Multiscale methods for evolution problems
Doktorsavhandling, 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.

localized orthogonal decomposition


stochastic partial differential equations

Strongly damped wave equation

finite element method

spatial network models

parabolic equations

Opponent: Olof Runborg, KTH, Sweden


Per Ljung

Chalmers, Matematiska vetenskaper, Tillämpad matematik och statistik

Partial differential equations (PDEs) find a wide range of applications in various fields, and are commonly used to model and analyze complex phenomena such as heat transfer and wave propagation. When such phenomena are modeled on strongly heterogeneous domains, such as composite materials or porous media, the data of the corresponding PDE contains rapid variations. This type of equations is commonly referred to as multiscale problems, for which well-established approximation techniques such as the finite element method are typically ill-suited as they are unable to resolve large fluctuations. For this purpose, several so-called multiscale methods have been developed. This thesis focuses on the application of such methods in the context of evolution problems.





Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 5308




Opponent: Olof Runborg, KTH, Sweden

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