An Adaptive Algorithm Based on Stochastic Discontinuous Galerkin for Convection Dominated Equations with Random Data

In this paper, we propose an adaptive approach, based on mesh refinement or parametric enrichment, for convection diffusion equations containing randomness in their coefficients. A parametric system of convection diffusion equations obtained by an application of stochastic Galerkin approach is discretized by using a symmetric interior penalty Galerkin (SIPG) method with upwinding for the convection term in the spatial domain. We show the reliability of the proposed residual-based error estimator in the energy norm contributed by the error due to the SIPG discretization, the error due to the data oscillations, and the error due to the (generalized) polynomial chaos discretization in the parametric space. To illustrate the performance of the proposed estimator, several benchmark examples including a random diffusivity parameter, a random velocity parameter, random diffusivity/velocity parameters, and a random (jump) discontinuous diffusivity parameter, are tested.