Model Reduction Techniques and Parareal Algorithm for Multiscale problems

Abstract

A broad range of scientific and engineering problems involve multiple scales. For example, composite material properties and subsurface properties can vary over many length scales. Direct numerical methods of multiscale problems is often difficult due to the fact that a very fine mesh of the domain is required to reflect the heterogeneous coefficients. From a computational point of view, the major challenge to solve these problems is the size of the computation, even with the aid of supercomputers. On the other hand, from an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple scale systems, such as the effective conductivity, permeability, elastic moduli and eddy diffusivity. Therefore, it is desirable to develop fast and effective numerical methods that capture the small scale effect on the large scales, but do not require resolving all the small features.

There has been extensive research effort devoted to developing computational methods for multiscale problems. Among the most popular and developed techniques are homogenization method, multiscale finite element methods and parareal algorithm. The goal of homogenization methods and multiscale finite element methods is to construct numerical solvers on the coarse grid. Their resulting linear systems are typically much smaller than using fine grid. Parareal algorithm facilitates speeding up the numerical solver to time dependent equations on the condition of sufficient processors. Typically, parareal algorithm could result in less wall-clock time than sequentially computing.

In this dissertation, we will design and apply model reduction techniques to time-fractional diffusion equations, parabolic equations and stokes equations in heterogeneous media. Homogenization approach is studied for the time-fractional diffusion equation. We discuss constraint energy minimizing generalized multiscale finite element method for the incompressible Stokes flow problem in a perforated domain. In this dissertation, we present two methodologies for parabolic problems with heterogeneous coefficients: a novel approach coupling multiscale methods with parareal algorithm and an efficient numerical solver coupling space-time finite element method and Non-local multi-continua technique. The former aims for time-independent permeability field and the latter for time-dependent permeability field.