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Adaptive stochastic Galerkin FEM

机译：自适应随机Galerkin有限元

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摘要

A framework for residual-based a posteriori error estimation and adaptive mesh refinement and polynomial chaos expansion for general second order linear elliptic PDEs with random coefficients is presented. A parametric, deterministic elliptic boundary value problem on an infinite-dimensional parameter space is discretized by means of a Galerkin projection onto finite generalized polynomial chaos (gpc) expansions, and by discretizing each gpc coefficient by a FEM in the physical domain. An anisotropic residual-based a posteriori error estimator is developed. It contains bounds for both contributions to the overall error: the error due to gpc discretization and the error due to Finite Element discretization of the gpc coefficients in the expansion. The reliability of the residual estimator is established. Based on the explicit form of the residual estimator, an adaptive refinement strategy is presented which allows to steer the polynomial degree adaptation and the dimension adaptation in the stochastic Galerkin discretization, and, embedded in the gpc adaptation loop, also the Finite Element mesh refinement of the gpc coefficients in the physical domain. Asynchronous mesh adaptation for different gpc coefficients is permitted, subject to a minimal compatibility requirement on meshes used for different gpc coefficients. Details on the implementation with the open-source software framework alea are presented; it is generic, and is based on available stiffness and mass matrices of a FEM for the deterministic, nonparametric nominal problem evaluated in the FEniCS environment. Preconditioning of the resulting matrix equation and iterative solution are discussed. Numerical experiments in two spatial dimensions for membrane and plane stress boundary value problems on polygons are presented. They indicate substantial savings in total computational complexity due to FE mesh coarsening in high gpc coefficients.
机译：提出了一种基于残差的后验误差估计，自适应网格细化和具有随机系数的一般二阶线性椭圆PDE的多项式混沌扩展的框架。无限维参数空间上的参数化确定性椭圆边值问题通过Galerkin投影离散到有限广义多项式混沌（gpc）展开上，并通过有限元法在物理域中离散化每个gpc系数。开发了一种基于各向异性残差的后验误差估计器。它包含了对总体误差的两个贡献的界线：由于gpc离散化引起的误差和因展开中gpc系数的有限元离散化引起的误差。建立了残差估计器的可靠性。基于残差估计器的显式形式，提出了一种自适应细化策略，该策略允许在随机Galerkin离散化中控制多项式度自适应和维数自适应，并嵌入到gpc自适应环路中，还对有限元网格进行细化。物理域中的gpc系数。允许对不同gpc系数进行异步网格自适应，但要满足对用于不同gpc系数的网格的最低兼容性要求。本文详细介绍了使用开源软件框架的实现。它是通用的，并且基于FEM的可用刚度和质量矩阵，用于FEniCS环境中评估的确定性，非参数名义问题。讨论了所得矩阵方程的预处理和迭代解。提出了二维空间中多边形的膜和平面应力边界值问题的数值实验。它们表明，由于高gpc系数的FE网格变粗，可大大节省总的计算复杂度。

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