panel-bie

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

PyPI version Python versions License CI Documentation codecov

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

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

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])

The package includes a CLI for quick experiments and diagnostics.

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

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

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.

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.

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

Double-layer potential solver for interior Dirichlet problems.

Single-layer potential solver for Neumann problems.

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.

• 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.

MIT License. See LICENSE for details.