Degenerate scale for 2D Laplace equation with Robin boundary condition
Résumé
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.
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