A Fokker-Planck type approximation of parabolic PDEs with oblique boundary data (Q2790713)

The authors consider solutions to quasilinear parabolic partial differential equations with zero oblique boundary data in a bounded domain. They approximate its solutions by solutions to a Fokker-Planck type equation in the whole space. The equation corresponds to the quasilinear one inside of the domain and a Fokker-Planck equation with a penalizing drift term outside of the domain. The penalization parameter \(N\) is the strength of the drift term. The convergence is locally uniform as \(N\to \infty\) and error estimates in the penalization parameter \(N\) are obtained.NEWLINENEWLINEThe result is first derived for the linear case, which is a Fokker-Planck equation with Neumann boundary data but not necessarily constant diffusion matrix. Here the construction of a supersolution for the approximate problem is based on the solution of the limit problem inside of the domain as well as an extension using the explicit form of the drift term. The general case is dealt with by a second approximation step, which interpolates between the linear second-order operator with diffusion matrix matching the boundary data outside of the domain and the quasilinear operator inside of the domain.NEWLINENEWLINEThe analysis is complemented by a numerical investigation, in which the error bound is numerically verified.