In this article, we analyse the domain mapping method approach to approximate statistical moments of solutions to linear elliptic partial differential equations posed over random geometries including smooth surfaces and bulk-surface systems. In particular, we present the necessary geometric analysis required by the domain mapping method to reformulate elliptic equations on random surfaces onto a fix deterministic surface using a prescribed stochastic parametrisation of the random domain. An abstract analysis of a finite element discretisation coupled with a Monte-Carlo sampling is presented for the resulting elliptic equations with random coefficients posed over the fixed curved reference domain and optimal error estimates are derived. The results from the abstract framework are applied to a model elliptic problem on a random surface and a coupled elliptic bulk-surface system and the theoretical convergence rates are confirmed by numerical experiments.

中文翻译：

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在随机体域和表面域上提出的椭圆方程的域映射方法

在本文中，我们分析了用于近似线性椭圆偏微分方程解的统计矩的域映射方法方法，该方程在随机几何结构（包括光滑表面和体表面系统）上构成。特别是，我们提出了域映射方法所需的必要几何分析，以使用随机域的规定随机参数化将随机表面上的椭圆方程重新表述为固定确定性表面。有限元离散化与蒙特卡罗采样结合的抽象分析被呈现为所得到的椭圆方程具有在固定弯曲参考域上提出的随机系数，并且导出了最佳误差估计。