Finite elements for elliptic problems with stochastic coefficients
Finite elements for elliptic problems with stochastic coefficients
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DOI：
10.1016/j.cma.2004.04.008
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发表时间：
2005-01-01
影响因子：
7.2
通讯作者：
Todor, RA
中科院分区：
文献类型：
--
作者：
Frauenfelder, P;Schwab, C;Todor, RA
我们描述了随机椭圆边值问题（sbvp）的确定性有限元（FE）求解算法，其系数被假设为具有有限二阶矩和已知的分段平滑两点空间相关函数的随机场。随机变量和确定性变量的分离（不确定性的参数化）是通过 Karhunen-Loeve (KL) 展开实现的。提出了一种基于核无关快速多极子方法 (FMM) 的 KL 特征值计算 O(NlogN) 算法。 KL 展开式的截断给出了随机系数的 (M, 1) 维纳多项式混沌 (PC) 展开式，并且被证明会导致高维、确定性边值问题 (dbvp)。其解在具有解析域锐界的随机变量中的解析性用于规定近似解的 (M, r) Wiener PC 展开中的变量随机多项式次数 r = r(M))。给出了 KL 特征对的 FEM 近似的逐点误差界限、KL 展开式的截断以及 dbvp 的 FE 解。数值例子表明，M取决于随机扩散系数的空间相关长度。 PC 随机伽辽金 FEM 中的可变多项式次数 r 允许在适度的时间内处理 M 高达 30 和 r 高达 10 的 KL 展开。 (C) 2004 Elsevier B.V. 保留所有权利。
We describe a deterministic finite element (FE) solution algorithm for a stochastic elliptic boundary value problem (sbvp), whose coefficients are assumed to be random fields with finite second moments and known, piecewise smooth two-point spatial correlation function. Separation of random and deterministic variables (parametrization of the uncertainty) is achieved via a Karhunen-Loeve (KL) expansion. An O(NlogN) algorithm for the computation of the KL eigenvalues is presented, based on a kernel independent fast multipole method (FMM). Truncation of the KL expansion gives an (M, 1) Wiener polynomial chaos (PC) expansion of the stochastic coefficient and is shown to lead to a high dimensional, deterministic boundary value problem (dbvp). Analyticity of its solution in the stochastic variables with sharp bounds for the domain of analyticity are used to prescribe variable stochastic polynomial degree r = r(M)) in an (M, r) Wiener PC expansion for the approximate solution. Pointwise error bounds for the FEM approximations of KL eigenpairs, the truncation of the KL expansion and the FE solution to the dbvp are given. Numerical examples show that M depends on the spatial correlation length of the random diffusion coefficient. The variable polynomial degree r in PC-stochastic Galerkin FEM allows to handle KL expansions with M up to 30 and r, up to 10 in moderate time. (C) 2004 Elsevier B.V. All rights reserved.