Boundary element method

The boundary element method reformulates a partial differential equation as an integral equation over the boundary using Green's functions. For a linear operator L, the solution u in the domain Ω can be represented as an integral of boundary values of u and its normal derivative against the fundamental solution (Green's function). This leads to a boundary integral equation that is discretized by dividing the boundary into elements, yielding a system of linear equations for the unknown boundary data.

The method is most effective for problems with constant coefficients and linear behavior, as the fundamental solution must be known analytically. For nonlinear or heterogeneous problems, BEM requires iterative schemes or coupling with domain methods.

The roots of BEM trace back to classical potential theory and the work of George Green and Carl Friedrich Gauss. In the 1960s, the method emerged as a practical computational tool through the work of Frank Rizzo and others, who applied it to elastostatics. The term 'boundary element method' was coined in the 1970s, and the method gained popularity in engineering for problems in fracture mechanics, acoustics, and electromagnetics.

Early implementations suffered from dense matrices, limiting problem sizes. The development of fast multipole methods in the 1980s and 1990s, pioneered by Leslie Greengard and Vladimir Rokhlin, reduced the computational cost from O(N²) to O(N), enabling large-scale simulations.

BEM excels in problems with unbounded domains, such as external acoustics, wave scattering, and geophysics, where domain methods require artificial truncation. It is also advantageous for problems with high aspect ratios, like crack propagation, where boundary meshing is simpler than volume meshing.

In engineering, BEM is used for stress analysis, heat transfer, and fluid flow. In biomedical engineering, it models electrical activity in the heart and brain. The method's boundary-only nature reduces mesh generation effort and is particularly suited for shape optimization, as the interior solution can be computed post-hoc.

BEM has several niche variants: the dual boundary element method handles degenerate boundaries; the symmetric Galerkin BEM produces symmetric matrices; and the fast multipole BEM accelerates computations. A lesser-known fact is that BEM can be derived from the method of fundamental solutions, which uses fictitious sources outside the domain.

In the 1970s, BEM was applied to the analysis of the human heart's electrical activity, contributing to electrocardiography. Also, BEM has been used in archaeology to model groundwater flow around ancient structures. The method's accuracy can be compromised at 'fictitious eigenfrequencies' in acoustics, a problem addressed by the Burton–Miller method.

Green's function

A fundamental solution that gives the response of a linear system to a point source.

Fundamental solution

The solution of the PDE with a Dirac delta forcing, used in BEM to construct integral equations.

Fast multipole method

An algorithm to accelerate the computation of long-range interactions, reducing complexity from O(N²) to O(N).