标题： 变分多尺度方法在对流扩散反应方程问题有限元分析中的目标误差估计 显示英文标题

标题： A variational multiscale approach to goal-oriented error estimation in finite element analysis of convection-diffusion-reaction equation problems

摘要： 本文提出了一种针对线性泛函的目标型后验误差估计框架，适用于稳定化有限元离散化的定态对流扩散反应（CDR）方程。 误差估计的理论框架基于变分多尺度（VMS）概念，在该概念中，解被分解为可解析的（有限元）尺度和不可解析的（子网格）尺度。 在这项工作中，我们提出了一个正交子网格尺度（OSGS）方法用于VMS离散化中的目标型误差估计。 在OSGS方法中，子网格尺度（SGS）空间与有限元空间正交。 误差是在感兴趣的量中估计的，由未知量$Q(u)$的线性泛函$u$给出。 如果子网格尺度$u'$被估计，则感兴趣量的误差可以通过$Q(u')$近似。 我们的方法与基于伴随的后验误差估计方法进行了比较，后者需要求解一个额外的辅助问题。 结果显示两种方法产生的误差估计相似，而基于VMS的显式方法比基于伴随的隐式方法计算成本更低。 数值测试表明，我们提出的目标型误差估计技术在感兴趣的量泛函方面是有效的。 显示英文摘要

摘要： This paper presents a goal-oriented a posteriori error estimation framework for linear functionals in the stabilized finite element discretization of the stationary convection-diffusion-reaction (CDR) equation. The theoretical framework for error estimation is based on the variational multiscale (VMS) concept, where the solution is decomposed into resolved (finite element) and unresolved (sub-grid) scales. In this work, we propose an orthogonal sub-grid scale (OSGS) method for a goal-oriented error estimation in VMS discretizations. In the OSGS approach, the space of the sub-grid scales (SGSs) is orthogonal to the finite element space. The error is estimated in the quantity of interest, given by the linear functional $Q(u)$ of the unknown $u$. If the SGS $u'$ is estimated, the error in the quantity of interest can be approximated by $Q(u')$. Our approach is compared with a duality-based a posteriori error estimation method, which requires the solution of an additional auxiliary problem. The results indicate that both methods yield similar error estimates, whereas the VMS-based explicit approach is computationally less expensive than the duality-based implicit approach. Numerical tests demonstrated the effectiveness of our proposed error estimation techniques in terms of the quantity of interest functionals.