Boundary element method

Definition

Let $\Omega \subset \R^n$ be a bounded domain with piecewise smooth boundary $\Gamma$ and outward unit normal $\nu$. Let $L$ be a linear elliptic differential operator with constant coefficients and a known fundamental solution $G(x, y)$. Consider $Lu = 0$ in $\Omega$ with boundary conditions on $\Gamma$. Write $q = \partial u / \partial \nu$ (or the conormal derivative of $L$, such as the traction in elasticity). The boundary element method consists of the following steps.

1. Boundary integral equation: Green's representation formula, taken as $x \in \Omega$ tends to the boundary, gives for $x \in \Gamma$ $$c(x)\, u(x) + \int_\Gamma \frac{\partial G}{\partial \nu_y}(x, y)\, u(y)\, dS_y = \int_\Gamma G(x, y)\, q(y)\, dS_y,$$ where $c(x) = \tfrac12$ at smooth points of $\Gamma$. This holds for the Laplacian with $G$ normalised so that $-\Delta_y G(x, \cdot) = \delta_x$.
2. Boundary mesh: $\Gamma$ is divided into finitely many boundary elements, and $u$ and $q$ are approximated on each element by polynomials with unknown nodal values.
3. Discretisation: the integral equation is imposed at the nodes (collocation) or against test functions (Galerkin). This gives a linear system $H \mathbf{u} = G \mathbf{q}$ with dense matrices $H$ and $G$.
4. Solution: the known boundary data fill half of the unknowns, and the system is solved for the rest. Then $u$ inside $\Omega$ is computed from the representation formula.