BoundaryElementMethoden

Boundary element refers to a class of techniques used to solve boundary value problems by reformulating domain equations as boundary integral equations. In practice, the boundary element method (BEM) is the most common realization. It relies on fundamental solutions, or Green's functions, of the governing differential operator to express the solution inside the domain in terms of quantities on the boundary.

The method works by representing the interior field through boundary integrals. The unknowns are typically boundary values such as displacements and tractions (for elasticity) or potential and flux (for heat or electrostatics). The boundary is discretized into elements or panels, and the integral equations are turned into a system of algebraic equations using collocation or Galerkin techniques. The resulting system is usually dense, in contrast to the sparse systems of many volume-discretization methods, and solving it yields the boundary solution from which interior fields can be recovered if needed.

Advantages include dimensionality reduction (reducing a 2D or 3D problem to its boundary), natural treatment of unbounded exterior domains, and high accuracy near boundaries. Limitations include the need for explicit fundamental solutions, difficulties with nonlinear or strongly inhomogeneous problems, and greater complexity in meshing curved geometries. Large 3D problems may require fast summation or compression methods to manage dense matrices.

Applications span acoustics, electromagnetics, elastostatics, fluid mechanics, heat conduction, and potential theory. The boundary element approach remains a specialized but widely used tool for problems where boundary effects dominate and the domain is homogeneous or linearly heterogeneous.