The boundary element method for solving rigid inclusion problems

Electrostatics (or steady-state temperature) probing fields in a domain containing a rigid inclusion, e.g., the domain outside an underground pipe, can be described by a Dirichlet problem for the Laplace equation in an annulus in which on the inner boundary the solution takes an unknown, but constant, value. For such a problem, the boundary element method (BEM) is well-suited. The extra unknown represented by the constant value of the electric potential on the inner boundary is determined by imposing that the integral of the current flux over the outer boundary of the annular domain vanishes. Numerical results are presented and discussed showing the convergence and accuracy of the BEM.