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We develop a new multiscale finite element method for theLaplace equation with oscillating Neumann boundaryconditions 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.

A Multiscale Finite Element Method for Oscillating Neumann Problem on Rough Domain