Functional Analysis of Time Domain Boundary Element Methods: A Review

As we mentioned in the introduction to this section, we use the classical approach for the analysis of the time domain boundary layer potentials and the corresponding time domain boundary integral operators that was introduced by Bamberger and Ha-Duong (1986). This means that we analyse the potentials and operators in the Laplace domain first, after mapping the original problem (P) to the Laplace domain by means of the Laplace transform. Reviews on this method have been published in the form of research papers, such as the ones by Ha-Duong (2003) or Laliena and Sayas (2009) [103], or in the form of lecture notes, such as the ones by Becache (1994) or, mast recently, Sayas (2011). the time harmonic Helmholtz problem first. Assume that u solves (P), and let u:= Ct[u]. Then u solves the exterior Helmholtz boundary value problem Equation 5a) is known as the homogeneous Helmholtz equation, while condition (5c) is called the Sommerfeld radiation condition. We note that there are at least two different versions of the Sommerfeld radiation condition by which (5c) could be replaced. For the definition of the boundary layer potentials and integral operators, we state the fundamental solutions of the Helmholtz equation in Lemma 3. Lemma 3 (Fundamental Solutions of the Helmholtz Equation)

(6) (7) (8)

with where denotes the Hankel function of order zero of the first, kind. The boundary potentials for the Helmholtz Problem (HH) are defined similarly to the time domain boundary layer potentials given in Definition 3. Definition 4 (Boundary Layer Potentials for the Helmholtz Problem) Let For appropriate densities the Helmholtz Single Layer potential is given by

(9)

and the Helmholtz Double Layer potential by (10)

the dependency on the wave numberinto account. We further refer to Costabel, M. (1988) for mapping properties of arbitrary elliptic boundary integral operators. The corresponding boundary integral operators and are defined analogously to Definition. Note that the Helmholtz boundary potentials and operators are the Laplace transforms of their transient counterparts, e.g. In the rest of this study, we restrict ourselves to the three dimensional case, even though we conduct our studies on a posteriori error estimates and our numerical experiments in two space dimensions. This distinction can be justified by the fact that the fundamental solution of the two dimensional Helmholtz equation is much more complicated than its three dimensional counterpart, and therefore dealing with it would lead to many additional technical difficulties. Other authors have followed these two different routes in theory and practice for the same reason. The new generalised mapping properties studied, however, also hold in two space dimensions. Note that we omit the hat while working in the Laplace domain and write u instead ofif its correct meaning is clear and there is no danger of confusion.

For some domain , the energy of u in is given

(11)

By the Parseval-Plancherel identity (Lemma 2),

(12)

This relation motivates the definition of the following energy-related norms. Recall the definition of the usual

(13)

Now, to guarantee let for some Then the norm

(14)

is equivalent to the We often refer to these norms as the wavenumber-dependent norms or -dependent noims. Remark (Equivalence Estimates for the Classical and |u;|-Dependent Norms) Note that the equivalence of the norms Ls -dependent. We have, for but only and hence, in summary,

(15)

Correspondingly, for

(16)

(15) and (16) can be summarised as

(17)

for any Norm equivalences of the type

(18)

for can be concluded from (17). Example (Norm Equivalences) a) For s= 1 Similarly, for s = 2, b) Since, for there holds and therefore equivalence of these two norms. Due to the -dependence of the equivalence estimates illustrated in Remark, we cannot deduce, for instance, the trace theorem for the -dependent norms directly from the results for the standard norms without any dosses' regarding powers of

The proofs of the following results are all similar to the ones of the original results given in McLean, W. (2000). For the particular case s = 1, a result similar

standard norms. These are also collected in Lemma. We give outlines of the proofs here for completeness and to demonstrate the (minor) differences to the original proofs. The term ‗generalisation‘ Ls appropriate in the sense that we obtain the classical theorems again for Lemma 4 : a) For the trace operator given by has a unique extension to a bounded linear operator where the continuity constant with respect to the - dependent noimdepends on but not on , i.e. there exists a -independent constant such that

(19)

for b) Let be domain and Then the trace operator defined by has an extension to a linear bounded operator where the continuity constant with respect to the -dependent norms depends on s, and but not. on in the sense of part a).

Part b) is a consequence of part a), using a technique called ‗flattening of the boundary‘ described in the proof of McLean, W. (2000), It is thus enough to show part a). We write and for Since there holds, by the definition of the inverse Fourier transform, By the definition, and thus

(20)

By the Cauchy-Schwarz inequality for integrals, we obtain where

(22)

via the substitution that gives dt and By McLean, W. (2000),, there holds, for where the constant is obviously -independent. Combining (21) and (22), we obtain

(23)

Taking the integralin (23) proves the claimed result via the definitions of the respective norms. Since a Lipschitz boundary is the result above can only be applied for in this case. In fact it can be extended to the range as the following result shows. The proof is again similar to the original ones of McLean, W. (2000),. Lemma 5 : Let be a Lipschitz domain and Then the trace opemtor defined in Lemma 4 b) is bounded independently of as an operator mapping to in the same way as in Lemma 4 b), i.e. with a continuity constant that depends only on s, and The following result is an immediate consequence of Lemma 4.5 and the fact that the norms of a linear operator and of its dual operator are equal; see, for instance, McLean, W. (2000). Corollary 1: Let be a Lipschitz domain andThen the adjoint operator to the trace operator defined in Lemma 4 b) is bounded independently of as an operator mapping to in the same way as in Lemma 5. Proof (of Lemma 5) First we define an anisotropic Sobolev space via the norm By the definitions of the norms, there holds

where, as in the proof of Lemma 4, Writing and for again, we define whereLs the Lipschitz-continuous function whose graph is. Then, by the definition,

(24)

Analogously to the proof of McLean, W. (2000) theorem, there hold the inequalities

(25)

and

(26)

where Ls -independent. Further, by the definition and (20), and by the Cauchy-Schwarz inequality for integrals, Using the same substitution as in the proof of Lemma 4, we obtain where the constant is obviously -independent . Thus

(27)

Combining estimates (24), (27), (25) and (26), we obtain which proves the claim. The next result is about the inverse to the so-called extension operator. As before, the proofs follow the respective ones given in McLean, W. (2000), closely. bounded linear operator where the continuity constant with respect to the - dependent norm depends on s, but not on in the sense of Lemma 7), i.e. there exists a -independent constant such that

(28)

for b) Let be domain and Then there exists a -dependent bounded linear operator which is a right inverse to The continuity constant with res-pect to the - dependent normsdepends on s, and but not on in the sense of part a). is sometimes called the extension operator to Proof Part b) is a consequence of part a); taking and using the same technique as in the proof of Lemma 7 b). It is thus enough to show part a). Take such that for The operator is defined by, for

(29)

where, in this case. Then and thus Substituting we obtain, with

and hence Now Using the same substitution as in the proof of Lemma 4, we obtain where the integral is bounded independently of for all and thus finally This completes the proof.

We close our observations on the -dependent norms with some brief remarks on interpolation. We use the notation with for interpolation spaces here, where X, Y are Banach spaces with Without providing any technical details, we first cite a result for future reference. Lemma 7 : For any there holds We now return to our specific setup. It is well established that for and . In what follows, we need: Theorem 1 (Interpolation for Sobolev Spaces Let the linear operator A be bounded as and as with for j = 0,1. Then

(30)

for and i.e. is also a bounded map for these numbers s,t. but it is cited here for the special case of Sobolev spaces. In what follows, it is shown that Theorem 4.1 still holds without any additional -dependent factors in (30) if the wave number dependent norms are used. To do this, we follow the proofs in McLean, W. (2000),. Let for and It is shown in McLean, W. (2000),] that with and and. where is a weighted L2 norm, Note that is the norm of the interpolation space that corresponds to . Repeating the proof of McLean, W. (2000),line by line, we find that, for there holds with and We can thus use the interpolation result Theorem in the usual way for the -dependent norms, without any additional factors of appearing in the interpolation estimates.

In this section, we deal with the mapping properties of the boundary layer potentials given in Definition. and of the corresponding boundary integral operators for the Helmholtz problem. The bounds are explicit with respect to the wave number cj, which is going to prove to be important in the next section, where we consider the mapping properties of the time domain

We first collect estimates with respect to the natural (or energy) norms that have been proven in the literature. By natural norms, we mean the spaces and equipped with either the classical norms or with the -dependent norms introduced. We then use Costabels technique to generalise these results to a wider range of Sobolev spaces. The mapping properties themselves are well known; the new contribution is the explicitness in

We state the mapping properties in the classical Sobolev norms and in the equivalent dependent norms. The differences we observe are merely a result of the differences in part b) and c) of the following lemma. In the estimates, is the argument of the Laplace transform. The differences in the estimates with respect to this variable are crucial, as their powers correspond to the orders of the time derivatives in the space-time estimates via the Parseval-Plancherel identity.

Let be a Lipschitz domain. Then the following results hold. a) Trace Theorem (i) For with (ii) For with b) Trace Extension Lemma (i) For them exists an extension into for which. with. (i) For there exists an extension into for which with. c) Bound for the normal derivative (i) Let u solve the homogeneous Helmholtz equation (4.5a). Then, with. A similar result is given in Melenk, J. M. (2010), but there the factor on the right hand side is instead of (i) Let u solve the homogeneous Helmholtz equation (5a). Then, with Remark : a) These estimates am optimal with mspect to b) Note that estimates b) (ii) and c) (ii) yield estimates b) (i) and c) (i) in Lemma 4.8, mspec- tively, via Remark. The following two lemmas are consequences of Lemma 8. Lemma 9 (Mapping Properties, Classical Norms, Helmholtz Problem) Let be a Lipschitz domain. Let and . Then

(31) (32) (33) (34) (35) (36)

with The Single Layer operator is further bounded independently of when it is considered as an operator mapping from to in three space dimensions: For there holds

(37)

with In two space dimensions, one can replace the bound in (37) by with again.

two space dimensions, with Lemma 10 (Mapping Properties, -Dependent Norms, Helmholtz Problem) Let be a Lipschitz domain. Let and Then

(38) (39) (40) (41) (42) (43)

with We define bilinear forms and by

(44)

for and

(45)

for Then there hold the coercivity estimates stated in Lemma 11. Lemma 11 (Coercivity Estimates, Helmholtz Problem Bilinear Forms) a) Let Then, with (i) (ii) b) Let Then, with (i) (ii) The continuity estimates stated in Corollary are an immediate consequence of Lemmas 9 and 10. Bilinear Forms) a) Let Then, with (i) (ii) b) Let Then, with (i) (ii) We note that we gain one power of in the coercivity estimates for but none for the continuity estimate when -dependent Sobolev spaces are used instead of classical Sobolev spaces. Regarding it is the other way round. In both cases, the presence of powers of in the continuity estimates and their absence in the coercivity estimates means that we have coercivity and continuity on two different spaces in the space-time framework. In the frequency domain though,is just another constant, and one can apply the Lax-Milgram Theorem as usual in this context. The inverse operator to Ls denoted by in A. Bamberger and T. Ha-Duong (1986). As mentioned in the proof of, there hold and and hence we simply write instead of here, in particular to avoid confusion with the Newton potential, which is defined below. Some properties of are collected in Lemma 12.

Let Then

(46) (47) (48) (49)

with

Proof (of equations (47) and (49) in Lemma 12) (49) follows just as in the proof, by using Lemma 8 a) (ii) instead of (i). To prove (47), we modify the proof, which is Lemma 8 c) (i), we take (ii). The term can be estimated. Hence we obtain and thus (47). With regard to the inhomogeneous Helmholtz equation the Newton potential (or volume potential) is given

(50)

for Correspondingly, the Newton potential for the wave equation is

(51)

for To simplify the notation, and in contrast to (P) and (HH), denotes a bounded domain here. In the context of (P) and (HH),would be the scatterer, which is denoted bythere. Melenk and Sauter show that for with respectively As an immediate consequence we obtain a bound on

(52)

Up to now we have only presented results on the boundedness of the Helmholtz boundary layer potentials and boundary integral operators with respect to their respective natural energy spaces and However, generalised mapping properties are of interest as well, in particular in the context of a posteriori error estimation. The groundbreaking work on the boundary integral operators for a class of elliptic problems that includes the Laplace, Helmholtz and Lame problem was done. He subsequently extended his analysis to the boundary integral operators for the heat equation. It is well known that the Helmholtz boundary integral operators have the same mapping properties as their numberwhen the spaces are not the respective natural energy spaces. Here we mimic Costabel‘s arguments in order to obtain bounds which are explicit in We begin with a generalisation of Lemma 13.

This proposition fundamentally makes two contributions to the field of time domain Boundary Element Methods. On the hypothetical side, it gives summed up mapping results to the administrators included and acquaints a posteriori mistake gauges with this field. On the implementational side, it gives a full integration scheme that can be utilized with non-uniform networks and presents an adaptable self-adaptive algorithm that permits refinements in both the spatial and worldly direction.

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Sayas, F.-J (2011). Retarded potentials and time domain boundary integral equations: a road-map. Lecture Notes, Workshop on Theoretical and Numerical Aspects of Inverse Problems and Scattering Theory, Available for download at http://www.math.udel.edu/~fjsayas/TDBIE.pdf.

Research Scholar of OPJS University, Churu, Rajasthan