Spatio-stochastic adaptive discontinuous Galerkin methods

Abstract

We introduce a goal-oriented, adaptive framework for the uncertainty quantification of systems modeled by stochastically parametrized nonlinear hyperbolic and convection-dominated partial differential equations. Of particular interest are conservation laws in aerodynamics that may have a small number of stochastic parameters but exhibit strong nonlinearity and a wide range of scales. Our framework exploits localized structure in the spatio-parameter space to enable rapid, reliable uncertainty quantification for output quantities of interest. Our formulation comprises the following technical components: (i) a discontinuous Galerkin finite element method, which provides stability for convection-dominated problems; (ii) element-wise polynomial chaos expansions, which capture the parametric dependence of the solution in a way amenable to adaptation; (iii) the dual-weighted residual method, which provides global and element-wise error estimates for quantities of interest; and (iv) a projection-based anisotropic error indicator along with the associated adaptation mechanics that can detect and refine strongly directional features in the physical and/or parameter spaces simultaneously in an efficient manner. Both the spatial and stochastic discretization errors are controlled through the adaptive refinement of the spatial mesh or polynomial chaos expansion degree based on these anisotropic error indicators. We analyze stability, approximation properties, and a priori error bounds of the spatio-stochastic adaptive method. We finally demonstrate the effectiveness of our formulation for engineering-relevant transonic turbulent aerodynamics problems with uncertainties in flow conditions and turbulence parameters.