Degenerate scale for 2D Laplace equation with Robin boundary condition. (July 2017) - Corfdir, A.
Abstract: It is well known that the 2D Laplace Dirichlet boundary value problem with a specific contour has a degenerate scale for which the boundary integral equation (BIE) has several solutions. We study here the case of the Robin condition (i.e. convection condition for thermal conduction problems), and show that this problem has also one degenerate scale. The cases of the interior problem and of the exterior problem are quite different. For the Robin interior problem, the degenerate scale is the same as for the Dirichlet problem. For the Robin exterior problem, the degenerate scale is always larger than for the Dirichlet problem and has some asymptotic properties. The cases of several simple boundaries like ellipse, equilateral triangle, square and rectangle are numerically investigated and the results are compared with the analytically predicted asymptotic behavior. An important result is that avoiding a contour leading to a degenerate Robin problem cannot be achieved as simply as in the case of Dirichlet boundary condition by introducing a large reference scale into the Green's function.