Dual boundary integral equations at a corner using contour approach around singularity

A dual integral formulation for the Laplace equation problem with a corner is derived by using the contour approach surrounding the singularity. It is found that using the contour approach the jump term comes half and half from the free terms in the L and M kernel integrations, which is different from the limiting process from an interior point to a boundary point where the jump term comes from the L kernel only. Thus, the definition of the Hadamard principal value for hypersingular integration at the collocation point of a corner is extended to a generalized sense for both the tangent and normal derivative of double layer potentials in comparison with the conventional definition. Two regularized versions of dual boundary integral equations with corners are proposed to avoid the boundary effect and are tested by an example. The numerical implementation is incorporated in the BEPO2D program. Also, a numerical example with a Dirichlet boundary condition on the corner is verified to determine the validity of the dual integral formulation.