Reliable Uncertainty Quantification using Adaptive Stochastic Discontinuous Galerkin Methods

This thesis presents a reliable and efficient algorithm for combined model uncertainty and discretization error quantification of partial differential equation based simulations. Specifically, we present an adaptive solution method for stochastic partial differential equations that (i) propagates the effect of prescribed input parameter uncertainties to the output quantity of interest and (ii) effectively estimates and controls the discretization errors associated with the propagation process. Our framework builds on a high-order discontinuous Galerkin method, element-wise localized polynomial chaos expansions, the dual-weighted residual error estimate, and a spatio-stochastic anisotropic adaptation strategy. We present \textit{a priori} error bounds for our spatio-stochastic approximation and demonstrate the effectiveness of the formulation for a sample two-dimensional scalar equation, as well as compressible flow problems with uncertainties in flow conditions.M.A.S