Sparse tensor product finite element method for some linear and nonlinear multiscale problems - Tan, Wee Chin (2018)
Partial differential equations with multiple scales arise from a wide range of scientific and engineering problems, such as composite materials, oil flow and seismology. Solving multiscale partial differential equations are extremely difficult. Traditionally, numerical methods have to use a mesh size that is at most the order of the smallest scale to extract all the information in the microscopic scale. There has been extensive work to study multiscale partial differential equations and develop numerical methods to approximate the solutions. Although there is a large literature on the topic, there are still many challenges and open problems. This thesis contributes novel essentially optimal numerical methods for solving locally periodic multiscale monotone parabolic problems, multiscale linear and nonlinear equations that depend on microscopic scales in both space and time, and multiscale nonlinear monotone variational inequalities. We develop the sparse tensor product finite element (FE) method to approximate the solutions to the problems. The method approximates all the necessary microscopic and macroscopic information with a prescribed accuracy using an essentially optimal number of degrees of freedom. For locally periodic problems, this is better than other methods available in the literature. Chapter 2 studies linear parabolic equations that depend on a microscopic time scale and a microscopic spatial scale. We consider the most interesting critical similarity case where the derivative with respect to the fast time variable plays a role in the cell problems. Using multiscale convergence, we deduce a high dimensional multiscale homogenized equation that provides the solution to the homogenized equation, which provides the macroscopic information, and the time-space corrector term, which encodes the microscopic information. We develop the sparse tensor product FE method for solving this time-space multiscale homogenized equation. The method provides an approximation to all the necessary information with a required accuracy, using an essentially optimal number of degrees of freedom. From the FE solution of the time-space multiscale homogenized equation, we construct a numerical corrector for the solution of the original time-space multiscale equation. Chapter 3 develops a new method for solving monotone parabolic equations that depend on $n$ separable microscopic scales. For nonlinear problems, forming the homogenized equation is practically impossible, as for each vector in $\spa$, we have to solve a nonlinear monotone cell problem. Applying multiscale convergence, we obtain a high dimensional multiscale homogenized equation which when solved, the solution will provide the necessary macroscopic and microscopic information. We develop the backward Euler and Crank-Nicholson methods for sparse tensor product FE spaces. In both approaches, the sparse tensor product FE method provides an approximation with an essentially optimal level of accuracy, but uses an essentially optimal number of degrees of freedom. For two scale problems, a new homogenization error is derived in terms of the microscopic scale. A numerical corrector is deduced with an explicit error. For general multiscale problems, a numerical corrector is derived, without an explicit error. This is because of the unavailability of a homogenization error. Chapter 4 considers multiscale monotone parabolic equations which depend on a microscopic time scale and a microscopic spatial scale. As in Chapter \ref{chap:chapter2}, we consider the most interesting case when the derivative with respect to the fast time variable plays a part in the limiting equation. Applying multiscale convergence, a high dimensional multiscale homogenized equation is obtained. This chapter develops a new numerical method to solve the multiscale monotone parabolic problems with an essentially equal level of accuracy as the full tensor product FE method. The backward Euler and Crank-Nicholson methods are developed for the sparse tensor product FE spaces. The sparse tensor product FE method uses an essentially optimal number of degree of freedom to get an approximation with the desired level of accuracy. A numerical corrector is also derived from the FE solutions. Chapter 5 studies locally periodic multiscale variational inequalities which depend on a macroscopic and $n$ microscopic scales. Using multiscale convergence, we deduce a multiscale homogenized variational inequality in a high dimensional tensorized domain. To the best of our knowledge, numerical method for multiscale variational inequalities have not been developed. We develop the sparse tensor product FE method for locally periodic multiscale problems which attains an essentially equal level of accuracy to that of the full tensor product FE method but requires only an essentially optimal number of degrees of freedom which is essentially equal to that for solving a problem in $\spa$. In the two scale case, we deduce a new homogenization error for the nonlinear monotone variational inequalities. A numerical corrector is constructed with an explicit error in terms of the FE error and the homogenization error. In the multiscale case, we construct a numerical corrector without an explicit error. In all the chapters, we apply the methods developed to solve some multiscale problems numerically to verify the theory.