Mathematical Theory

Boundary Integral Equations

The Dirichlet Problem

Given a bounded domain \(\Omega \subset \mathbb{R}^2\) with smooth boundary \(\partial\Omega\), the interior Dirichlet problem is:

\[\begin{split}\begin{cases} \Delta u = 0 & \text{in } \Omega \\ u = g & \text{on } \partial\Omega \end{cases}\end{split}\]

Double-Layer Representation

We represent the solution as a double-layer potential:

\[u(x) = \int_{\partial\Omega} \mu(y) \frac{\partial G}{\partial n_y}(x, y) \, ds(y)\]

where:

• \(G(x,y) = -\frac{1}{2\pi}\log|x-y|\) is the 2D Laplace Green’s function

• \(n_y\) is the outward unit normal at \(y\)

• \(\mu\) is the unknown density function

Jump Relations

As \(x\) approaches the boundary from inside:

\[\lim_{x \to x_0^-} u(x) = -\frac{1}{2}\mu(x_0) + \int_{\partial\Omega} \mu(y) \frac{\partial G}{\partial n_y}(x_0, y) \, ds(y)\]

This gives the integral equation:

\[-\frac{1}{2}\mu(x) + K\mu(x) = g(x)\]

where \(K\) is the double-layer operator.

Nyström Discretization

Panel Decomposition

The boundary is divided into panels, with Gauss-Legendre nodes on each panel. For a panel parameterized by \(t \in [-1, 1]\):

• Nodes: \(t_j\) (Gauss-Legendre points)

• Weights: \(w_j\) (Gauss-Legendre weights)

• Jacobian: \(J(t) = |r'(t)|\)

Kernel Evaluation

The double-layer kernel:

\[K(x, y) = \frac{1}{2\pi} \frac{(x - y) \cdot n_y}{|x - y|^2}\]

For \(x \neq y\), this is evaluated directly.

Diagonal Correction

When \(x = y\), the kernel is singular but integrable. The limiting value depends on curvature:

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

where \(\kappa\) is the signed curvature at \(x\).

Convergence

For smooth boundaries, the Nyström method with \(n\) Gauss-Legendre nodes per panel achieves:

• Spectral convergence as nodes per panel increases

• The error decays as \(O(h^{2n})\) where \(h\) is the panel size

For boundaries with corners, specialized quadrature (dyadic refinement, graded meshes) is needed.