Multiscale maxwell equations : homogenization and high dimensional finite element method - Chu, Van Tiep (2016)
Solving multiscale partial differential equations is exceedingly complex. Traditional methods have to use a mesh size of at most the order of the smallest scale to produce accurate approximations. The thesis contributes rigorous study of mathematical homogenization of multiscale Maxwell equations. It includes new homogenization errors when the solution to the homogenized equation possesses low regularity. The thesis develops the sparse tensor finite element approach, using edge finite elements, for solving the high dimensional multiscale homogenized Maxwell equations. It obtains the solution to the homogenized equation, which describes the solution to the multiscale equation macroscopically, and the scale interacting (corrector) terms, which encode the microscopic information, at the same time. The method achieves essentially optimal complexity. From the finite element solutions, we construct a numerical corrector for the solution of the multiscale problem, with an explicit error in terms of the homogenization error and the finite element error in the two-scale cases.