Numerical solutions of some multiscale problems

• Main Author: Pang, Chen Hui
• Other Authors: Hoang Viet Ha
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2023
• Subjects: Science::Mathematics::Applied mathematics
• Online Access: https://hdl.handle.net/10356/172901
• Institution: Nanyang Technological University

Physical problems that involve several spatial length scales that differ by multiple orders of magnitude occur in many applications of engineering, biology, seismology, etc. Solving such problems presents a challenge of high computational cost. Classical finite element (FE) methods require a mesh of size smaller than or equal to the order of the finest scale to approximate the solution accurately, resulting in an astronomical number of degrees of freedom that makes these problems computationally prohibitive. This thesis tackles some locally periodic two-scale problems with sparse tensor product finite element method, which has a reduced level of complexity that is essentially optimal. Moreover, all coarse scale and fine scale information is obtained in a single step of solving the two-scale homogenized equation. Chapter 2 examines a two-scale elasticity problem in the Hellinger-Reissner mixed form that enables the two-scale stress tensor to be solved directly. We first consider the case when the domain is polygonal. Two-scale convergence is employed to obtain the two-scale homogenized equation that contains both the solution to the homogenized problem and the corrector term, i.e., all coarse and fine scale information. We construct the full and the sparse tensor product FE approximation schemes and prove that the complexity of the sparse tensor product FE approach is essentially optimal for this mixed problem under some regularity conditions for the solution. Next, we develop the sparse tensor product FEs for the case of smooth domains where the displacement field of the elasticity problem normally possesses the desired H2-regularity. This construction is new in the literature. Sparse tensor product approximations are achieved by considering a hierarchy of polygonal domains that are inscribed in the smooth domain. We achieve essentially optimal complexity for this case. We specify the required regularity of the solutions to the two-scale homogenized problem in order to achieve the optimal convergence rate with the sparse tensor product FEs. A numerical corrector for the solution of the original two-scale mixed elasticity problem is constructed from the FE solution with an explicit error in terms of the fine scale and the FE mesh. Chapter 3 studies a two-scale fractional elliptic equation. The problem is challenging due to its non-locality. We consider the Caffarelli-Silvestre localized extended two-scale problem whose trace at the boundary of the extended variable z (i.e., at z=0) is the solution for the original two-scale fractional problem. By extending the notion of two-scale convergence to weighted Sobolev spaces in the extended domain, we derive the two-scale homogenized equation for the extended two-scale equation. We then construct a sparse tensor product finite element between the slow variable x, the fast variable y and the extended variable z and prove the essential optimality of the method in terms of accuracy and degree of freedom. Furthermore, due to the high regularity of the solution with respect to the extended variable, we also develop another essentially optimal approximating approach where hp-finite element is used for the extended variable, with the sparse tensor product FEs between the spatial slow and fast variables. The essentially optimal convergence rate is proved rigorously. We also prove the required regularity in weighted Sobolev spaces of the solution to the extended two-scale homogenized problem. A numerical corrector is developed for the solution of the extended two-scale problem, which implies a numerical corrector for the solution of the original fractional two-scale problem via the trace operator, with an error estimate in terms of the FE mesh-width and the fine scale. In the existing literature on sparse tensor product FEs for multiscale problems, the multiscale coefficient is assumed to be continuous. This excludes the important case where the coefficient is not continuous, which is observed in many applications, i.e., composites with a periodic array of inclusions whose conductivity coefficient differs from the conductivity coefficient of the matrix hosting them. This is the focus of Chapters 4, 5, and 6. Chapter 4 considers a time-independent two-scale elliptic problem, while Chapters 5 and 6 investigate time-dependent two-scale parabolic and hyperbolic problems, respectively. The difficulty in analysing these problems is attributed to the fact that the corrector term only possesses piecewise regularity and the need to approximate the discontinuous coefficient according to the spatial discretization. For these problems, we developed the sparse tensor product FEs for the two-scale homogenized equation, where the two-scale coefficient is approximated by one that depends on the polygonal approximation of the inclusion. We derive the essentially optimal convergence rate for the sparse tensor product FEs, which in general exhibit a slower convergence rate than that of a two-scale problem with a continuous two-scale coefficient in the literature, by a log factor of the mesh size. For time-dependent problems, standard time discretization schemes are employed. The piecewise regularity with respect to the fast variable that is required to achieve the essentially optimal convergence rate is proved rigorously. For each problem, numerical correctors for the two-scale problems with a discontinuous coefficient are constructed from the FE solutions, with rigorously justified error estimates in terms of the mesh size and the microscopic scale. In each chapter, we present numerical examples to substantiate the essentially optimal theoretical rates of convergence of the sparse tensor product FEs.