In physics and engineering there are many important applications where it is essential to obtain information on material properties inside (large) solid objects, e.g., the detection of oil reservoirs, the investigation of the interior of rocks and soil to understand its stability properties, or the assessment of the ice volume in glaciers to name just a few of them. For this purpose, typically, a wave is sent into the solid. Then the scattered wave is recorded and used to solve the governing mathematical equations for the quantity of interest.
Our goal is to employ retarded potential integral operators to reformulate the scalar wave equation as a system of space-time boundary integral equations (RPIE); standard references on this topic include , , , . The Cauchy data of the wave (or boundary densities if an indirect formulation is employed) is determined as the solution of an equation which involves a system of retarded potential integral operators. To investigate well-posedness we employ the Laplace transform and prove continuity and coercivity with respect to the frequency variable. These techniques in the context of numerical analysis go back to the pioneering works , , , ; a monograph on this topic is  and some further developments can be found, e.g., in  and .
We emphasize that the derivation of coercive and continuous integral equations in the Laplace domain is the key for their discretization by convolution quadrature. However, here we will focus on the continuous formulation and prove its well-posedness.
The new mathematical aspect of our setting is the presence of an interface and general mixed boundary conditions of Dirichlet, Neumann, and impedance type. We do not impose restrictions on where the interface meets the domain boundary. In a first step, we use a single-trace ansatz (cf. ), i.e., we employ Cauchy data (with continuous values and continuous fluxes across the interface) and the full Calderón projector. We prove continuity and stability of the formulation which implies well-posedness in certain anisotropic space-time Sobolev spaces.
The paper is organized as follows.
In Section 2 we introduce the wave equation with transmission condition at an interface and mixed boundary conditions.
Section 3 is devoted to the derivation of the system of RPIEs; the retarded acoustic single and double layer potentials are defined and the corresponding boundary integral operators are introduced by applying the trace and normal trace operator to these potentials. We end up with a system of integral equations for the unknown Cauchy data. Note that we employ a single-trace ansatz which involves a single Cauchy data across the interface in accordance with the transmission conditions.
In Section 4 we propose to incorporate the impedance boundary condition by keeping both Cauchy data in the equation. The advantage of this approach is that only boundary integral operators are involved which are defined on closed surfaces. We will prove well-posedness of the system of integral equations by showing coercivity and continuity of this system of RPIEs. This allows us to determine the analyticity class of the Laplace-transformed system and implies existence and uniqueness.
2 Problem Statement
First, we describe the geometry of the computational domain and use Fig. 1 as an illustration. The domain is split into three parts: the disjoint open sets such that , corresponding to two objects composed of different materials, and their interface ; as a convention we set although this unbounded domain will not appear in the formulation of the integral equations but in the analysis. The results of this paper also hold if the problem to solve is the exterior problem, and the interface is in the exterior domain, and when splitting in any number of subdomains; in order to simplify the notations, we restrict ourselves to the case of the interior problem and two domains.
For we set (thus ) and we employ the convention that
is the unit normal vector field atpointing into the exterior of . The skeleton manifold is defined by .
We also introduce a partition of the boundary, corresponding to different types of boundary conditions (see again Fig. 1): we split ; then we impose transmission conditions at , Dirichlet boundary conditions at , Neumann boundary conditions at , and an impedance condition at ; we do not require to be connected.
The functions and are defined on :
where the material-dependent constant coefficients , are extended to positive functions , to the exterior domain , such that are continuous across the interface . The resulting transmission Initial-Boundary value problem is
where denotes the convolution in time. For the boundary data we assume111We postpone the introduction of the relevant Sobolev spaces to Section 3.2.
Here and in the following, we employ the shorthand for . In the third equation in (2.2) the normal derivative can be taken either with respect to or ; denotes the jump of a function across the interface . The temporal convolution operator is a Dirichlet-to-Neumann () operator or an approximation to it. The simplest approximation is given by impedance boundary conditions: , where is the Dirac distribution. At this point we are vague concerning the function spaces which are mapped by in a continuous way but postpone this to Section 4.1 (Assumption 4.1), where also a dissipative condition will be imposed on .
3 Space-time Boundary Integral Equations for the Wave Transmission Problem
In this section we will formulate single-trace boundary integral operators and associated Calderón projectors.
3.1 Background: Layer Potentials and Boundary Integral Operators
We recall retarded potentials on two-dimensional compact, orientable manifolds in . We start by fixing some notations:
We will write , for , i.e. the index corresponds to the domain while indicates the type of boundary conditions imposed.
Recall the definition of as in (2.1). Let . For , the Dirichlet (D) and Neumann (N) trace operators are denoted by and are given by
i.e. the index means that the limit is taken from the subdomain , and the unit normal vector points outside . We also need a notation for the case where the limit is taken from the unbounded complement (where the unit normal still points outside ):
Finally, we will use the same symbols for the continuous extensions of the trace operators to appropriate Sobolev spaces.
For vector-valued functions , sufficiently smooth in , we define the normal component trace by222For we define .
Let be a function on ; for , we assume that the traces applied to are well-defined. Then the jump and co-normal jump across are defined by
The averages are defined by
This allows us to introduce the following boundary integral operators. We recall the fundamental solution of the wave equation in (which is the inverse Laplace transform of the fundamental solution in the Laplace domain, found in e.g. [8, Section 3]):
Let the coefficient functions , be as in (2.1). For a function and we define the retarded acoustic single and double layer 333In [7, Eq. (10)] an explicit expression for the integrand of the double layer potential is given. potentials for all by
where denotes the co-normal derivative with respect to the -variable. These potentials give rise to the following boundary integral operators. For functions we set
on . For , it holds almost everywhere on
At this stage we are vague concerning the appropriate function spaces and mapping properties of trace operators; for the precise setting we refer to Section 3.2.
For , let
Throughout this paper, denotes a fixed positive constant. The
constants in the estimates in this paper will depend continuously on
denotes a fixed positive constant. The constants in the estimates in this paper will depend continuously onand . These constants, possibly, tend to infinity if one or more of the quantities , , , , tend to zero or infinity. We will suppress this dependence in our notation.
For the convolution quadrature, we apply the Laplace transform with respect to time and obtain operators in the frequency variable . Thus, we end up with the Laplace transformed potentials for and :
and corresponding boundary integral operators on given by
We recall the formal definition of the Laplace transform and its inverse by444We employ the convention that, if and appear in the same context, then is the Laplace transform of .
Note that the Laplace transform applied to the convolution potentials satisfies
and analogous relations hold for the boundary integral operators in the time and Laplace domain. It is also well known that the following jump relations hold (see [14, Section 1.3]):
3.2 Mapping Properties of the arising Boundary Integral Operators
To formulate the mapping properties for the arising integral operators we first introduce Sobolev spaces on manifolds with boundary – standard references are , . For an open subset we denote the -scalar product and norm by
and suppress the subscript if the domain is clear from the context.
Let be a bounded Lipschitz domain with boundary . The unit normal vector field on is chosen to point into the exterior of and exists almost everywhere. For , let denote the usual Sobolev spaces with norm and is the closure of with respect to the norm. Its dual space is denoted by . On the boundary , we define the Sobolev space , in the usual way (see, e.g., ). Note that the range of for which is defined may be limited, depending on the global smoothness of the surface ; for Lipschitz surfaces, can be chosen in the range ; for , the space is the dual of .
We also define, for , the Sobolev spaces
where denotes the extension of to by zero.
In the following we will recall mapping properties of the single and double layer potentials and their corresponding integral equations.
For , the proofs of the following propositions (Prop. 3.3 and the 3rd and 6th inequality in Prop. 3.2, (3.8)) go back to . We have used here the estimates for the boundary integral operators as in .
Let and recall (2.1). Then, for , the operators , , , , , satisfy the following mapping properties: for all and there is some constant independent of such that
We denote by the dual pairing between and (without complex conjugation) so that is the continuous extension of the
scalar product. We can thus also introduce the symmetric and skew-symmetric dual pairing: for
Let and recall (2.1). Then, for , the operator
satisfies the coercivity estimate
for all , for some and for all
Proof. Fix . A straightforward calculation shows that
This operator was analyzed in [3, Lem. 3.1]: it maps continuously into and satisfies the coercivity estimate
for all .
3.3 Representation Formula
Let be as in Section 2 and let for with outward unit normal field . Note that the trace operators can be extended to continuous operators acting on functions in the Sobolev space . We collect the range of these traces into the space of Cauchy traces, and the multi-trace space:
and equip this spaces with the graph norm:
The single trace space is a subspace of and defined by555The components of are denoted by , .
In the context of the wave equation, these (spatial) trace spaces will be considered as the spaces of values of time-depending functions (distributions). To define the relevant function space we first consider the Schwartz class
where denote the space of polynomials (with complex coefficients). can be equipped with a metric that makes this space complete. A tempered distribution with values in a Banach space is a continuous linear map . A causal distribution with values in is a tempered -valued distribution such that
and we write
following the notation in .
The space666 for “time domain”. consists of all (possibly distributional) derivatives of continuous causal -valued functions with, at most, polynomial growth.
The corresponding Cauchy trace operator are given by
It is known [5, Lem. 3.5] that the range of is dense in . Since the spaces and are dual to each other, we have that the Cauchy trace spaces are in self-duality with respect to the symmetric dual pairing .
We employ the direct method to transform the wave equation into a space-time boundary integral equation and start with the Kirchhoff representation formula. The key potential is given by
for and as in (3.3)
Then, every that satisfies and also satisfies the representation formula
We introduce the Calderón projector by
and recall the property: solves the homogeneous wave equation in and if and only if
This equation will be our starting point for the formulation of problem (2.2) as a system of integral equations. More precisely we transform this equation to the Laplace domain since the temporal discretization by convolution quadrature is defined on the “Laplace side”. The Laplace transform of (3.12) is given by
Finally, the Calderón operator is given by (with the identity operator ) and can be expressed by the classical four boundary integral operators as
It turns out, that this form of is not optimal from the viewpoint of stability. We employ a further transformation and first define the frequency dependent diagonal matrix
4 Space-time Boundary Integral Equations for the Wave Transmission Problem with Mixed Boundary Conditions
To deal with problem (2.2) we incorporate Dirichlet and Neumann boundary conditions into the space . For this we extend the Dirichlet and Neumann parts of the boundary to a closed boundary (cf. Fig. 1) which is the boundary of a bounded domain such that , for . Then we set
and define the space of Cauchy traces of global fields, whose Dirichelet and Neumann components vanish on and respectively777This space naturally arises when offsetting Cauchy traces with the boundary data.:
4.1 First-kind Boundary Integral Equations
Assume that solves (2.2). Then, the Laplace transform with , , satisfies
Note that the transmission conditions (third equation in (2.2)) are built already in the function space ; we take into account the boundary conditions on and next.
4.2 Treatment of the Neumann and Dirichlet boundary conditions
To obtain a variational formulation for the unknown Cauchy data of the transmission problem (2.2) with balanced test and trial spaces we consider an offset function