  We develop a new multiscale finite element method for Laplace equation with
oscillating Neumann boundary conditions on rough boundaries. The key point is
the introduction of a new boundary condition that incorporates both the
microscopically geometrical and physical information of the rough boundary. We
prove the method has optimal convergence rate in the energy norm with a weak
resonance term for periodic roughness. Numerical results are reported for both
periodic and nonperiodic roughness.
