A multiscale finite element method for oscillating Neumann problem on rough domain
Provides a new numerical method for solving PDEs on rough domains, which is important for applications in materials science and engineering.
Developed a multiscale finite element method for Laplace equation with oscillating Neumann boundary conditions on rough boundaries, achieving optimal convergence rate in energy norm with a weak resonance term for periodic roughness.
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.