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 online.

中文翻译：

---

随机搭配有限元的误差估计和自适应性第 I 部分：单级近似

本文提出并分析了一种求解具有随机数据的线性椭圆偏微分方程的通用自适应细化策略。自适应策略扩展了 Guignard 和 Nobile 在 2018 年 (SIAM J. Numer. Anal., 56, 3121--3143) 引入的后验误差估计框架，以涵盖具有非仿射参数系数依赖性的问题。该策略的次优但仍然可靠且方便的实现涉及使用公共有限元逼近空间逼近解耦 PDE 问题。本文（第一部分）介绍了使用这种单级策略获得的计算结果。使用可能更有效的多级近似策略获得的结果将在本工作的第二部分中讨论，其中网格是单独定制的。