In this paper, we propose an adaptive approach, based on mesh refinement or parametric enrichment with polynomial degree adaption, for numerical solution of convection dominated equations with random input data. A parametric system emerged from 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 derive a residual-based error estimator contributed by the error due to the SIPG discretization, the (generalized) polynomial chaos discretization in the stochastic space, and data oscillations. Then, the reliability of the proposed error estimator, an upper bound for the energy error up to a multiplicative constant, is shown. Moreover, to balance the errors stemmed from spatial and stochastic spaces, the truncation error coming from Karhunen--Lo\`{e}ve expansion is also considered in the numerical simulations. Last, 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 to illustrate the performance of the proposed estimator.

翻译：本文提出一种基于网格细化或多项式阶数自适应参数富集的自适应方法，用于求解含随机输入数据的对流占优方程。通过随机伽辽金方法获得的参数系统，采用带对流项迎风格式的对称内罚伽辽金（SIPG）方法进行空间离散。我们推导出基于残差的误差估计器，该估计器由SIPG离散误差、（广义）多项式混沌在随机空间中的离散误差以及数据振荡误差共同构成。随后证明了所提误差估计器的可靠性——即能量误差的上界（至多相差一个乘法常数）。此外，为平衡空间与随机空间产生的误差，数值模拟中还考虑了Karhunen-Loève展开的截断误差。最后，通过包含随机扩散系数、随机速度参数、随机扩散/速度参数以及随机（跳跃）不连续扩散系数在内的多个基准算例，验证了所提估计器的性能。