Multiscale methods for evolution problems

Doctoral thesis, 2023

In this thesis we develop and analyze generalized finite element methods for time-dependent partial differential equations (PDEs). The focus lies on equations 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 (LOD) 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 approximation properties.

At first, the localized orthogonal decomposition framework is extended to the strongly damped wave equation, where two different highly varying coefficients are present (Paper~I). The dependency of the solution on the different coefficients vary with time, which the proposed method accounts for automatically. 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. Furthermore, we study wave propagation problems posed on spatial networks (Paper~III). Such systems are characterized by a matrix with large variations inherited from the underlying network. For this purpose, an LOD based approach adapted to general matrix systems is considered. Finally, we analyze the framework for a parabolic stochastic PDE with multiscale characteristics (Paper~IV). In all papers we prove error estimates for the methods, and confirm the theoretical findings with numerical examples.