In this paper we consider one-dimensional partial differential equations of parabolic type involving a divergence form operator with a discontinuous coefficient and a compatibility transmission condition. We prove existence and uniqueness result by stochastic methods which also allow us to develop a low complexity Monte Carlo numerical resolution method. We get accurate pointwise estimates for the derivatives of the solutionfrom which we get sharp convergence rate estimates for our stochastic numerical method.
"One-dimensional parabolic diffraction equations: pointwise estimates and discretization of related stochastic differential equations with weighted local times." Electron. J. Probab. 17 1 - 30, 2012. https://doi.org/10.1214/EJP.v17-1905