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We introduce Monte Carlo methods to compute the solution of ellipticequations with pure Neumann boundary conditions. We first prove that thesolution obtained by the stochastic representation has a zero mean value withrespect to the invariant measure of the stochastic process associated to theequation. Pointwise approximations are computed by means of standard and newsimulation schemes especially devised for local time approximation on theboundary of the domain. Global approximations are computed thanks to astochastic spectral formulation taking into account the property of zero meanvalue of the solution. This stochastic formulation is asymptotically perfect interms of conditioning. Numerical examples are given on the Laplace operator ona square domain with both pure Neumann and mixed Dirichlet-Neumann boundaryconditions. A more general convection-diffusion equation is also numericallystudied.

Monte Carlo approximations of the Neumann problem