Abstract
Many problems arising in computational science and engineering are described by mathematical models of high complexity - involving multiple disciplines, characterized by a large number of parameters, and impacted by multiple sources of uncertainty. The central theme of accurate, reliable and efficient forward solvers investigated in this dissertation concerns with Reduced Basis Method (RBM) for model order reduction, and Hybridizable Discontinuous Galerkin (HDG) for Finite Element Method (FEM) scheme. The essential ingredients are described in the following two aspects. The first aspect includes RBM and its application to uncertainty quantification (UQ) problems. In order to tackle the challenge in solving large-scale and high-dimensional uncertainty quantification (UQ) problems, we proposed a weighted version of a mathematically rigorous and computationally efficient model reduction strategy, RBM, and synergistically integrate it with the non-intrusive generalized Polynomial Chaos (gPC) framework in a goal-oriented fashion. Besides the application, another main contribution we made in this dissertation is the offline improvement of RBM. The offline stage of RBM relies on residual based error indicators over a discrete "training" set replacing the parameter domain continuum. However, large training sets diminish the attractiveness of RBM since this proportionally increases the cost of the offline phase. In addition, the standard algorithm for the residual norm computation suffers from premature stagnation at the level of the square root of machine precision. These two issues underline the need for offline-enhancing strategies to mitigate the computational difficulty. To address the former problem, two different types of techniques are provided: 1. A surrogate training set on which to perform the greedy algorithm; 2. A residual-free error indicator. The latter problem is resolved by a robust strategy allowing RBM algorithms to enrich the solution subspace with accuracy beyond root machine precision.Together with extensive discussion of high-fidelity and fast solvers, the second aspect is mainly focused on the development and analysis of the first HDG method for solving fifth-order Korteweg-de Vries (KdV) type equations. The semi-discrete scheme is shown stable with proper choices of the stabilization functions in the numerical traces. For the linearized fifth-order equations, the optimal convergence rates are proved theoretically and demonstrated numerically. For the nonlinear case, the demonstration is validated by the numerical experiments.