A general adaptive refinement strategy for solving linear elliptic partial differential equation with random data is proposed and analysed herein. The adaptive strategy extends the a posteriori error estimation framework introduced by Guignard and Nobile in 2018 (SIAM J. Numer. Anal., 56, 3121--3143) to cover problems with a nonaffine parametric coefficient dependence. A suboptimal, but nonetheless reliable and convenient implementation of the strategy involves approximation of the decoupled PDE problems with a common finite element approximation space. Computational results obtained using such a single-level strategy are presented in this paper (part I). Results obtained using a potentially more efficient multilevel approximation strategy, where meshes are individually tailored, will be discussed in part II of this work. The codes used to generate the numerical results are available on GitHub
翻译:本文提出并分析了解决线性椭圆部分差异方程的适应性完善总体战略,并随机提供数据。适应性战略扩展了Guignard和Nobile在2018年推出的事后误差估计框架(SIAM J.Numer.Anal., 56, 3121-3143),以涵盖非按成因参数系数依赖性的问题。战略执行不理想,但可靠和方便,包括将脱钩的PDE问题与共同的有限元素近似空间相近。本文件介绍了使用这种单级战略取得的计算结果(第一部分)。使用可能更加有效的多级近似值战略取得的结果,其中将逐个量度地讨论。在GitHub上提供用于产生数字结果的代码。