Monte Carlo approximations of the Neumann problem

Abstract

We introduce Monte Carlo methods to compute the solution of elliptic\nequations with pure Neumann boundary conditions. We first prove that the\nsolution obtained by the stochastic representation has a zero mean value with\nrespect to the invariant measure of the stochastic process associated to the\nequation. Pointwise approximations are computed by means of standard and new\nsimulation schemes especially devised for local time approximation on the\nboundary of the domain. Global approximations are computed thanks to a\nstochastic spectral formulation taking into account the property of zero mean\nvalue of the solution. This stochastic formulation is asymptotically perfect in\nterms of conditioning. Numerical examples are given on the Laplace operator on\na square domain with both pure Neumann and mixed Dirichlet-Neumann boundary\nconditions. A more general convection-diffusion equation is also numerically\nstudied.\n