I'm solving the following equation in the unit square using finite differences: epsilon(u_xx+u_yy)+u_x+u_y=0, where epsilon is a singular perturbation parameter. I need to use domain decomposition to isolate the corner singularity in the outflow corner. My subdomain in this corner is a quarter disk so I translate the d.e. to polar coordinates. If I don't use the translation: v=exp(M(x+y))u, M=1/2epsilon, before translating to polar coordinates, then my solution has a concave shape, but it should be convex. I find that even when epsilon=1 I need to use this integrating factor. Does anyone know why this is? Have searched high and low with no success.