Multiscale Methods for Elliptic Partial Differential Equations and Related Applications

Citation

Chu, Chia-Chieh (2010) Multiscale Methods for Elliptic Partial Differential Equations and Related Applications. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/PFGA-YY17. https://resolver.caltech.edu/CaltechETD:etd-07312009-095021

Abstract

Multiscale problems arise in many scientific and engineering disciplines. A typical example is the modelling of flow in a porous medium containing a number of low and high permeability embedded in a matrix. Due to the high degrees of variability and the multiscale nature of formation properties, not only is a complete analysis of these problems extremely difficult, but also numerical solvers require an excessive amount of CPU time and storage. In this thesis, we study multiscale numerical methods for the elliptic equations arising in interface and two-phase flow problems. The model problems we consider are motivated by the multiscale computations of flow and transport of two-phase flow in strongly heterogeneous porous media. Although the analysis is carried out for simplified model problems, it does provide valuable insight in designing accurate multiscale methods for more realistic applications.

In the first part, we introduce a new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. The method is H¹-conforming, and has an optimal convergence rate of O(h) in the energy norm and O(h²) in the L₂ norm, where h is the mesh diameter and the hidden constants in these estimates are independent of the "contrast" (i.e. ratio of largest to smallest value) of the PDE's coefficients. The new interior boundary conditions depend not only on the contrast of the coefficients, but also on the angles of intersection of the interface with the element edges. We conduct some numerical experiments to confirm the optimal rate of convergence of the proposed method and its independence from the aspect ratio of the coefficients.

In the second part, we propose a flow-based oversampling method where the actual two-phase flow boundary conditions are used to construct oversampling auxiliary functions. Our numerical results show that the flow-based oversampling approach is several times more accurate than the standard oversampling method. A partial theoretical explanation is provided for these numerical observations.

In the third part, we discuss "metric-based upscaling" for the pressure equation in two-phase flow problem. We show a compensation phenomenon and design a multiscale method for the pressure equation with highly oscillatory permeability.

Thesis Files