Boundary Element Methods

Boundary element methods solve boundary value problems by representing the solution with layer potentials on the boundary.

Typical model problems:

• Laplace,

• Helmholtz,

• Maxwell,

• elasticity / Lamé.

Main tradeoff:

Pro	Contra
unknowns live only on the boundary	integral kernels are singular
exterior domains are handled naturally by the fundamental solution	discrete operators are dense
solution can be evaluated anywhere from the layer potential	volume source terms need additional potentials

Solution Representation

Model Helmholtz problem:

\[ \Delta u + \kappa^2 u = 0 \quad \text{in } \Omega, \qquad u = g \quad \text{on } \Gamma, \]

BEM represents the solution by a layer potential with an unknown boundary density. For the single layer ansatz in the domain,

\[ u(x) = (\widetilde V\rho)(x) = \int_\Gamma G_\kappa(x,y)\,\rho(y)\,d\sigma_y, \qquad x \notin \Gamma, \qquad G_\kappa(x,y) = \frac{e^{i\kappa |x-y|}}{4 \pi |x-y|} \]

Taking the boundary trace gives the boundary integral equation for the density

\[ V\rho = g \quad \text{on } \Gamma. \]

Boundary Discretization

For the single layer ansatz, the boundary integral equation is

\[ V\rho = g \quad \text{on } \Gamma, \qquad (\widetilde V\rho)(x) = \int_\Gamma G_\kappa(x,y)\,\rho(y)\,d\sigma_y. \]

Choose a finite dimensional boundary space

\[ X_h = \operatorname{span}\{\varphi_1,\dots,\varphi_N\}. \]

The Galerkin variational formulation is: find \(\rho_h \in X_h\) such that

\[ \langle V\rho_h, \eta_h \rangle_\Gamma = \langle g, \eta_h \rangle_\Gamma \qquad \forall\,\eta_h \in X_h, \]

with system matrix

\[ A_{ij} = \langle V\varphi_j, \eta_i \rangle_\Gamma = \int_\Gamma \int_\Gamma \eta_i(x)\,G_\kappa(x,y)\,\varphi_j(y) \,d\sigma_y\,d\sigma_x. \]

A matrix vector product contains a potential evaluation: density values on source quadrature points are passed through the kernel \(G_\kappa(x,y)\) and accumulated at target quadrature points.

This is the computational bottleneck: far away boundary elements still interact through the Green’s function, so the discrete operator is dense.

FMM keeps near interactions direct and replaces far interactions by compressed Green’s function expansions.