Numerical solution of the nonlinear Dirichlet and Neumann problems based on the probabilistic approach

In this chapter we apply the probabilistic approach of Chap. 7 to nonlinear problems with Dirichlet and Neumann boundary conditions [210,212]. We recall that the probabilistic approach is based on making use of the well-known probabilistic representations of solutions to linear parabolic equations and the ideas of weak sense numerical integration of SDEs. Despite their probabilistic nature these methods are nevertheless deterministic. The probabilistic approach takes into account a coefficient dependence on the space variables and a relationship between diffusion and advection in an intrinsic manner. In particular, the layer methods allow us to avoid difficulties stemming from essentially changing coefficients and strong advection.