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Solution of stochastic partial differential equations (SPDEs) using Galerkin method: Theory and applications.

机译：使用Galerkin方法求解随机偏微分方程（SPDE）：理论与应用。

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Stochastic equations arise when physical systems with uncertain data are modeled. This dissertation focuses on elliptic stochastic partial differential equations (SPDEs) and systematically develops theoretical and computational foundations for solving them. The numerical problem is posed on D × Ω, where D is the physical space domain and Ω denotes the space of all the admissible elementary events. Two types of SPDEs are considered here: (a) Random-RHS type, i.e. only the source term contains randomness and (b) Random-LHS type, i.e. only the PDE coefficient is random. It is assumed that mean and covariance of the input stochastic functions are given and similar quantities need to be obtained for the solution. This study identifies the necessary function spaces, details the weak forms and discusses their properties, develops a priori error estimates for the solution and its statistical moments and constructs finite-element-based solution schemes for these SPDEs. Computer codes are developed that implement the finite element schemes in order to carry out a suite of numerical experiments. Additional strategies comprising of Monte-Carlo simulations, analytical solutions developed as part of this study, and alternate differential equations for direct evaluation of statistical moments are employed here in order to validate the results obtained from the finite element schemes. Direct applicability of the current SPDEs to practical engineering problems such as stochastic porous media flow is discussed. Theoretical predictions of convergence rates of the solution and its statistical moments are verified from the numerical calculations. A limited effort is also devoted to a posteriori error estimation and h-adaptive computation, however, this is left as a potential future work. In summary, this study establishes the necessary extensions to the theory and solution schemes of conventional Galerkin approximation-based finite element method to stochastic equations, thus, motivating application of vast amount of existing knowledge in the deterministic finite element community to the solution of SPDEs.
机译：当对具有不确定数据的物理系统进行建模时，就会出现随机方程。本文主要研究椭圆型随机偏微分方程（SPDEs），系统地为解决这些问题提供了理论和计算基础。数值问题放在 D ×Ω上，其中 D 是物理空间域，而Ω表示所有允许的基本事件的空间。这里考虑两种类型的SPDE：（a）随机RHS类型，即仅源项包含随机性;以及（b）随机LHS类型，即仅PDE系数是随机的。假设给出了输入随机函数的均值和协方差，并且对于该解需要获得相似的数量。这项研究确定了必要的函数空间，详细介绍了弱形式，并讨论了它们的性质，为解决方案及其统计矩开发了先验误差估计，并为这些SPDE构建了基于有限元的解决方案。开发了执行有限元方案的计算机代码，以进行一系列数值实验。为了验证从有限元方案获得的结果，此处采用了包括蒙特卡洛模拟，作为本研究一部分开发的分析解决方案以及用于统计矩直接评估的替代微分方程的其他策略。讨论了当前SPDE对实际工程问题（如随机多孔介质流）的直接适用性。数值计算验证了该解及其统计矩收敛速度的理论预测。有限的努力也专门用于后验误差估计和 h 自适应计算，但是，这留作了潜在的未来工作。综上所述，本研究为传统的基于Galerkin逼近的有限元方法对随机方程组的理论和求解方案建立了必要的扩展，从而激发了确定性有限元社区中大量现有知识在SPDE求解中的应用。

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