UC San Diego

Many science and engineering applications are impacted by a significant amount of uncertainty in the model. Examples include groundwater flow, microscopic biological system, material science and chemical engineering systems. Common mathematical problems in these applications are elliptic equations with stochastic data. In this dissertation, we examine two types of stochastic elliptic partial differential equations(SPDEs), namely nonlinear stochastic diffusion reaction equations and general linearized elastostatic problems in random media. We begin with the construction of an analysis framework for this class of SPDEs, extending prior work of Babuska [3] in 2010. We then use the framework both for establishing well- posedness of the continuous problems and for posing Galerkintype numerical methods. In order to solve these two types of problems, single integral weak formulations and stochastic collocation methods are applied. Moreover, a priori error estimates for stochastic collocation methods are derived, which imply that the rate of convergence is exponential, along with the order of polynomial increasing in the space of random variables. As expected, numerical experiments show the exponential rate of convergence, verified by a posterior error analysis. Finally, an adaptive strategy driven by a posterior error indicators is designed