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Robust a posteriori error estimation for parameter-dependent linear elasticity equations
Mathematics of Computation ( IF 2 ) Pub Date : 2020-11-16 , DOI: 10.1090/mcom/3572
Arbaz Khan , Alex Bespalov , Catherine E. Powell , David J. Silvester

Abstract:The focus of this work is a posteriori error estimation for stochastic Galerkin approximations of parameter-dependent linear elasticity equations. The starting point is a three-field partial differential equation model with the Young modulus represented as an affine function of a countable set of parameters. We introduce a weak formulation, establish its stability with respect to a weighted norm and discuss its approximation using stochastic Galerkin mixed finite element methods. We motivate an a posteriori error estimation scheme and establish upper and lower bounds for the approximation error. The constants in the bounds are independent of the Poisson ratio as well as the spatial and parametric discretisation parameters. We also discuss proxies for the error reduction associated with enrichments of the approximation spaces and we develop an adaptive algorithm that terminates when the estimated error falls below a user-prescribed tolerance. The error reduction proxies are shown to be reliable and efficient in the incompressible limit case. Numerical results are presented to supplement the theory. All experiments were performed using open source (IFISS) software that is available online.


中文翻译:

参数依赖的线性弹性方程的鲁棒后验误差估计

摘要:这项工作的重点是基于参数的线性弹性方程的随机Galerkin逼近的后验误差估计。起点是一个三场偏微分方程模型,其杨氏模量表示为一组可数参数的仿射函数。我们介绍了一个弱公式,建立了它在加权范数方面的稳定性,并讨论了使用随机Galerkin混合有限元方法进行的近似。我们提出了一种后验误差估计方案,并为近似误差确定了上限和下限。边界中的常数与泊松比以及空间和参数离散化参数无关。我们还讨论了与近似空间的丰富化相关的减少错误的代理,并且我们开发了一种自适应算法,该算法在估计的误差降至用户指定的容差以下时终止。在不可压缩的极限情况下,减少错误的代理被证明是可靠且有效的。数值结果是对理论的补充。所有实验均使用在线提供的开源(IFISS)软件进行。
更新日期:2021-01-05
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