The Helmholtz Dirichlet and Neumann problems on piecewise smooth open curves

Abstract

A numerical scheme is presented for solving the Helmholtz equation with Dirichlet or Neumann boundary conditions on piecewise smooth open curves, where the curves may have corners and multiple junctions. Existing integral equation methods for smooth open curves rely on analyzing the exact singularities of the density at endpoints for associated integral operators, explicitly extracting these singularities from the densities in the formulation, and using global quadrature to discretize the boundary integral equation. Extending these methods to handle curves with corners and multiple junctions is challenging because the singularity analysis becomes much more complex, and constructing high-order quadrature for discretizing layer potentials with singular and hypersingular kernels and singular densities is nontrivial. The proposed scheme is built upon the following two observations. First, the single-layer potential operator and the normal derivative of the double-layer potential operator serve as effective preconditioners for each other locally. Second, the recursively compressed inverse preconditioning (RCIP) method can be extended to address "implicit" second-kind integral equations. The scheme is high-order, adaptive, and capable of handling corners and multiple junctions without prior knowledge of the density singularity. It is also compatible with fast algorithms, such as the fast multipole method. The performance of the scheme is illustrated with several numerical examples.

Figures (12)

• Figure 1: Convergence of $u(r_{\rm targ})$ with $n_{\rm sub}$ for the Helmholtz Dirichlet problem on the straight line segment (\ref{['eq:straight']}).
• Figure 2: Number of GMRES iterations and computation times as a function of wavenumber for the Helmholtz problem on a straight line segment. Top: Dirichlet condition. Bottom: Neumann condition. Left: Number of GMRES iterations as a function of $L/\lambda$. Right: timing results as a function of $N$ -- the total number of discretization points on $\Gamma$. Here, $\Gamma$ is divided into equi-sized panels in the parameter space and $16$ Gauss-Legendre nodes are used to discretize each panel. $N=8L/\lambda$ for this example. The relative $l_2$ error is measured on a $300\times 300$ tensor grid over the square $[-1.3, 1.3]\times[-1.5, 1.1]$. The largest relative $l_2$ errors are about $4\times 10^{-7}$ for the Dirichlet problem and $6\times 10^{-7}$ for the Neumann problem.
• Figure 3: Same as Figure \ref{['fig:line_iter']}, but for the spiral defined by \ref{['eq:spiral']}. $N=16L/\lambda$ for this example.
• Figure 4: The magnitude of the total field $|u^{\rm tot}|$ and the estimate abolute error for the spiral with $L/\lambda =200$, computed on a $3000\times 3000$ tensor grid over the rectangle $[-2.5, 1.75]\times [-2.75, 1.5]$. Top: Dirichlet condition. Bottom: Neumann condition. Left: $|u^{\rm tot}(r)|$. Right: $\log_{10}$ of the estimated absolute pointwise error in $u^{\rm tot}(r)$.
• Figure 5: Boundary with corners. Left: one-corner curve defined by \ref{['eq:corner']}. Right: eight-corner curve by tiling the curve on the left four times horizontally.

Theorems & Definitions (2)

• Remark 1
• Remark 2