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.

翻译：本文提出一种自适应方法，基于网格细化或参数富集，用于求解系数含有随机性的对流扩散方程。通过应用随机伽辽金法得到的参数化对流扩散方程组，在空间域采用含迎风处理的对流项对称内罚伽辽金（SIPG）方法进行离散化。我们证明了所提出的基于残差的误差估计器在能量范数下的可靠性，该误差由SIPG离散化误差、数据振荡误差以及参数空间（广义）多项式混沌离散化误差共同构成。为说明所提估计器的性能，测试了多个基准算例，包括随机扩散系数、随机速度参数、随机扩散/速度参数以及随机（跳跃）不连续扩散系数等情形。