The subject of this work is an adaptive stochastic Galerkin finite element method for parametric or random elliptic partial differential equations, which generates sparse product polynomial expansions with respect to the parametric variables of solutions. For the corresponding spatial approximations, an independently refined finite element mesh is used for each polynomial coefficient. The method relies on multilevel expansions of input random fields and achieves error reduction with uniform rate. In particular, the saturation property for the refinement process is ensured by the algorithm. The results are illustrated by numerical experiments, including cases with random fields of low regularity.

翻译：本文研究一种用于参数型或随机椭圆型偏微分方程的自适应随机伽辽金有限元方法，该方法可生成关于参数变量的稀疏乘积多项式展开。对于相应的空间逼近，每个多项式系数采用独立加密的有限元网格。该方法基于输入随机场的多级展开，并以统一速率实现误差削减。特别地，算法确保了加密过程的饱和性质。数值实验验证了该方法的有效性，其中包括低正则性随机场的算例。