt{3}<1$, we can choose $0<\delta _1<\delta _0$ small enough so that

$$\begin{aligned} |{{\widetilde{\chi }}}_\theta (\delta _1)|=\max _{r\ge \delta _1}|{{\widetilde{\chi }}}_\theta (r)|. \end{aligned}$$

(7.3)

Let $h_1\le {{\hat{h}}}_{1}$ be a positive constant such that $\mu _{\min }h_1|\sigma _0|<\delta _1$. Now, it suffices to prove (7.2) for $h\le h_1$ and $|w|\ge \delta _1$. For $|w|\ge \delta _1$, let ${{\hat{w}}}=\delta _1w/|w|$, which can be written as ${{\hat{w}}}=h{{\hat{\mu }}}_{h,\ell }\sigma _0$ with ${\hat{\mu }}_{h,\ell }=\mu _{h,\ell }\delta _1/|w|$. Then ${\hat{\mu }}_{h,\ell }$ satisfies $\mu _{\min }< |{\hat{\mu }}_{h,\ell }|$ as

$$\begin{aligned} \mu _{\min }h\le \mu _{\min } h_1 < \frac{\delta _1}{|\sigma _0|} = \frac{|{{\hat{w}}}|}{|\sigma _0|} = |{{\hat{\mu }}}_{h,\ell }|h. \end{aligned}$$

(7.4)

Since $|{{\hat{w}}}|=\delta _1\le |w|$, (7.3) gives $|\chi (w)|\le |\chi ({{\hat{w}}})|$, which in turn together with (7.2) and the fact that $|{{\hat{w}}}|<\delta _0$ and $|{{\hat{\mu }}}_{h,\ell }| >\mu _{\mathrm{{min}}}-\epsilon _h$ obtained from (7.4) shows that

$$\begin{aligned} |\chi (w)|^{2N_p} \le |\chi ({{\hat{w}}})|^{2N_p} \le (1+\epsilon )e^{-2\sigma _\mu \beta }, \end{aligned}$$

and the proof is completed. $\square$