Generalized finite element methods for time-dependent multiscale problems

Abstract: In this thesis we develop and analyze generalized finite element methods for time-dependent partial differential equations (PDEs). The focus lies on equa- tions with rapidly varying coefficients, for which the classical finite element method is insufficient, as it requires a mesh fine enough to resolve the data. The framework for the novel methods are based on the localized orthogonal decomposition technique. The main idea of this method is to construct a modified finite element space whose basis functions contain information about the variations in the coefficients, hence yielding better ap- proximation properties. At first, the localized orthogonal decomposition framework is extended to the strongly damped wave equation, where two different highly varying coeffi- cients are present (Paper I). The dependency of the solution on the different coefficients vary with time, which the proposed method accounts for automat- ically. Then we consider a parabolic equation where the diffusion is rapidly varying in both time and space (Paper II). Here, the framework is extended so that the modified finite element space uses space-time basis functions that contain the information of the diffusion coefficient. In both papers we prove error estimates for the methods, and confirm the theoretical findings with numerical examples.