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Singularity Analysis of Laplace Boundary Integral Equations on Three-Dimensional Domains with Corners
Wenmao Hua, Jun Lai, Baoling Xie, Lu Zhang
TL;DR
Elliptic boundary value problems on nonsmooth three-dimensional domains have singular behavior near edges and vertices, while general conical geometries lack explicit singular-exponent formulas. This paper analyzes canonical wedge and circular-cone models and develops a Nyström-based nonlinear eigenvalue solver for general cones. Numerical experiments confirm root-exponential convergence and illustrate interactions between edge and vertex singularities.
Problem
The paper addresses singular behavior of boundary densities near edges and vertices, including coupled edge and vertex effects on general conical geometries where explicit formulas are unavailable.
Source: §1 · pp. 1–2Method
The paper derives singular-exponent characterizations for wedge and circular-cone models, then formulates a boundary-integral operator pencil and Nyström-based nonlinear eigenvalue solver for general cones.
Source: §1 · p. 2, §6 · p. 22Results
Numerical experiments confirmed root-exponential convergence, with exponential convergence for integrated-sine grading, and illustrated interactions between edge and vertex singularities.
Source: §5.1 · p. 19, §6 · p. 22Takeaways & Limitations
Conical models capture leading local singular behavior near edges and vertices and provide canonical problems for studying their interaction.
Source: §6 · p. 22The integral in the circular-cone identity converges only for Re(µ) in (−1, 0), while the x→1 endpoint requires a limiting interpretation.
Source: §3.2.1 · p. 8
Abstract
from arXiv · showhide
Solutions to elliptic boundary value problems on three-dimensional nonsmooth domains exhibit singular behavior near edges and vertices. Using layer potential theory, we investigate boundary integral equations for the Laplace equation and characterize the leading singular exponents of the surface density on conical domains. For two canonical geometries, we derive explicit formulas for the two families of edge singular exponents on a dihedral wedge, and characterize the modewise vertex singular exponents on an axisymmetric circular cone through the zeros of associated Legendre functions and their derivatives. Building on these two prototypes and the dilation-invariance property, we formulate the associated operator pencil for general conical domains and develop a nonlinear eigenvalue solver. This solver employs the Nystr\"om method with graded meshes, logarithmically corrected trapezoidal rules, and Brent's method to compute these singular exponents accurately and efficiently. Numerical examples for trihedral cones, regular m-hedral cones, and droplet-type cones verify the effectiveness of our approach and illustrate the interactions between edge and vertex singularities.
1. Introduction
Elliptic problems on nonsmooth three-dimensional domains develop edge and vertex singularities, yet systematic treatment of their interaction on general piecewise smooth cones is lacking. The paper characterizes singular exponents for canonical geometries and develops a computational framework for general conical geometries.
Motivation
- Corners and edges produce physical effects and distinctive scattering signatures, while accurate local singularity analysis matters for forward solvers and inverse problems.Examples include field enhancement near sharp tips, stress concentration in fracture mechanics, and Meixner-type edge conditions in electromagnetic scattering.Source: §1 · pp. 1–2
Background
- Layer potentials recast elliptic problems as second-kind boundary integral equations, reducing dimensionality and supporting fast algorithms for three-dimensional problems.They also naturally satisfy far-field radiation conditions in unbounded domains.Source: §1 · p. 2
Related work
- For piecewise smooth domains, singular expansions and loss of compactness complicate boundary-integral analysis; existing computational work leaves systematic treatment of interacting edges and vertices on general cones lacking.Prior high-order methods have concentrated largely on two dimensions, isolated conical points, smooth surfaces, or axisymmetric configurations.Source: §1 · p. 2
Contribution
- The paper studies singular behavior of the double-layer density for the three-dimensional Laplace Dirichlet problem, deriving canonical-geometry characterizations and a Nyström-based solver for general conical geometries.The solver targets general piecewise smooth cones, where edge and vertex effects are coupled and explicit formulas are unavailable.Source: §1 · p. 2
Paper organization
- The paper proceeds from wedge and circular-cone analysis to a nonlinear eigenvalue solver and numerical examples on trihedral, regular m-hedral, and droplet-type cones.The canonical models yield explicit singular exponents and local coefficient expansions; later sections present numerical examples and conclude with future directions.Source: §1 · pp. 2–3
2. Mathematical preliminaries
The preliminaries model local nonsmooth geometry by tangent cones and use dilation-invariant cones to study leading singular behavior. They then formulate the Laplace Dirichlet problem as a boundary integral equation, whose density can become singular on piecewise smooth boundaries.
2.1. Dilation-invariant conical domains
- Tangent cones classify boundary points as smooth faces, dihedral-wedge edges, or vertices with spherical polygonal cross-sections, and capture their leading local singular behavior.The analysis uses wedge and axisymmetric circular-cone prototypes for edges and vertices, respectively.Source: §2.1 · p. 3
- Dilation invariance preserves layer-potential kernel homogeneity under radial scaling, enabling separated-variable forms for singular boundary densities.The general model allows piecewise smooth cones with curvilinear cross-sections, subject to geometric assumptions on their arcs and corners.Source: §2.1 · pp. 3–4
2.2. Boundary integral equations
- The double-layer potential and its boundary trace, together with jump relations, yield second-kind boundary integral equations for the prescribed Dirichlet data.The setup permits bounded interior or unbounded exterior domains; the exterior problem includes a far-field decay condition.Source: §2.2 · pp. 4–5
- Unlike on smooth surfaces, where the double-layer operator is compact and density inherits boundary-data regularity, piecewise smooth boundaries generally produce edge and vertex density singularities.For the exterior equation, solvability assumes the right-hand side is orthogonal to the adjoint kernel; Neumann data can be treated analogously.Source: §2.2 · p. 5
3. Two canonical models
This section analyzes double-layer density singularities on two tractable geometries: a dihedral wedge for edges and an axisymmetric circular cone for vertices. It derives singular exponents and local coefficient expansions.
3.1. Infinite dihedral wedge
- On a dihedral wedge, the edge density has two fractional-power families, and its truncated expansion can match the boundary integral equation with residual O(r^N+1).The expansion has unique coefficients under the theorem's assumptions and away from a finite exceptional set of opening angles.Source: §3.1 · pp. 6–7
3.2.2. Leading singular behavior
- On a circular cone, modewise singular exponents are determined by roots of associated Legendre functions and their derivatives, yielding two exponent branches.The roots determine the leading density behavior near the vertex.Source: §3.2.2 · pp. 9–10
- The cone's overall leading singularity is the minimum of the leading exponents across Fourier modes.The modewise and overall leading exponents are distinguished in the stated characterization.Source: §3.2.2 · p. 10
3.2.3. An explicit coefficient mapping
- For smooth boundary data, each Fourier mode's first N Taylor powers can be matched to the first N singular terms to construct a local coefficient map.The mth mode's powers begin at r^|m| and continue through r^|m|+N−1.Source: §3.2.3 · pp. 10–11
- Outside a finite exceptional set of cone angles, the truncated density has unique coefficients and satisfies the boundary integral equation with residual O(r^N+1) near the vertex.The result applies to the first N noninteger exponents for each mode under the theorem's smooth-data assumption.Source: §3.2.3 · pp. 12–13
4. Singularities of general piecewise smooth cones
The section models general piecewise smooth cones as conical patches and formulates their vertex singular exponents as a nonlinear eigenvalue problem. It analyzes the kernel and develops a Nyström solver that addresses corner behavior and logarithmic self-interaction.
4.1. General piecewise smooth cones
- For general piecewise smooth cones, singular exponents are values for which the one-dimensional integral equation admits a nontrivial solution.Source: §4 · p. 13, §4.1 · p. 14
- Dilation invariance reduces the vertex singularity analysis to a one-dimensional integral equation for the leading term of order r^μ.Source: §4.1 · pp. 13–14
4.2. Analysis of the kernel function
- On each smooth patch, the kernel has a logarithmic diagonal singularity with smooth coefficients and a continuous diagonal limit.Source: §4.2 · pp. 14–15
4.3. An eigenvalue solver for piecewise smooth cones
- The Nyström solver combines corner grading, logarithmically corrected trapezoidal rules, and Brent’s method; constant-density subtraction improves quadrature accuracy and convergence.Source: §4.3 · pp. 15, 17
5. Numerical experiments
Numerical experiments test the solver on trihedral, regular m-hedral, and smooth droplet-type cones, examining convergence and the edge–vertex singularity behavior.
5.1. Example 1: Interactions between edge and vertex singularities
- For the right-angled trihedral cone, the computed leading exponent is approximately 0.45417373430136, with root-exponential self-convergence to round-off and endpoint behavior matching the edge exponent 2/3.The reported benchmark is accurate to about eight digits.Source: §5.1 · p. 19
5.2. Example 2: Asymptotics of m-hedral cones toward circular cones
- For regular m-hedral cones, the leading exponent increases toward the circular-cone value from below, with relative error fitted by εrel(m) ≈ 1.38 m^-1.935.At fixed m, the self-convergence error decays root-exponentially.Source: §5.2 · p. 20
5.3. Example 3: Non-polyhedral droplet-type cones
- For basic and wavy droplet cones, leading exponents converge rapidly, while endpoint plots match the inherited edge slope 3/4.Convergence is root-exponential for the basic droplet and exponential for the wavy droplet; the computations use algebraic-power and integrated-sine grading, respectively.Source: §5.3 · pp. 20–21
6. Conclusion
The paper presents a Nyström framework for computing singular exponents on three-dimensional nonsmooth domains and reports numerical tests across conical geometries. It positions tangent cones as models for local singular behavior and identifies extensions to broader domains and problems as future work.
6. Conclusion
- The framework derives two dihedral edge-exponent families, characterizes modewise exponents for axisymmetric circular cones, and formulates a nonlinear eigenvalue solver for general dilation-invariant cones.The general-cone exponents are computed through a boundary integral equation on the spherical cross-section.Source: §6 · p. 22
- Experiments report root-exponential self-convergence, exponential convergence with integrated-sine grading, the predicted trihedral edge exponent 2/3, and O(m^-2) convergence toward the circular-cone limit.The framework is also tested on non-polyhedral droplet cones.Source: §6 · p. 22
- The authors expect tangent cones to govern leading singular behavior on general cornered domains and identify high-order boundary integral solvers and wave scattering as future directions.The stated framework provides a foundation for future analysis of broader piecewise smooth domains.Source: §6 · p. 22
Appendix A. Properties of Legendre functions
Appendix A collects Legendre-function formulations used in the main text, including conventions for associated functions and standard identities.
Appendix A. Properties of Legendre functions
- The appendix specifies the associated Legendre function convention for nonnegative integer orders and defines negative integer orders by a connection formula.It also states the associated Legendre differential equation.Source: Appendix A · p. 23
- It records a hypergeometric series representation for Pμ and a logarithmic expansion involving the digamma function.The expansion is stated for μ outside the integers.Source: Appendix A · p. 23
- The collected standard results include the addition theorem, generating function, and Wronskian of Legendre functions.The appendix points readers to for further details.Source: Appendix A · pp. 23–24
Appendix B. Proof of the dihedral coefficient expansion
Appendix B establishes a dihedral-wedge coefficient expansion under stated boundary-data assumptions and relates its modes to explicit wedge exponents.
Appendix B. Proof of the dihedral coefficient expansion
- The symmetric and antisymmetric wedge modes have explicit exponent families, and the exterior problem preserves the exponent set while interchanging their roles.The listed families are given in terms of the opening angle α and integer k.Source: Appendix B · p. 24
- The restatement follows Serkh and Rokhlin; finitely many excluded opening angles are treated in by allowing terms of the form r^μ ln r.The cited extension applies across α ∈ (0, 2π).Source: Appendix B · p. 24
- For all but finitely many opening angles, convergent Taylor series for the even and odd boundary-data parts yield unique coefficients through each positive integer order n.The resulting boundary integral equation has residual order O(r^(n+1)).Source: Appendix B · pp. 24–25
Appendix C. Subtraction technique
The appendix describes a subtraction technique for modifying the jump relation near conical corners during Nyström discretization. It also analyzes the principal-value double-layer operator for a constant density on a truncated cone, separating side and spherical-cap contributions.
Appendix C. Subtraction technique
- The subtraction technique modifies the jump relation near conical corners during Nyström discretization.The passage does not specify the subtraction formula.Source: Appendix C · p. 25
- The setup takes a target on the conical side and examines the angular kernel as its parameter tends to zero.The conical side is parameterized using the boundary curve of the spherical cross-section.Source: Appendix C · p. 25
- For a constant double-layer density on a truncated cone, the appendix evaluates the principal-value operator by considering the conical side and spherical cap separately.The side contribution is analyzed first, followed by the cap contribution.Source: Appendix C · pp. 25–26