panel-bie

Panel-based boundary integral equation (BIE) solvers for 2D Laplace and Stokes problems.

Overview

This package provides Nyström discretization methods for solving boundary integral equations on smooth 2D curves. It includes:

• Double-layer potential for Laplace (Dirichlet problems)
• Single-layer potential for Laplace (Neumann problems)
• Combined-field formulations for robust exterior problems
• Curvature-corrected diagonal for improved conditioning
• Near-field evaluation with Helsing-Ojala quadrature corrections

Installation

pip install panel-bie

# With plotting support
pip install panel-bie[plot]

Quick Start

from panel_bie import DoubleLayerLaplace, solve_dirichlet
from superellipse import Superellipse

# Create geometry
curve = Superellipse(a=1, b=1, p=8)
disc = curve.panel_discretization(panels_per_quadrant=4, nodes_per_panel=16)

# Set up boundary data
g = lambda pt: pt[0]**2 - pt[1]**2  # harmonic function

# Solve
solver = DoubleLayerLaplace(disc)
mu = solver.solve(g)

# Evaluate at interior point
u_0 = solver.evaluate(mu, point=[0.0, 0.0])

Command-Line Interface

The package includes a CLI for quick experiments and diagnostics.

Commands

solve - Solve a Dirichlet problem on a superellipse geometry:

# Create a geometry file
echo '{"a": 1.0, "b": 1.0, "p": 4.0}' > curve.json

# Solve with harmonic boundary data u = x² - y²
panel-bie solve -g curve.json --bc "x**2 - y**2" --eval 0,0

# With custom discretization
panel-bie solve -g curve.json --bc "x" --eval 0.5,0 --panels 8 --nodes 16

condition - Compute the condition number of the discretized operator:

# Default squircle (p=4)
panel-bie condition

# Higher-order Lamé curve
panel-bie condition --p 8 --panels 8

# Custom geometry
panel-bie condition --a 2.0 --b 1.0 --p 4 --panels 4 --nodes 16

convergence - Test convergence with increasing panel count:

# Default p=4 superellipse
panel-bie convergence

# Higher exponent
panel-bie convergence --p 8

CLI Options

Command	Option	Description
solve	--geometry, -g	JSON file with geometry (a, b, p, q)
solve	--bc	Boundary condition as Python expression
solve	--eval	Evaluation point as x,y
solve	--panels	Panels per quadrant (default: 4)
solve	--nodes	Nodes per panel (default: 16)
solve	--beta	Panel grading parameter (default: auto)
condition	--a, --b, --p	Superellipse parameters
condition	--panels, --nodes	Discretization parameters
condition	--beta	Panel grading parameter (default: auto)
convergence	--a, --b, --p	Superellipse parameters

The --beta parameter controls panel grading near corners and is automatically computed to ensure numerical stability (beta must be less than p/2).

Theory

Double-Layer Potential

For Laplace's equation, the double-layer potential representation:

$$u(x) = \int_\Gamma \mu(y) \frac{\partial G(x,y)}{\partial n_y} , ds_y$$

where $G(x,y) = \frac{1}{2\pi} \log|x-y|$ is the fundamental solution.

The density $\mu$ satisfies the second-kind integral equation:

$$\left(-\frac{1}{2}I + K\right)\mu = g$$

where $K$ is the double-layer operator.

Diagonal Correction

For smooth boundaries, the kernel $K(x,y)$ has a finite limit as $y \to x$:

$$K(x,x) = \frac{\kappa(x)}{4\pi}$$

where $\kappa(x)$ is the signed curvature. This package uses this analytic limit rather than numerical regularization.

Near-Field Evaluation

For target points close to the boundary, standard quadrature loses accuracy. This package implements the Helsing-Ojala method for accurate near-boundary evaluation.

API Reference

DoubleLayerLaplace(discretization)

Double-layer potential solver for interior Dirichlet problems.

Method	Description
assemble()	Build the discretized operator matrix
solve(g)	Solve for density given boundary data g
evaluate(mu, point)	Evaluate solution at interior point
evaluate_grid(mu, grid)	Evaluate on a grid of points

SingleLayerLaplace(discretization)

Single-layer potential solver for Neumann problems.

Geometry protocol

Any geometry object providing:

• points: (N, 2) boundary nodes
• normals: (N, 2) outward unit normals
• weights: (N,) quadrature weights
• curvature: (N,) signed curvature

Compatible with superellipse.PanelDiscretization.

References

• Kress, R. (2014). Linear Integral Equations, 3rd ed. Springer.
• Helsing, J. & Ojala, R. (2008). On the evaluation of layer potentials close to their sources. J. Comput. Phys. 227:2899–2921.

License

MIT License. See LICENSE for details.