We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We perform a posteriori error analysis of Galerkin approximations and derive a reliable and efficient estimate for the energy error in these approximations. Practical versions of this error estimate are discussed and tested numerically for a model problem with non-affine parametric representation of the coefficient. Furthermore, we use the error reduction indicators derived from spatial and parametric error estimators to guide an adaptive solution algorithm for the given parametric PDE problem. The performance of the adaptive algorithm is tested numerically for model problems with two different non-affine parametric representations of the coefficient.

我们考虑具有一般一致有界参数系数的线性椭圆偏微分方程（PDE）。在随机Galerkin有限元方法的框架中，可以近似地解决此PDE问题。我们对Galerkin近似进行后验误差分析，并得出这些近似中能量误差的可靠而有效的估计。对于具有系数的非仿射参数表示的模型问题，讨论了此误差估计的实际版本并对其进行了数值测试。此外，我们使用从空间和参数误差估计量得出的误差减少指标来指导针对给定参数PDE问题的自适应求解算法。