Solution of stochastic partial differential equations (SPDEs) using Galerkin method: Theory and applications

Stochastic equations arise when physical systems with uncertain data are modeled. This dissertation focuses on elliptic stochastic partial differential equations (SPDEs) and systematically develops theoretical and computational foundations for solving them. The numerical problem is posed on D × Ω, where D is the physical space domain and Ω denotes the space of all the admissible elementary events. Two types of SPDEs are considered here: (a) Random-RHS type, i.e. only the source term contains randomness and (b) Random-LHS type, i.e. only the PDE coefficient is random. It is assumed that mean and covariance of the input stochastic functions are given and similar quantities need to be obtained for the solution. This study identifies the necessary function spaces, details the weak forms and discusses their properties, develops a priori error estimates for the solution and its statistical moments and constructs finite-element-based solution schemes for these SPDEs. Computer codes are developed that implement the finite element schemes in order to carry out a suite of numerical experiments. Additional strategies comprising of Monte-Carlo simulations, analytical solutions developed as part of this study, and alternate differential equations for direct evaluation of statistical moments are employed here in order to validate the results obtained from the finite element schemes. Direct applicability of the current SPDEs to practical engineering problems such as stochastic porous media flow is discussed. Theoretical predictions of convergence rates of the solution and its statistical moments are verified from the numerical calculations. A limited effort is also devoted to a posteriori error estimation and h-adaptive computation, however, this is left as a potential future work. In summary, this study establishes the necessary extensions to the theory and solution schemes of conventional Galerkin approximation-based finite element method to stochastic equations, thus, motivating application of vast amount of existing knowledge in the deterministic finite element community to the solution of SPDEs.