  In this paper we prove an approximation result for the viscosity solution of
a system of semi-linear partial differential equations with continuous
coefficients and nonlinear Neumann boundary condition. The approximation we use
is based on a penalization method and our approach is probabilistic. We prove
the weak uniqueness of the solution for the reflected stochastic differential
equation and we approximate it (in law) by a sequence of solutions of
stochastic differential equations with penalized terms. Using then a suitable
generalized backward stochastic differential equation and the uniqueness of the
reflected stochastic differential equation, we prove the existence of a
continuous function, given by a probabilistic representation, which is a
viscosity solution of the considered partial differential equation. In
addition, this solution is approximated by solutions of penalized partial
differential equations.
