A MULTISCALE FINITE ELEMENT METHOD FOR OSCILLATING NEUMANN PROBLEM ON ROUGH DOMAIN

We develop a new multiscale finite element method for the 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. Our approach applies to problems posed on a domain with a rough boundary as well as oscillating boundary conditions. We prove the method has a linear convergence rate in the energy norm with a weak resonance term for periodic roughness. Numerical results are reported for both periodic and nonperiodic roughness.