Corners and stable optimized domain decomposition methods for the Helmholtz problem

Abstract

We construct a new Absorbing Boundary Condition (ABC) adapted to solving the Helmholtz equation in polygonal domains in dimension two. Quasi-continuity relations are obtained at the corners of the polygonal boundary. This ABC is then used in the context of domain decomposition where various stable algorithms are constructed and analysed. Next, the operator of this ABC is adapted to obtain a transmission operator for the Domain Decomposition Method (DDM) that is well suited for broken line interfaces. For each algorithm, we show the decrease of an adapted quadratic pseudo-energy written on the skeleton of the mesh decomposition, which establishes the stability of these methods. Implementation within a finite element solver (GMSH/GetDP) and numerical tests illustrate the theory.

Appendices

Geometric interpretation of the ABC

To provide a geometric intuition of the different terms of the bilinear form a from (17), we consider the simple situation where the domain is a K sided regular polygon approximating the disc ${\mathcal {D}}=\{x^2+y^2\le R^2\}$ from the interior as K goes to infinity. The corners of the polygon are denoted $A_k=R(\cos \frac{2k\pi }{K},\sin \frac{2k\pi }{K})$ with k defined modulo K. The middle of the edges are the $A_{k+1/2}=R(\cos \frac{(2k+1)\pi }{K},\sin \frac{(2k+1)\pi }{K})$. With our convention, see Fig. 2, the interior angle is $\theta _K=\frac{2\pi }{K}-\pi \in (-\pi ,0)$. The arclength between two successive corners is $\ell _K=\frac{2\pi }{K}R$.

Let $\varphi$ and $\psi$ be two regular functions defined on the circle $\partial {\mathcal {D}}=\{x^2+y^2=R^2\}$. First, the Hermitian part of a, defined on the first line of (17), is a simple broken approximation of the integral

$$\begin{aligned} {\mathcal {I}}_1(\varphi ,\psi ) := \int _{\partial {\mathcal {D}}} \left( \varphi (s)\overline{\psi (s)}+\frac{1}{2\omega ^2}\varphi '(s)\overline{\psi '(s)}\right) ds, \end{aligned}$$

where s is the curvilinear abscissa. Consider now the anti-Hermitian part of a, defined on the second line of (17). We study the two quantities

$$\begin{aligned} {\mathcal {A}}_K&:=\sum _k \cos \left( \frac{\theta _K}{2}\right) (\varphi (A_{k+1/2})+\varphi (A_{k-1/2}))\overline{(\psi (A_{k+1/2})+\psi (A_{k-1/2}))},\\ {\mathcal {B}}_K&:=\sum _k \frac{\cos \left( {\theta _K}\right) }{\cos \left( \frac{\theta _K}{2}\right) }(\varphi (A_{k+1/2})-\varphi (A_{k-1/2}))\overline{(\psi (A_{k+1/2})-\psi (A_{k-1/2}))}. \end{aligned}$$

Lemma 11

As $K\rightarrow \infty$, one has ${\mathcal {A}}_K\rightarrow {\mathcal {I}}_2(\varphi ,\psi ):= \frac{2}{R}\int _{\partial {\mathcal {D}}}\varphi (s)\overline{\psi (s)}ds$.

Proof

For large K, $\cos \left( \frac{\theta _K}{2}\right) \sim \frac{\ell _K}{2R}$. Therefore ${\mathcal {A}}_K= \sum _k \frac{\ell _K}{2R} 2\varphi (A_k)\overline{2\psi (A_k)}+\text {high order terms}$. One recognizes a Riemann sum, and passing to the limit yields the claim. $\square$

Lemma 12

As $K\rightarrow \infty$, one has ${\mathcal {B}}_K\rightarrow {\mathcal {I}}_3(\varphi ,\psi ):= -2R\int _{\partial {\mathcal {D}}}\varphi '(s)\overline{\psi '(s)}ds$.

Proof

For large K, $\frac{\cos \theta _K}{\cos \left( \frac{\theta _K}{2}\right) }\sim \frac{-2R}{\ell _K}$. Therefore ${\mathcal {B}}_K= \sum _k \frac{-2R}{\ell _K} \ell _K\varphi '(A_k)\overline{\ell _K\psi '(A_k)}+\text {high order terms}$. Again, passing to the limit yields the claim. $\square$

The sesquilinear form $\varphi ,\psi \mapsto a(\varphi ,\psi )$ is thus an approximation of the sesquilinear form

$$\begin{aligned} {{\widetilde{a}}}(\varphi ,\psi ):={\mathcal {I}}_1(\varphi ,\psi )-\frac{i}{4\omega }\left( {\mathcal {I}}_2(\varphi ,\psi )+{\mathcal {I}}_3(\varphi ,\psi )\right) , \end{aligned}$$

where the Hermitian part ${\mathcal {I}}_1$ is independent of the curvature radius R, and where the anti-Hermitian part $-\frac{i}{4\omega }({\mathcal {I}}_2+{\mathcal {I}}_3)$ depends on the curvature radius via terms proportional to R and 1/R. Similarly, the sesquilinear form $\varphi ,\psi \mapsto a^*(\varphi ,\psi )$ approximates the sesquilinear form ${{\widetilde{a}}}^*(\varphi ,\psi ):={\mathcal {I}}_1(\varphi ,\psi )+\frac{i}{4\omega }\left( {\mathcal {I}}_2(\varphi ,\psi )+{\mathcal {I}}_3(\varphi ,\psi )\right)$. In classical planar differential geometry, a smooth curve $\varGamma$ locally separates the domain into two regions $\varOmega _\pm$. The curvature radius R of $\varGamma$ seen from $\varOmega _+$ is the opposite of the curvature radius $-R$ of $\varGamma$ seen from $\varOmega _-$. Therefore, another possible interpretation of the difference between ${{\widetilde{a}}}$ and ${{\widetilde{a}}}^*$ is that they correspond to the same curve $\varGamma$ seen from one side or the other.

Remark 7

Considering Remark 1, the same interpretation holds for T and $T^*$: they account for the curvature radius, and a change of sign of the curvature radius changes T in $T^*$ and reciprocally.

The ABC associated to ${{\widetilde{a}}}(\varphi ,\psi )=(u,\psi )_{L^2(\varGamma )}$, noticing that $\varphi \simeq ({\mathbf {i}}\omega )^{-1}\partial _{{\mathbf {n}}}u$, writes in its strong form

$$\begin{aligned} \left( 1-\frac{1}{2\omega ^2}\partial _{ss} - \frac{{\mathbf {i}}}{4\omega }\left( \frac{2}{R}+2R\partial _{ss}\right) \right) \partial _{{\mathbf {n}}}u- {\mathbf {i}}\omega u=0. \end{aligned}$$

In polar coordinates and on the border of the disk ${\mathcal {D}}$, the normal derivative is the derivative along r ($\varphi \simeq ({\mathbf {i}}\omega )^{-1}\partial _{r}u$) and the curvilinear derivative is given by $\partial _{ss} = \frac{1}{R^2}\partial _{\theta \theta }$. Hence the following strong form of the ABC in polar coordinates:

$$\begin{aligned} \left( 1-\frac{1}{2R^2\omega ^2}\partial _{\theta \theta } - \frac{{\mathbf {i}}}{2\omega R}\left( 1+\partial _{\theta \theta }\right) \right) \partial _{r}u- {\mathbf {i}}\omega u=0. \end{aligned}$$

When the radius of the disk R goes to infinity, the border of ${\mathcal {D}}$ tends to be (locally) straight and the ABC converges towards the classical low order ABC $\partial _{r}u- {\mathbf {i}}\omega u=0$.

Subdomains and unknowns decoupled: DDM-3

In this appendix, we modify algorithm DDM-2 to decouple (22) from (23), at the price of introducing a new auxiliary unknown on each edge $\varGamma _k^i$. This unknown, denoted $\psi _{i,k}$, represents the Dirichlet trace of $u_i$ on $\varGamma _k^i$. The interest is that $u_i^{p+1}$ can be obtained by solving a classical Helmholtz boundary value problem, where the boundary condition involves $(\varphi _{i,k}^{p})_k$ and $(\psi _{i,k}^{p})_k$ at the previous iteration index p.

Initialize $u_i^0\in H^1(\varOmega _{i})$ with square integrable normal derivatives on each subdomain, and $(\varphi _{i,k}^0)_k\in \oplus _k H^1(\varGamma _k^i)$, $(\psi _{i,k}^0)_k\in \oplus _k L^2(\varGamma _k^i)$ on the exterior boundary of each subdomain. For $p=0,1,\ldots$, solve for each subdomain

$$\begin{aligned} \left\{ \begin{array}{ll} (-\varDelta - \omega ^2)u_i^{p+1}= f &{}\quad \text {in }\varOmega _{i} ,\\ \displaystyle \left( \partial _{{\mathbf {n}}^i}- {\mathbf {i}}\omega \right) u_i^{p+1} =\displaystyle - \left( \partial _{{\mathbf {n}}^j}+ {\mathbf {i}}\omega \right) u_j^{p} &{}\quad \text {on }\partial \varOmega _{i} \cap \partial \varOmega _{j},\forall j\ne i, \\ \displaystyle \left( \partial _{{\mathbf {n}}^i} - {\mathbf {i}}\omega \right) u_i^{p+1} = \displaystyle {\mathbf {i}}\omega \left( \varphi _{i,k}^{p} - \psi _{i,k}^{p} \right) &{}\quad \text {on }\varGamma _k^i, \forall k, \end{array} \right. \end{aligned}$$

(32)

and for each edge

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle \left( 1- \frac{1}{2\omega ^2} \partial _{{\mathbf {t}}_k{\mathbf {t}}_k} \right) \varphi _{i,k}^{p+1}({\mathbf {x}})=\psi _{i,k}^{p+1}({\mathbf {x}}), &{}\quad {\mathbf {x}}\in \varGamma _k^i,\\ \displaystyle \varphi _{i,k}^{p+1}({\mathbf {x}})+\psi _{i,k}^{p+1}({\mathbf {x}})= \frac{1}{{\mathbf {i}}\omega }\frac{\partial u_i^{p}}{\partial {\mathbf {n}}^i}({\mathbf {x}})+u_i^p({\mathbf {x}}), &{}\quad {\mathbf {x}}\in \varGamma _k^i,\\ \\ \displaystyle \left( \left( 1+ \frac{{\mathbf {i}}\beta }{\omega } \frac{ \cos \left( \frac{\theta _{k\ell }}{2}\right) }{\cos \theta _{k\ell }}\right) \frac{\partial \varphi _{i,k}^{p+1}}{\partial \varvec{\tau }_k} +\left( \beta - {\mathbf {i}}\omega \cos \left( \frac{\theta _{k\ell }}{2}\right) \right) \varphi _{i,k}^{p+1} \right) ({\mathbf {A}}_{k\ell }^{ij}) &{}\\ \displaystyle \quad = \left( \left( -1+ \frac{{\mathbf {i}}\beta }{\omega } \frac{ \cos \left( \frac{\theta _{k\ell }}{2}\right) }{\cos \theta _{k\ell }}\right) \frac{\partial \varphi _{j,\ell }^{p}}{\partial \varvec{\tau }_\ell } +\left( \beta + {\mathbf {i}}\omega \cos \left( \frac{\theta _{k\ell }}{2}\right) \right) \varphi _{j,\ell }^{p} \right) ({\mathbf {A}}_{k\ell }^{ij}), &{}\quad {\mathbf {A}}_{k\ell }^{ij}\in {\mathcal {C}}_{k}^i,\\ \\ \left( {\mathbf {i}}\omega \varphi _{i,k}^{p+1} + \partial _{\varvec{\tau }_k}\varphi _{i,k}^{p+1}\right) ({\mathbf {B}}_k^{ij}) =\left( {\mathbf {i}}\omega \varphi _{j,k}^p-\partial _{\varvec{\tau }_k}\varphi _{j,k}^p\right) ({\mathbf {B}}_k^{ij}), &{}\quad {\mathbf {B}}_k^{ij}\in {\mathcal {F}}_{k}^i, \end{array} \right. \nonumber \\ \end{aligned}$$

(33)

with the same $\beta$ as before to remove the singularity, see Remark 3. We now show the algorithm is endowed with a decreasing energy for $f=0$. Define

$$\begin{aligned} G^p&:=\overset{N_{\text {dom}}-1}{\underset{i=0}{\sum }} \left( \int _{\partial \varOmega _{i}\setminus \varGamma } \left| \left( \partial _{{\mathbf {n}}^i}-{\mathbf {i}}\omega \right) u_i^{p} \right| ^2\mathop {}\!\mathrm {d}\gamma + \overset{K-1}{\underset{k=0}{\sum }}\int _{\varGamma _k^i}\omega ^2\left| \varphi _{i,k}^p+\psi _{i,k}^p\right| ^2\mathop {}\!\mathrm {d}\gamma \right. \\&\quad +\, \overset{K-1}{\sum _{\underset{{{\mathbf {A}}_{k\ell }^{ij}\in {\mathcal {C}}_{k}^i}}{k=0}}} \frac{1}{2|\omega |\sin ^2\left( \frac{\theta _{k\ell }}{2}\right) } \left| \left( 1+ \frac{{\mathbf {i}}\beta }{\omega } \frac{ \cos \left( \frac{\theta _{k\ell }}{2}\right) }{\cos \theta _{k\ell }}\right) \frac{\partial \varphi _{i,k}^p}{\partial \varvec{\tau }_k} \right. \\&\quad \left. +\, \left( \beta -{\mathbf {i}}\omega \cos \left( \frac{\theta _{k\ell }}{2}\right) \right) \varphi _{i,k}^{p} \right| ^2 ({\mathbf {A}}_{k\ell }^{ij})\\&\quad +\, \overset{K-1}{\sum _{\underset{{\mathbf {B}}_k^{ij}\in {\mathcal {F}}_{k}^i}{k=0}}}\left. \frac{1}{2\omega } \left| \omega \varphi _{i,k}^p+\partial _{\varvec{\tau }_k}\varphi _{i,k}^p \right| ^2 ({\mathbf {B}}_k^{ij})\right) = F^p+\overset{N_{\text {dom}}-1}{\underset{i=0}{\sum }} \overset{K-1}{\underset{k=0}{\sum }}\int _{\varGamma _k^i}\omega ^2\left| \varphi _{i,k}^p+\psi _{i,k}^p\right| ^2\mathop {}\!\mathrm {d}\gamma . \end{aligned}$$

Lemma 13

The algorithm (32)–(33) is stable. For $f=0$, it has decreasing energy

$$\begin{aligned}G^{p+1} = G^p -2\overset{N_{\text {dom}}-1}{\underset{j=0}{\sum }} \overset{K-1}{\underset{\ell =0}{\sum }}\int _{\varGamma _{\ell }^j}\left( 2\omega ^2|\varphi _{j,\ell }^p|^2+|\partial _{{\mathbf {t}}_\ell }\varphi _{j,\ell }^p|^2\right) \mathop {}\!\mathrm {d}\gamma .\end{aligned}$$

Proof

Similar computations to the ones of the proof of Lemma 8 give

$$\begin{aligned} G^{p+1}=G^p-4{{\,\mathrm{Re}\,}}\overset{N_{\text {dom}}-1}{\underset{j=0}{\sum }} \overset{K-1}{\underset{\ell =0}{\sum }}\int _{\varGamma _{\ell }^j}\omega ^2\varphi _{j,\ell }^p\overline{\psi _{j,\ell }^p}\mathop {}\!\mathrm {d}\gamma -2{{\,\mathrm{Re}\,}}\overset{N_{\text {dom}}-1}{\underset{j=0}{\sum }} \overset{K-1}{\underset{\underset{{\mathbf {A}}\in {\mathcal {C}}_{\ell }^j\cup {\mathcal {F}}_{\ell }^j}{\ell =0}}{\sum }}\frac{\partial \varphi _{j,\ell }^p}{\partial \varvec{\tau }_\ell }({\mathbf {A}})\overline{\varphi _{j,\ell }^p}({\mathbf {A}}). \end{aligned}$$

Integrating the first equation of system (33) at iteration p on $\varGamma _{\ell }^j$ against $\varphi _{j,\ell }^p$ and taking the sum over all subdomain and edge indices j and $\ell$ gives the result:

$$\begin{aligned}&\overset{N_{\text {dom}}-1}{\underset{j=0}{\sum }} \overset{K-1}{\underset{\ell =0}{\sum }}\int \limits _{\varGamma _{\ell }^j}\left( |\varphi _{j,\ell }^p|^2+\frac{1}{2\omega ^2}\left| \frac{\partial \varphi _{j,\ell }^p}{\partial {\mathbf {t}}_\ell }\right| ^2\right) \mathop {}\!\mathrm {d}\gamma \\&\qquad -\, \frac{1}{2\omega ^2}\overset{N_{\text {dom}}-1}{\underset{j=0}{\sum }}\!\! \overset{K-1}{\underset{\underset{{\mathbf {A}}\in {\mathcal {C}}_{\ell }^j\cup {\mathcal {F}}_{\ell }^j}{\ell =0}}{\sum }} \frac{\partial \varphi _{j,\ell }^p}{\partial \varvec{\tau }_\ell }({\mathbf {A}})\overline{\varphi _{j,\ell }^p}({\mathbf {A}})\\&\quad =\!\!\!\!\overset{N_{\text {dom}}-1}{\underset{j=0}{\sum }} \overset{K-1}{\underset{\ell =0}{\sum }}\int \limits _{\varGamma _{\ell }^j}\psi _{j,\ell }^p\overline{\varphi _{j,\ell }^p}\mathop {}\!\mathrm {d}\gamma . \end{aligned}$$

$\square$