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In this paper we consider multi-dimensional partial differential equations ofparabolic type involving divergence form operators that possess a discontinuouscoefficient matrix along some smooth interface. The solution of the equation isassumed to present a compatibility transmission condition of its conormalderivatives at this interface (multi-dimensional diffraction problem). We provean existence and uniqueness result for the solution and construct a lowcomplexity numerical Monte Carlo stochastic Euler scheme to approximate thesolution of the parabolic partial differential equation in divergence form. Inparticular, we give new estimates for the partial derivatives of the solution.Using these estimates, we prove a convergence rate for our stochastic numericalmethod when the initial condition belongs to an iterated domain of thedivergence form operator. Our method presents the same convergence rate as thestochastic numerical schemes elaborated for the same problem in theone-dimensional context. Finally, we compare our results to classicaldeterministic numerical approximations and illustrate the accuracy of ourmethod.

A transformed stochastic Euler scheme for multidimensional transmission PDE