behaviour of iterative solvers for the linear systems that arise in Galerkin methods. A recent review of these issues in the context of BIEs for high- frequency scattering is that of Antoine and Darbas (2012).
5.2. Relevant classical scattering results
In this subsection we collect some classical results about the high-frequency Helmholtz equation. Whereas in Section 3 we discussed the high-frequency asymptotics of the solution, here we discuss some classical results that bound the solution in terms of the data (in suitable norms).
These results are taken from a large body of work concerned with both the Helmholtz equation and the wave equation. The problem involving the Helmholtz equation that has received most interest is the question of
how the solutions behave as k → ∞. For scattering problems involving
the wave equation the most-studied problem is the long-time behaviour of the energy in a neighbourhood of the obstacle. These two problems are obviously related, since when one takes the Fourier transform in time of the wave equation one obtains the Helmholtz equation; however, to understand rigorously the relationship between the two problems one must understand how solutions of the wave equation with singularities in the initial data (such as a delta function) behave when they hit the obstacle. Understanding this so-called propagation of singularities was one of the main motivations for the development of the tools of microlocal analysis.
Some classic books and monographs describing this body of work in scat- tering theory from different perspectives are Babich and Buldyrev (2008),
Borovikov and Kinber (1994), H¨ormander (1985a, 1985b), Lax and Phillips
(1989), Melrose (1995), Morawetz (1975b) and Vainberg (1989) (with the Russian texts translations of earlier Russian originals).
A key concept in scattering theory is the geometrical classification of domains as either trapping or non-trapping. We give the non-technical def- inition; see, for instance, Lax and Phillips (1989, Epilogue). For a more
mathematical, but technical, definition see Melrose and Sj¨ostrand (1982,
Definition 7.20) or Melrose (1979, Definition 1.3). In this definition and the
rest of the section, as in Section 2, BR:={x ∈ Rd,|x| < R}.
Definition 5.4. (Trapping and non-trapping for smooth domains)
Assume that Ω+ is smooth, that is, Γ is C∞. For any R > 0 such that
Ω− ⊂ BR, consider all possible rays starting in Ω+ ∩ BR (i.e., starting
ray hits Γ, continue it according to the law of reflection (angle of incidence
equals angle of reflection) until it leaves BR. We call Ω+ trapping if there
are arbitrarily long paths or closed paths of rays; otherwise Ω+ is called
non-trapping.
(Note that Definition 5.4 does not cover rays that hit the boundary at a tangent, and there are additional subtleties with these. Indeed, Taylor (1976) showed that, in some cases, one cannot uniquely define the reflection of rays tangent to the boundary.)
We have defined trapping and non-trapping for C∞ domains since the
propagation of singularities of the wave equation on C∞ domains is fully
understood by the results of Melrose and Sj¨ostrand (1978, 1982). However,
when Ω+ is not C∞, and especially when it is not C1, understanding how
a ray reflects is more complicated. Following the development of the very effective but non-rigorous Geometrical Theory of Diffraction (Keller 1962), there has been much work on rigorously understanding the propagation of singularities on domains with corners and edges. The recent papers by Melrose, Vasy and Wunsch (2008, 2012) both contain good overviews of this work.
To illustrate the difficulty involved in formulating a definition of trapping for non-smooth domains, consider the particular example of rays hitting a convex polygon. When a ray hits a corner it produces diffracted rays emanating from the corner, and in particular some that travel along the sides of the polygon (understanding these rays on the boundary is implicit in the approximation results for the convex polygon in Section 3.3). This means that there exist ‘glancing rays’ that travel around the boundary of the polygon (hitting a corner and then either continuing on the next side or travelling back) and do not escape to infinity; thus the exterior of a convex polygon is, in this sense, a trapping domain. At each diffraction from a corner, however, these rays lose energy, and thus the trapping is in a weaker sense than having a closed path of rays (as in Definition 5.4).
We state below some results about solutions of the Helmholtz equation in non-trapping domains (Theorems 5.6 and 5.7) that depend on the prop-
agation of singularities results of Melrose and Sj¨ostrand (1978, 1982) for
C∞ domains. Using the recent results about propagation of singularities on
manifolds with corners and edges from the programme of work by Melrose, Vasy and Wunsch, it is reasonable to believe that analogous (or slightly weaker) results to Theorems 5.6 and 5.7 would hold for the exterior of a non-trapping polygon (in the weaker sense discussed above), for example. However, although the relevant technical tools now exist, such results have not yet been proved. (Actually, we note that, in the special case of the
exterior of a polygon, either the results of G´erard and Lebeau (1993), about
results of Hillairet (2005), about propagation of singularities on Euclidean surfaces with conic singularities, may be sufficient to achieve this goal, again with some non-trivial technical work outstanding.)
A slightly different, but related, question is whether there exists the ap- propriate analogue of the propagation of singularities results of Melrose and
Sj¨ostrand (1978, 1982) for, say, C2 domains (instead of C∞ domains), pos-
sibly allowing for higher-order diffractive effects. To the authors’ knowledge this does not appear to be the case, although we note that Ivrii’s work on the Weyl law and spectral asymptotics on domains slightly better than
C1 (Bronstein and Ivrii 2003, Ivrii 2003) contains some results for non-
tangent rays.
The preceding paragraphs hopefully give an indication of some of the subtleties associated with the concepts of trapping and non-trapping on non- smooth domains. For the remainder of this section we will consider domains such as the rectangular and elliptical cavities in Figure 5.1 as trapping
domains, despite not satisfying Definition 5.4, since they are not C∞, since
for these obstacles there exist closed paths of rays that hit only smooth parts of the boundary.
We have just encountered a slight ‘culture clash’ between the classic scat- tering literature and the numerical analysis literature: classical scattering theory is happiest in smooth domains, whereas from the point of view of practical applications we want to consider domains with corners and edges. What then should we aim for in the numerical analysis of scattering prob-
lems? An important class of C∞ domains for which the non-trapping con-
dition holds consists of domains that are star-shaped in the following sense.
Definition 5.5. (Star-shaped) We say that Ω− is star-shaped with re-
spect to the point x0 ∈ Ω− if there exists c > 0 such that (x− x0)· n ≥ c
for almost every x ∈ Γ, where n is the normal vector pointing outwards
from Ω−. We say that Ω−is star-shaped if it is star-shaped with respect to
some x0 ∈ Ω−.
If Ω− is star-shaped and C∞ then Ω+ is non-trapping in the sense of
Definition 5.4 (Lax and Phillips 1989, Chapter 5, Proposition 3.1) (star- shapedness guarantees that the rays can only be reflected a finite number of times before escaping from the obstacle). Since the normal vector is defined almost everywhere on Lipschitz boundaries, star-shapedness is a well-defined property for Lipschitz domains, and thus a reasonable aim from the point of view of numerical analysis of scattering problems is to prove results for star-shaped Lipschitz domains, and then also for (smooth) non-trapping domains.
Another ‘culture clash’ is that classical scattering theory is happiest in 3D (essentially because the fundamental solution for the Helmholtz equation
numerical analysis perspective one often starts in 2D, where computations are easier, and then progresses to 3D.
One way to understand the significance of trapping and non-trapping is the following. Recall that the Helmholtz operator with k > 0 in a bounded domain can have non-trivial solutions for certain homogeneous boundary conditions (these are eigenfunctions of the Laplacian). In an unbounded domain the Helmholtz equation with k > 0 cannot have any non-trivial solutions satisfying homogeneous boundary conditions, provided an appro-
priate radiation condition is prescribed (Corollary 2.9). However, as k→ ∞,
solutions of the Helmholtz equation behave more and more like rays (this is the whole notion of ray theory), and thus if the domain is trapping then, for certain k, there will be solutions of the Helmholtz equation localized in the trapping part of the obstacle and behaving almost like non-trivial solutions satisfying homogeneous boundary conditions. This informal discussion can be made mathematically precise through the concepts of resonances and quasimodes (see the references in Section 5.6.2).
In Section 5.6.2 we take a closer look at the type of behaviour that so- lutions of the Helmholtz equation can exhibit in trapping domains (and define the concepts of a resonance and a quasimode); for the remainder of this subsection we look at bounds on the solution operator of the Helmholtz equation in non-trapping domains.