Layer Potential Methods for Doubly-Periodic Harmonic Functions

TL;DR

This work addresses solving Laplace-type problems on doubly-periodic, finitely-connected tori by developing layer-potential representations built from a doubly-periodic Green's function expressed via the Jacobi theta function or a modified Weierstrass sigma function. The authors prove compactness of the single- and double-layer operators, characterize the nontrivial nullspace of the second-kind Fredholm operator when multiple holes are present, and formulate Dirichlet, Neumann, and Steklov problems in this framework, including a flux-enforcing modification for multi-hole domains. They implement Nyström discretization and demonstrate spectral convergence and high accuracy in numerical experiments across Dirichlet, Neumann, and Steklov problems on tori with irregular holes, showing no lattice-sum overhead and effective handling of complex geometries. The results advance robust, high-precision solutions for harmonic BVPs on doubly-periodic domains, with potential extensions to Helmholtz, Stokes, and higher-genus surfaces for broader physical applications.

Abstract

We develop and analyze layer potential methods to represent harmonic functions on finitely-connected tori (i.e., doubly-periodic harmonic functions). The layer potentials are expressed in terms of a doubly-periodic and non-harmonic Green's function that can be explicitly written in terms of the Jacobi theta function or a modified Weierstrass sigma function. Extending results for finitely-connected Euclidean domains, we prove that the single- and double-layer potential operators are compact linear operators and derive relevant limiting properties at the boundary. We show that when the boundary has more than one connected component, the Fredholm operator of the second kind associated with the double-layer potential operator has a non-trivial null space, which can be explicitly constructed. Finally, we apply our developed theory to obtain solutions to the Dirichlet and Neumann boundary value problems, as well as the Steklov eigenvalue problem. We implement the developed methods using Nyström discretization and find approximate solutions for these problems in several numerical examples. Our method avoids a lattice sum of the free space Green's function, is shown to be spectrally convergent, and has a faster convergence rate than the method of particular solutions for problems on tori with irregularly shaped holes.

Figures (10)

• Figure 1: (left) An illustration of a finitely-connected torus, $\Omega$ as in \ref{['e:Omega']}. (right) A plot of the level sets of the Green's function $z \mapsto G\left(z - \frac{1}{2} (1 + \tau) \right)$, given in \ref{['eqn: green']}, on the torus $\mathbb{T}_\tau$ with $\tau = \frac{1}{3} + \frac{2}{3}i$.
• Figure 3: Eigenfunctions $u_k$ of the Steklov EVP \ref{['eqn: steklov']} corresponding to $\sigma_k$, for $k=2$, $52$, $105$, and $500$, are computed on a square torus with $M=25$ holes. The corresponding eigenvalues are tabulated in Table \ref{['tabl: steklov many err']}. Each hole is discretized with $N=200$ boundary points, resulting in a total of $5000$ degrees of freedom. Using an aposteriori estimate, the eigenvalue approximations have a relative error less than $10^{-14}$. Eigenfunctions are normalized to have a range $[-1,1]$, and the periodicity of each eigenfunction can be observed at the "boundary of the square". As anticipated, eigenfunctions $u$ corresponding to large $\sigma$ concentrate near $\partial \Omega$ and oscillate with wavelength $\approx \frac{2\pi}{\sigma}$. See Section \ref{['sec: steklovbvp numerical']} Example 6 for details.
• Figure 4: For a square torus, $\mathbb T_{\tau}$ with $\tau=i$, with $M=2$ circular holes, we plot $\psi_1 \in N(\mathcal{D}^{*}-\tfrac{1}{2} I )$(left) and $\mathcal{S}[\psi_1](z)$, $z \in \mathbb T_{\tau}$(right). Note that $\mathcal{S}[\psi_1]$ is constant on each hole, as proven in Lemma \ref{['lem: properties double layer']}(5). For details, see Remark \ref{['r:NullSpace']}.
• Figure 5: Approximate Dirichlet BVP solutions on a square torus (left) and an equilateral torus (right) with one circular hole. See Section \ref{['sec: dirichlet num']} Example 1.
• Figure 6: (top) Approximate Dirichlet BVP solutions on a square torus (left) and equilateral torus (right) with $M=3$ circular holes. These plots can be compared to Figure \ref{['fig: mult hole flux']}, where the solutions to a similar problem on a domain with three trefoil-shaped holes are plotted. (bottom) Convergence plots illustrate that the numerical methods converge spectrally for both the three circular holes and the three trefoil-shaped holes on square tori (left) and equilateral tori (right). The convergence rates for the two shaped holes are similar, but slightly better for circles. See Section \ref{['sec: dirichlet num']} Example 2 for more details.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Remark 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2: Adjoint of Double-Layer Potential
• Lemma 2.1: Gauss' Lemma for finitely-connected tori
• Lemma 2.2: Properties of Double-Layer Potentials