are valid provided |r| ≤1. The recently derived expression (see Appendix C) Z π
0
log(r−cos(σ))einσdσ = (−i)
−1−ω1n
n log|1−ω1|+ (−1)n−ωn1
n log|1 +ω1| +1
n
n−1
X
j=0
ω1j+ω2j(1−(−1)n−j)
n−j − 1−ω2n
n log|1−ω2| +(−1)n−ωn2
n log|1 +ω2|−iπωn2 n − 1
n2[1−(−1)n] + log(2)1−(−1)n n
,
(2.108)
where ω1 and ω2 are the roots of the polynomial
2ωr−ω2−1 = −(ω−ω1) (ω−ω2), (2.109) holds for all real values of r; the real and imaginary parts of this expression provide the necessary log-cosine and log-sine integrals.
In view of the high-order convergence of the FC method (cf. Section 4.5 and Appendix A), a high-order accurate algorithm for evaluation of Iq1,q2[ϕ] (and thus Ieq1,q2[ϕ]) on the solee basis of a uniform σ mesh results through application of equations (2.107) and (2.108) in conjunction with equations (2.102), (2.104), and (2.106).
Remark 2.5.3. In the following chapter we propose an algorithm that is applicable in the case Γ is a non-smooth but piecewise smooth curve Γ. While the methods of that chapter can also be used for smooth curvesΓ, the FC-based methods introduced in the present section are generally significantly more efficient for a given prescribed error and more accurate for a given discretization size. The improvements that result from use of the FC-based approach are demonstrated in Section 2.6 and a comparison with more general Lipschitz-boundary algorithm is carried out in Section 4.5.5 in the context of Zaremba eigenvalue problems.
exterior domains Ω and for which the right hand sides f and g are given by f =eikα·x =eik(cos(α)x1+sin(α)x2))
g =nx·eikα·x
(2.110)
(where α is the angle of incidence). Note, however, that as mentioned in Theorem 2.2.3 the integral equation system (2.4) is not uniquely solvable for a discrete set of values that correspond to Dirichlet eigenvalues of the complement R2 \Ω; the numerical approach we use to eliminate this difficuty is discussed in Appendix B.
In our first experiment we apply our scattering solver to the kite-shaped scatterer that is presented in Figure 2.6, whose smooth boundary is given by the parametrization
x1 = cos(t) + 0.65 cos(2t)−0.65 and x2 = 1.5 sin(t), (2.111) and we assume Neumann and Dirichlet boundary conditions in the interval t ∈ [π/2; 3π/2]
and its complement, respectively. In this figure and throughout this thesis Dirichlet and Neumann boundary segments ΓD and ΓN are color-coded in red and blue, respectively. Fig- ure 2.8 demonstrates the high-order convergence results for the value of the scattered field u(x0) at the particular pointx0 = (1,2), which lies in the exterior of the domain. Figures 2.6 and 2.7 depict the scattering pattern for the incident wave coming at an angle α=π/8 with wavenumberk = 40. Figure 2.5 displays the unknown currentψ(see equation (2.3)) obtained in the course of the present experiment. Note the indication of thed−1/2 behavior of the inte- gral density near the Dirichlet-Neumann junction. The character of the Dirichlet-Neumann singularity demonstrated in this image is consistent with the results of Theorem 2.3.3, but it suggests that densities on the Neumann segments may in fact be smoother than implied by that theorem.
Figure 2.5: Integral density along the entire boundary (top) and zoomed near the Dirichlet- Neumann junction (bottom). The character of the singularity at the Dirichlet-Neumann junction is consistent with the results of Theorem 2.3.3, but it suggests that densities on the Neumann segments may in fact be smoother than implied by that theorem.
Figure 2.6: Scattering from a kite-shaped domain under Zaremba boundary conditions.
Scattered field. In this figure and throughout this thesis Dirichlet and Neumann boundary segments ΓD and ΓN are color-coded in red and blue, respectively.
Figure 2.7: Scattering from a kite-shaped domain under Zaremba boundary conditions. Total field.
Figure 2.8: Convergence of the value u(x0) for a kite shaped domain with k = 10.
The following experiment concerns the unit disc (where Dirichlet and Neumann boundary conditions are prescribed on the left and right halves of the disc boundary). Figures 2.9 and 2.10 demonstrate the diffraction pattern for a total field solved in the domain exterior to the disc (incident wave angle α = π/8 and wavenumber k = 50.). Note the expected asymmetry in the scattered field, as well as the wave patterns near both transition points.
Figure 2.9: Scattering from a disc under Zaremba boundary conditions. Scattered field.
Figure 2.10: Scattering from a disc under Zaremba boundary conditions. Total field.
To conclude this section we present a brief comparison of the proposed solvers with one of the most efficient Zaremba solvers previously available [64]. The method introduced in reference [64] is based on iterative inverse preconditioning that solves Zaremba problems for
the Laplace and elasticity equation. This method, which applies to a variety of singular problems, is described in detail with examples in [65], and it has been implemented in a numerical MATLAB package which is freely available (http://www.maths.lth.se/na/
staff/helsing/Tutor/index.html). Zaremba boundary conditions are not implemented in the package, but even for the simpler Dirichlet problem the execution time required by this algorithm is not as favorable as those required by the solvers proposed in this thesis: a computing time of 0.46 seconds is required for the Dirichlet problem for Helmholtz equation, while with the FC-based solver presented in this thesis executes in 0.06 seconds for the significantly more challenging Zaremba problem for Helmholtz equation on the same domain (unit disc), with the same incident wave frequencyk = 2 and for the same relative error 10−13 in the solution. (All the numerical results presented in this thesis were obtained on a single core of a 2.4 GHz Intel E5-2665 processor.) Such time differences, a factor of eight in this case, can be very significant in practice, in contexts where thousands or even tens of thousands of solutions are necessary, as is the case in inverse problems as well as in our own solution of high- frequency eigenvalue problems, etc. The main reason for the difference in execution times is that even though the iterative solver requires a limited number of iterations, iteration- dependent matrix entries occur (in view of corresponding iteration dependent discretization points), which require large number of evaluations of expensive Hankel functions at each iteration, and, thus, a significantly increased computing cost.
Additional results demonstrating the high-order convergence of the FC-solver when it is applied to solve challenging eigenvalue problems are presented in Section 4.5.
Chapter 3
Integral equation solvers for the
Zaremba boundary value problem on Lipschitz domains
3.1 Preliminaries
We consider interior and exterior boundary value problems of the form
∆u(x) +k2u(x) = 0 x∈Ω, u(x) =f(x) x∈ΓD,
∂u(x)
∂nx =g(x) x∈ΓN
(3.1)
for u ∈ Hloc1 (Ω) (with a Sommerfeld radiation condition in case of exterior problems).
Throughout this chapter Ω⊂R2 denotes a bounded simply-connected domain with a Lips- chitz boundary Γ =∂Ω and the Dirichlet and Neumann boundary portions ΓD and ΓN are disjoint subsets of Γ.
Let the piecewise-smooth boundary Γ be expressed in the form
Γ =
QN+QD
[
q=1
Γq, (3.2)
whereQD andQN denote the numbers of smooth Dirichlet and Neumann boundary portions,
≤ ≤ ≤ ≤
(resp. Neumann) segment of the boundary curve Γ. Clearly, letting JD ={1, . . . , QD} and JN ={QD + 1, . . . , QD+QN} we have that
ΓD = [
q∈JD
Γq and ΓN = [
q∈JN
Γq
are the (piecewise smooth) portions of Γ upon which Dirichlet and Neumann boundary con- ditions are enforced, respectively. Note that in view of the assumption above both Dirichlet- Neumann junctions and non-smooth points in Γ necessarily occur at a common endpoint of two segments Γq1, Γq2 (1 ≤ q1, q2 ≤ QD +QN). Note, additionally, that consecutive values of the index q do not necessarily correspond to consecutive boundary segments (see, e.g., Figure 3.1).
Figure 3.1: Boundary decomposition illustration. Dashed line: Neumann boundary. Solid line: Dirichlet boundary.
Remark 3.1.1. Throughout this chapter the decomposition of the curve Γ is taken in such a way that no Dirichlet-Dirichlet or Neumann-Neumann junctions occur at a point at which the curve Γ is smooth. In other words, every endpoint of Γq is either a Dirichlet-Neumann junction or a non-smooth point of Γ. Clearly this is not a restriction: two Dirichlet (resp.
Neumann) segments Γq1 and Γq2 that meet at a point at which Γ is smooth can be combined into a single Dirichlet (resp. Neumann) segment.
We employ the integral equation approach described in detail in Section 2.4 and write here the integral equation system (2.4) for the problem 3.1:
Z
Γ
Gk(x, y)ψ(y)dsy =f(x) x∈ΓD, γψ(x)
2 +
Z
Γ
∂Gk(x, y)
∂nx ψ(y)dsy =g(x) x∈ΓN.
(3.3)