The Weak Form Implementation of the Boundary Integral Equations
The boundary equation system
on ∂Ω
on ∂Ω
in the unknowns u and ψ is known as the Calderón identities.
In order to get a systematic treatment of both the solution and the flux on the boundary, these equations are implemented using a weak formulation. This also allows for simultaneous handling of Dirichlet and Neumann boundary conditions.
To derive a weak formulation for this equation system, the boundary part of Green’s second identity (see above) can be used with test function υ:
Focusing on the boundary equation and using the notations υ = utest, ψ = n · ∇u, and ψtest = n · ∇υ gives
Using the Calderón identities gives
In the actual implementation the terms are rather collected in the right-hand side with the induced change of signs.
This formulation is one of several possible ways to formulate weak equations for BEM. It is known as Costabel’s symmetric coupling and has advantages with respect to enforcing solution continuity and flux balance on boundaries for coupling BEM with FEM.
Asymptotic Conditions
In the case of unbounded domains the representation of u in terms of single and double layer needs to be modified to take into account for possible contributions from the far-field boundary. For Laplace’s equation these contributions will give rise to a constant additive term
in Ω
In 3D the fundamental solution for the Laplace operator and its derivatives approaches 0 for the point approaching infinity and therefore the first two terms will vanish. Consequently uconst can be interpreted as an asymptotic value of the solution at infinity. In 2D the first two terms do not vanish when approaching infinity due to the logarithmic nature of fundamental solutions and therefore the value of uconst cannot be interpreted in the same way.
In both 2D and 3D one needs to specify this constant term either directly by specifying its value or indirectly by setting a constraint on the total flux through the boundary in order to uniquely determine the solution.