Generalized finite element methods for time-dependent multiscale problems
Licentiatavhandling, 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.

localized orthogonal decomposition


Strongly damped wave equation

parabolic equations.

finite element method

Online via Zoom
Opponent: Victor Ginting


Per Ljung

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Ljung, P. Målqvist, A. Persson, A. A generalized finite element method for the strongly damped wave equation with rapidly varying data

Ljung, P. Maier, R. Målqvist, A. A space-time multiscale method for parabolic problems





Online via Zoom

Opponent: Victor Ginting

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