Boosting the Maxwell double layer potential using a right spin factor
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Abstract
We construct new spin singular integral equations for solving scattering problems for Maxwell’s equations, both against perfect conductors and in media with piecewise constant permittivity, permeability and conductivity, improving and extending earlier formulations by the author. These differ in a fundamental way from classical integral equations, which use double layer potential operators, and have the advantage of having a better condition number, in particular in Fredholm sense and on Lipschitz regular interfaces, and do not suffer from spurious resonances. The construction of the integral equations builds on the observation that the double layer potential factorises into a boundary value problem and an ansatz. We modify the ansatz, inspired by a nonselfadjoint local elliptic boundary condition for Dirac equations.
Keywords
Maxwell scattering Singular integral equation Clifford algebraMathematics Subject Classification
45E05 78M15 15A661 Introduction
For smooth domains, K is a weakly singular integral, which gives a compact operator on many function space and invertibility can be deduced by classical Fredholm theory. For Lipschitz domains, K is a singular integral operator (modulo the factor \(\nu \)), and its \(L_p\) boundedness, \(1<p<\infty \), follows from the seminal work by Coifman, McIntosh and Meyer [8]. For the rest of this paper, we will restrict attention to the most fundamental function space for singular integrals: \(L_2\). On a strongly Lipschitz domain, that is when \(\partial \Omega \) is locally a Lipschitz graph, Rellich identities replaces the Fredholm arguments to show that \(I\pm K\) is a Fredholm operator on \(L_2(\partial \Omega )\).
The factorization into (1.3) and (1.4) explains the relation between the boundary value problem and K. However, we need to reinterpret the Dirichlet problem as a boundary value problem for analytic functions: We regard harmonic functions as real parts of analytic function, neglecting some possible minor topological obstructions. Having switched in this way from the Laplace equation to the Cauchy–Riemann system, the Dirichlet problem now amounts to the Hilbert problem of finding the analytic function in \(\Omega \) which has a prescibed real part at the boundary. Thus, in terms of operators, solving the boundary value problem means inverting the map (1.4).
In Sect. 5 we generalise (1.9) beyond the case of a perfect conductor, and formulate an integral equation with a spin ansatz for solving the Maxwell scattering problem against a finite number of objects with different scalar och constant permittivity, permeability and conductivity.
It is surprising that it is still a somewhat open problem to find a numerically well behaved boundary integral equation for solving scattering problems for Maxwell’s equation. See Epstein and Greengard [11] and Epstein, Greengard and O’Neil [12] for recent new Debye formulations, and Colton and Kress [9, 10] for the classical formulations. The spin integral equations that we propose in the present paper are based on the McIntosh singular integral approach with Clifford algebra from [2, 5, 15]. However, the integral formulations there are not suitable for numerical computations, since they suffer from spurious resonances and the same problems as the classical double layer potential equation. The work in the present paper began in [16], where an equation in the spirit of (1.9) was formulated for solving the Maxwell scattering problem against a perfect conductor, using the formalism from [3]. Both (1.9) and the spin integral equation from [16] have the advantages of not suffering from spurious resonances and having an improved condition number, at least in the Fredholm sense for Lipschitz boundaries, compared to classical formulations. However, the equation in [16] is for eight unknown scalar functions, as compared to the four unknown scalar functions for the equation in this paper. In both cases, the main novelty lies is the use of an auxiliary spin boundary condition 1.7, to obtain a singular integral operator with improved condition number. To our knowledge, this local nonselfadjoint Dirac boundary condition has not been exploited in this way before, with numerical computations in mind.
2 Higher dimension algebra
In this section, we fix notation and survey the higher dimensional algebra which we need for Dirac equations. See [2, 16] for more details.
In particular we recall in Example 2 how Maxwell’s equations fit into this framework. We denote by \(\Omega ^+\) a bounded domain in \({{\mathbf {R}}}^n\) with a strongly Lipschitz boundary \(\Sigma =\partial \Omega ^+\). This means that locally around each point on the boundary, \(\Sigma \) coincides with the graph of a Lipschitz regular function, suitably rotated. The unbounded exterior domain we denote by \(\Omega ^= {{\mathbf {R}}}^n{\setminus }\overline{\Omega ^+}\). Sometimes we abbreviate \(\Omega =\Omega ^+\). The unit normal vector field on \(\Sigma \) pointing into \(\Omega ^\) we denote by \(\nu \). The \({{\mathbf {R}}}^n\) standard basis is denoted \(\{e_j\}_{j=1}^n\).
We use the hermitean inner product \((\cdot ,\cdot )\) on \(\wedge {{\mathbf {C}}}^n\) for which the above basis multivectors is an ONbasis. The function space on \(\Sigma \) where we consider our integral equations, is the space \(L_2(\Sigma )= L_2(\Sigma ;\wedge {{\mathbf {C}}}^n)\) of square integrable functions \(f:\Sigma \rightarrow \wedge {{\mathbf {C}}}^n\) with inner product \((f,g)= \int _\Sigma (f(x),g(x)) d\sigma (x)\), where \(d\sigma \) denotes standard surface measure.
Example
Example
In \({{\mathbf {R}}}^3\), a multivector w can be viewed a collection of two scalars \(\alpha \) and \(\beta \), and two vectors a and b, where \(w= \alpha +a+*b+*\beta \in \wedge {{\mathbf {C}}}^3\). Here \(*\) denotes the Hodge star defined by \(*1=e_1\mathbin {\scriptstyle {\wedge }}e_2\mathbin {\scriptstyle {\wedge }}e_3\), \(*e_1=e_2\mathbin {\scriptstyle {\wedge }}e_2\), \(*e_2=e_1\mathbin {\scriptstyle {\wedge }}e_3\) and \(*e_3=e_1\mathbin {\scriptstyle {\wedge }}e_2\), which identifies \(\wedge ^0{{\mathbf {C}}}^3\) and \(\wedge ^3{{\mathbf {C}}}^3\), and \(\wedge ^1{{\mathbf {C}}}^3\) and \(\wedge ^2{{\mathbf {C}}}^3\) respectively. Following the setup in [16], we write the full electromagnetic field as the multivector field \(F= E+*H\), and note that Maxwell’s equations implies the Dirac equation 2.6 for this F. However, F is not a general solution to 2.6, but satisfies the constraint that the \(\wedge ^0{{\mathbf {C}}}^3\) and \(\wedge ^3{{\mathbf {C}}}^3\) parts of F vanishes.
Example
3 Known well posedness results
In this section we survey the known invertibility results from [2, 5, 15] for the maps (2.8)–(2.15) on the space \(L_2(\Sigma ;\wedge {{\mathbf {C}}}^n)\) on bounded strongly Lipschitz surfaces in \({{\mathbf {R}}}^n\). We include the proofs since they serve as background later in Sect. 4.
Proof
Proof
Theorem 3.3
The maps (2.8)–(2.15) all are Fredholm maps with index zero for any \({\mathrm{Im}}\,k\ge 0\).
Proof
Proposition 3.4
The maps (2.8)–(2.15) all are isomorphisms when \({\mathrm{Im}}\,k> 0\). The maps (2.10), (2.11), (2.12) and (2.13) also are isomorphisms when \(k\in {{\mathbf {R}}}{\setminus }\{0\}\), provided that \(\Omega ^\) is a connected domain.
Proof
4 The spin ansatz and new integral equations
Proposition 4.1
 (i)
The four restricted projections \(A^+: B^+{{\mathcal {H}}}\rightarrow A^+{{\mathcal {H}}}\), \(A^+: B^{{\mathcal {H}}}\rightarrow A^+{{\mathcal {H}}}\), \(A^: B^+{{\mathcal {H}}}\rightarrow A^{{\mathcal {H}}}\) and \(A^: B^{{\mathcal {H}}}\rightarrow A^{{\mathcal {H}}}\) are isomorphisms.
 (ii)
The four compressed projections \(A^+B^+: A^+{{\mathcal {H}}}\rightarrow A^+{{\mathcal {H}}}\), \(A^B^: A^{{\mathcal {H}}}\rightarrow A^{{\mathcal {H}}}\), \(A^+B^: A^+{{\mathcal {H}}}\rightarrow A^+{{\mathcal {H}}}\) and \(A^B^+: A^{{\mathcal {H}}}\rightarrow A^{{\mathcal {H}}}\) are isomorphisms.
 (iii)
The spectrum of the rotation operator AB does not contain \(+1\) or \(1\).
 (iv)
The spectrum of the cosine operator C does not contain \(+1\) or \(1\).
Note that (iv) is symmetric under swapping A and B, and therefore so is (i), (ii) and (iii).
Proof
Example
Proposition 4.2
The maps (4.1)–(4.8) are all Fredholm operators with index zero when \({\mathrm{Im}}\,k\ge 0\), and isomorphisms when \(k=0\). Moreover, the norms of these inverses (Fredholm inverses) are bounded by 2, when \(k=0\) \(({\mathrm{Im}}\,k\ge 0)\). The maps (4.2), (4.3), (4.6) and (4.7) are isomorphisms when \({\mathrm{Im}}\,k\ge 0\).
Proof
The bounds of the (Fredholm) inverses follows from the skewadjointness of ES by the formulas in the proof of Proposition 4.1. The fact that the index is zero follows from the method of continuity for \(k\ne 0\), since \(k\mapsto E_k\) is continuous.
Theorem 4.3
 The interior boundary value problem to find a solution f to \({{\mathbf {D}}}f=ikf\) in \(\Omega ^+\) with prescribed tangential/normal part \(N^\pm f=g\) at \(\Sigma \) is well posed in the sense that \(N^\pm : E_k^+L_2\rightarrow N^\pm L_2\) is invertible, if and only if the singular integral equationis uniquely solvable for \(h\in T^+L_2\). In this case the solution to the boundary value problem is \(f=E_k^+ S^h\) at \(\Sigma \).$$\begin{aligned} T^+S^N^\pm E_k^+S^ h=T^+S^ g \end{aligned}$$
 The exterior boundary value problem to find a solution f to \({{\mathbf {D}}}f=ikf\) in \(\Omega ^\pm \) with prescribed tangential/normal part \(N^\pm f=g\) at \(\Sigma \) is well posed in the sense that \(N^\pm : E_k^ L_2\rightarrow N^\pm L_2\) is invertible, if and only if the singular integral equationis uniquely solvable for \(h\in T^+L_2\). In this case the solution to the boundary value problem is \(f=E_k^ S^+h\) at \(\Sigma \).$$\begin{aligned} T^+S^+N^\pm E_k^S^+ h=T^+S^+ g \end{aligned}$$
Proof
For the interior boundary value problems, the ansatz \(E_k^+: S^L_2 \rightarrow E_k^+L_2\) is an invertible map for any \({\mathrm{Im}}\,k\ge 0\) by Proposition 4.2. For the exterior boundary value problems, the ansatz \(E_k^: S^+L_2 \rightarrow E_k^L_2\) is an invertible map for any \({\mathrm{Im}}\,k\ge 0\) by Proposition 4.2. We have also seen that \(T^+S^\pm : N^\pm L_2\rightarrow T^+L_2\) and \(S^\pm : T^+L_2\rightarrow S^\pm L_2\) are invertible maps. These invertible maps enable us to fomulate the boundary value problems as singular integral equations on the subspace \(L_2(\Sigma ;\wedge ^\text {ev}{{\mathbf {C}}}^n)\) as stated. \(\square \)
Example
We saw in Example 2 how the Dirichlet problem for the Laplacian in \(\Omega ^+\subset {{\mathbf {R}}}^2\), or equivalently the Hilbert boundary value problem for analytic functions with prescribed real part on \(\Sigma \), can be formulated in terms of invertibility of \(N^+: E^+L_2\rightarrow N^+L_2\).
Example
5 Maxwell scattering in piecewise constant media
We formulated in Example 4 a spin integral equation for solving the Maxwell scattering problem against a perfect conductor, which is a singular integral equation for four scalar functions. In this section, we fomulate a similar spin integral equation for solving more general scattering problems, for timeharmonic Maxwell’s equations at frequency \(\omega \). We do not aim to present a complete solvability theory in this section, since it requires a solution of fundamental open problems. Instead we fomulate the algorithm and describe the future work that is needed.

solving Maxwell’s equations (5.1) with wave number \(k_j\) in \(\Omega _j\), \(j=0,1,\ldots ,N\),

with \(E^0\), \(E^0\) satisfying the Silver–Müller radiation condition at infinity, see [16, eq. (4)],
 and wheresolve Maxwell’s equations in distributional sense across \(\Sigma \).$$\begin{aligned} E^\text {inc}+E^0+\cdots + E^N\quad \text {and}\quad H^\text {inc}+H^0+\cdots + H^N \end{aligned}$$
To show injectivity of \(B_\Sigma \) one can generalise the methods in Proposition 3.4. We omit the details and refer to [4].
6 Problems with the classical ansatz
We end this paper with an explicit computation on a conical domain, elaborating on Mellin transform techniques of Fabes, Jodeit and Lewis [13], that shows that the classical double layer potential equation may have a condition number which is significantly worse than that of the underlying boundary value problem. This in contrast to the spin integral equations proposed in Theorem 4.3, which typically is no worse than the boundary value problem numerically. By localising the result below, we obtain similar results for bounded domains which have corners.
Notes
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