Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods
Mixed finite element analysis of lognormal diffusion and multilevel Monte Carlo methods
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DOI：
10.1007/s40072-015-0051-0
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发表时间：
2016-03-01
影响因子：
1.5
通讯作者：
Ullmann, E.
中科院分区：
文献类型：
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作者：
Graham, I. G.;Scheichl, R.;Ullmann, E.
这项工作的动机是需要开发有效的工具来量化与放射性废物处理研究有关的地下流动的不确定性。我们考虑用相关对数正态分布描述的随机多孔介质中的单相流动问题。我们感兴趣的是这些问题的有限元离散化所带来的误差。与最近的一些关于标准节点有限元离散分析的工作相反，我们在这里考虑质量保守的最低阶Raviart-Thomas混合有限元。这是非常重要的，因为在实际的地下水流动问题中，局部质量守恒是非常可取的。由于有限的空间正则性和算子不具有一致的椭圆性和有界性，在对数正态随机场存在下，分析是非平凡的。我们建立了达西速度和压力的有限元误差边界，以及更准确的恢复压力近似。然后，我们应用误差界证明了用于估计这些量的统计量的多层蒙特卡罗算法的收敛性。此外，我们证明了一类具有达西速度的有界线性泛函的收敛性。一个重要的特例是二维流池中有效渗透率的近似。通过数值实验验证了算法的收敛性。
This work is motivated by the need to develop efficient tools for uncertainty quantification in subsurface flows associated with radioactive waste disposal studies. We consider single phase flow problems in random porous media described by correlated lognormal distributions. We are interested in the error introduced by a finite element discretisation of these problems. In contrast to several recent works on the analysis of standard nodal finite element discretisations, we consider here mass-conservative lowest order Raviart-Thomas mixed finite elements. This is very important since local mass conservation is highly desirable in realistic groundwater flow problems. Due to the limited spatial regularity and the lack of uniform ellipticity and boundedness of the operator the analysis is non-trivial in the presence of lognormal random fields. We establish finite element error bounds for Darcy velocity and pressure, as well as for a more accurate recovered pressure approximation. We then apply the error bounds to prove convergence of the multilevel Monte Carlo algorithm for estimating statistics of these quantities. Moreover, we prove convergence for a class of bounded, linear functionals of the Darcy velocity. An important special case is the approximation of the effective permeability in a 2D flow cell. We perform numerical experiments to confirm the convergence results.