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On the method of reflections

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Abstract

This paper aims at reviewing and analysing the method of reflections, which is an iterative procedure designed for solving linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on boundary integral representation. For both the sequential and parallel forms of the method appearing in the literature, we interpret the procedure in terms of projection operators. Using a Hilbert space setting and orthogonality, we prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. Several examples of boundary value problems that enter such a framework are given, an alternative proof of convergence is provided in a case which does not. A few numerical tests conclude the study.

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Notes

  1. One can for instance check the Ref. [10], in which different solution methods are used depending on the nature of the considered one-object problem.

  2. One may consider the first kind of boundary condition for the Laplace equation to be Dirichlet’s, the second, Neumann’s, and the third, Robin’s.

  3. The algorithm presented here is a reformulation of the method analysed in [47], as the auxiliary fields do not appear explicitly in the reference.

  4. The version of the method presented in Chapter 6 of [32] differs slightly from the above procedure by focusing on one specific object. The reflections with respect to this object are then computed as in the sequential form of the method, whereas the reflections with respect to the other \(N-1\) objects are computed as in the parallel form.

  5. In the same work, a sufficient condition for convergence, depending on the frequency, the diameters and the areas of the sub-scatterers, as well as the respective distances between the sub-scatterers, is established.

  6. To see this, one can compare the hierarchy of different levels of approximation given in [11], ranging from the Born approximation to the Foldy–Lax model, with the sequence of approximations produced by the method of reflections.

  7. The mapping T is said to be nonexpansive on the normed space H if it is a function from H to itself such that

    $$\begin{aligned} \forall (u,v)\in H^2,\ \Vert Tu-Tv\Vert _H\le \Vert u-v\Vert _H. \end{aligned}$$

  8. The mapping T is said to be asymptotically regular on the normed space H if it is a function from H to itself such that

    $$\begin{aligned} \forall u\in H,\ \lim _{k\rightarrow +\infty }\Vert T^{k+1}u-T^ku\Vert _H=0. \end{aligned}$$

  9. Indeed, one has trivially that \(M\subset {\text {Fix}}T\), while the converse follows from the idempotence and the self-adjointness of the orthogonal projectors.

  10. This result remains valid for the more general sum \(\sum _{j=1}^N\gamma _j\,P_{M_j}\), where the scalars \(\gamma _i\), with i in \(\{1,\dots ,N\}\), are the weights of a convex combination (that is, such that, \(\forall i\in \{1,\dots ,N\}\), \(\gamma _i>0\), and \(\sum _{j=1}^n\gamma _j=1\)).

  11. Some authors say the sequence converges uniformly (see [4]).

  12. According to the definition of Friedrichs [26], if \(M_i\) and \(M_j\) are closed subspaces in a Hilbert space H, the angle between \(M_i\) and \(M_j\) is the angle in \([0,\frac{\pi }{2}]\) whose cosine is defined by

    Another notion is that of the minimal angle between \(M_i\) and \(M_j\), given by Dixmier in [24], which is the angle in \([0,\frac{\pi }{2}]\) whose cosine is defined by

    These two definitions are different if \(M_i\cap M_j\ne \{0\}\), and they both coincide with (different) principal angles (as introduced by Jordan [39]) if \(H=\mathbb {C}^n\).

  13. For \(N=2\), it is known (see [19] for instance) that \(c(M_1,M_2)<1\), where \(c(M_1,M_2)\) denotes the cosine of the angle between the subspaces \(M_1\) and \(M_2\), if and only if \(M_1+M_2\) is closed, if and only if \({M_1}^\bot +{M_2}^\bot \) is closed, if and only if \((M_1\cap (M_1\cap M_2)^\bot )+(M_2\cap (M_1\cap M_2)^\bot )\) is closed. For \(N\ge 3\), a generalization of the Friedrichs angle to several subspaces is introduced in [4] and it is shown in the same paper that the method converges linearly if and only if \(c(M_1,\dots ,M_N)<1\), which is a weaker condition (see Example 4.5 in [4]) than the sufficient one, based on Theorems 2.1 in [21] and 4.1 in [4], that one of the cosines \(c_{ij}=c_0(M_i\cap (\cap _{\ell =1}^N M_\ell )^\bot ,M_j\cap (\cap _{\ell =1}^N M_\ell )^\bot )\) of the Dixmier angles involving two subspaces \(M_i\) and \(M_j\) is strictly less than one.

  14. We recall that the Green function G is such that

    $$\begin{aligned} \forall (x,y)\in \varOmega \times \varOmega ,\ x\ne y,\ G(x,y)=\varPhi (y-x)-\varphi _x(y), \end{aligned}$$

    the function \(\varPhi \) being the fundamental solution of the Laplace equation and \(\varphi _x\) being a corrector function which, for a fixed x in \(\varOmega \), satisfies \(\varDelta \varphi _x=0\) in \(\varOmega \) and \(\varphi _x=\varPhi (\cdot -x)\) on \(\partial \varOmega \).

  15. For instance, for suspensions sedimenting in a uniform gravitational field, one has \(F_{O_i}=m_{O_i}\varvec{G}\), where \(m_{O_i}\) is the mass of the particle adjusted for buoyancy and \(\varvec{G}\) is the gravitational acceleration, and \(\tau _{O_i}=0\).

  16. Integral Equation Solver (http://www.mathworks.com/matlabcentral/fileexchange/34241) by Alexandre Munnier and Bruno Pinçon, MATLAB Central File Exchange. Retrieved February 15, 2016.

  17. This fact can be commonly observed in practice (see [12] for instance) and is theoretically proved for \(N=2\) in [52].

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Acknowledgements

Philippe Laurent would like to thank Frédéric Boyer for introducing him to the method of reflections. Guillaume Legendre would like to thank Christophe Hazard for pointing him to the relevant paper [5], and Mikhael Balabane himself for an interesting discussion on the topic. Julien Salomon would like to thank Gabriele Ciaramella, Olivier Glass and Alexandre Munnier for helpful discussions. Finally, the authors collectively thank the anonymous reviewers whose comments and suggestions helped improve the manuscript.

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Authors and Affiliations

  1. Institut de Recherche en Communications et Cybernétique de Nantes, CNRS, UMR 6597, École des Mines de Nantes, 4, rue Alfred Kastler, 44307, Nantes, France

    Philippe Laurent

  2. CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Guillaume Legendre

  3. INRIA Paris, ANGE Project-Team, 75589, Paris Cedex 12, France

    Julien Salomon

  4. CNRS, Laboratoire Jacques-Louis Lions, Sorbonne Université, 75005, Paris, France

    Julien Salomon

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Laurent, P., Legendre, G. & Salomon, J. On the method of reflections. Numer. Math. 148, 449–493 (2021). https://doi.org/10.1007/s00211-021-01207-6

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