Generalized finite element methods for time-dependent multiscale problems

Licentiate thesis, 2021

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.