國立清華大學 數學系

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 talk, I will introduce 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, I will introduce a 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 has an optimal convergence rate of  in the energy norm and in the L_2 norm. The hidden constants in these estimates are independent of the ``contrast'' (i.e. ratio of largest to smallest value) of the PDE's coefficients. I 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 discuss a recent work on coupling a conservation law for mass at the continuum scale with a discrete network model that describes the pore scale flow in a porous medium.  We assume that over the same physical domain there exists an effective mass conservation equation at the continuum scale which could have been solved can be solved  efficiently. Our coupling method uses local simulations on sampled domains at network scale to evaluate the continuum equation and thus solve for the pressure in the domain.  In the case where classical homogenization applies, we prove convergence of the proposed multiscale solutions to the homogenized equations. We also extended the method to two-phase flow, for the situations in which the saturation profile go through a sharp transition from fully saturated to almost unsaturated states. Our coupling method for the pressure equation uses local simulations on small sampled network domains at the pore scale to evaluate the continuum equation and thus solve for the pressure in the domain. Tea Time: 3:30pm, R707