Efficient simulation of mixed boundary value problems and conformal mappings

TL;DR

The paper addresses numerical conformal mapping and conformal moduli for mixed Dirichlet–Neumann boundary value problems in polygonal and circular-arc domains by introducing the Reflected Walk-on-Spheres (RWoS) algorithm. This mesh-free, stochastic method extends the classical Walk-on-Spheres approach with reflections at Neumann boundaries using anti-conformal mappings, enabling simulation of harmonic functions and the construction of conformal maps to the canonical rectangle $R_h$. It combines the conjugate function method with probabilistic representations based on Brownian motion to compute moduli $\mathcal{M}(Q)$ and the mapping $f = u + i h\tilde{u}$, providing 2D and 3D demonstrations and validating results against known references. The approach offers a parallelizable alternative to meshed methods, handles complex boundary geometries like circular-arc quadrilaterals, and extends naturally to 3D heat-flow problems on insulated polyhedra, with open-source code for reproducibility.

Abstract

In this paper, we present a stochastic method for the simulation of Laplace's equation with a mixed boundary condition in planar domains that are polygonal or bounded by circular arcs. We call this method the Reflected Walk-on-Spheres algorithm. The method combines a traditional Walk-on-Spheres algorithm with use of reflections at the Neumann boundaries. We apply our algorithm to simulate numerical conformal mappings from certain quadrilaterals to the corresponding canonical domains, and to compute their conformal moduli. Finally, we give examples of the method on three dimensional polyhedral domains, and use it to simulate the heat flow on an L-shaped insulated polyhedron.

Figures (13)

• Figure 1: Boundary value problem \ref{['dir-neu']}. The Dirichlet boundary conditions are marked as thick lines. The Neumann boundary conditions are thin lines.
• Figure 2: An illustration of the Walk-on-spheres algorithm with an $\varepsilon$-shell.
• Figure 3: An illustration of the Reflected Walk-on-spheres algorithm. The original domain $\Omega$ is the region enclosed by a rectangle and a half circle. The left side of the rectangle $N_1$ and the half circle are the Neumann boundaries. $g_1, g_2$ are the anti-conformal mappings that reflects $\Omega$.
• Figure 4: The rectangle with vertices $(0, 0)$, $(1, 0)$, $(1, h)$, $(0, h)$.
• Figure 5: The canonical conformal mesh for rectangle.

Theorems & Definitions (12)

• Theorem 2.1: Riemann Mapping Theorem
• Definition 2.1: Conformal modulus of a quadrilateral
• Lemma 2.2: Reciprocal identity
• Definition 2.2: Dirichlet--Neumann problem in general
• Lemma 3.1
• Definition 4.1
• Definition 4.2: Harmonic measure
• Remark 4.1
• Definition 4.3: Anti-conformal Mapping
• Proposition 4.1