A Posteriori Error Estimation and Up-Scaling for Mixed Finite Element Methods

[707015] Todd Arbogast混合有限元方法的后置误差估计和上尺度问题。本课题研究了在逻辑矩形网格上用混合有限元方法逼近二阶椭圆型和抛物型偏微分方程。第一个目标是开发一个局部后验误差估计器或指示器，以便空间误差可以局部化。我们提出利用矩形单元混合和非协调方法之间的等效性来估计误差。第二个目标是为高度可变系数和点状源开发上尺度或均匀化技术，即用于解决模型系统中小于实际计算网格大小的精细长度尺度。与使用这些放大技术相关的误差必须是可量化的。我们建议将我们的技术建立在离散方程的基础上，并使用粗糙尺度的Raviart-Thomas投影算子，该算子保留了在任何单元面上的通量。第三个目标是证明这些技术在实际环境中的适用性；我们考虑地下流的模拟。前两个目标是互补的。误差估计将使我们能够在解决方案表现不佳的地方改进网格，如靠近尖锐前沿，局部异质性或源（即井），以便计算工作可以集中在解决数据和解决方案中的主要长度尺度上。向上缩放将允许我们进一步解决一些低于网格大小的尺度。我们对地下流体流动的了解对一系列活动都很重要，包括地下水污染的清理和石油和天然气的生产。地下水供应日益受到不当处置或意外排放而进入环境的污染物的威胁。近年来，美国的石油产量明显下降。这些问题可以通过复杂的工程过程得到改善，这些过程需要仔细的设计和监控，而这又依赖于我们在计算机上模拟地下流体运动的能力。我们的计算机模拟必须足够详细，这样我们才能进一步预测物理、化学和生物过程以及人类干预的后果。这种模拟要求我们精确地近似控制流体运动和相互作用的微分方程。由于许多原因，很难做到这一点，但最基本的原因是缺乏分辨率：我们只能使用数据并计算空间中少数网格点的流体速度。迄今为止，还没有有效的方法来估计流体速度近似值的误差。如果我们知道我们的错误很大，我们可以通过增加网格分辨率来采取纠正措施，直到可用的计算机资源的极限。在此限制之外，有必要用一些适当定义的平均量来代替网格尺度下的某些非常小的量（例如岩石性质的局部变化）。我们在本提案中解决了这些问题，并展示了我们的技术在实际环境中的适用性，以获得工程师的接受。

9707015 Todd Arbogast A POSTERIORI ERROR ESTIMATION AND UP-SCALING FOR MIXED FINITE ELEMENT METHODS This project concerns the approximation of second order elliptic and parabolic partial differential equations by mixed finite element methods on logically rectangular grids. The first objective is to develop a local a posteriori error estimator or indicator, so that spatial errors can be localized. We propose to estimate the error by exploiting the equivalence between mixed and non-conforming methods for rectangular elements. The second objective is to develop up-scaling or homogenization techniques for highly variable coefficients and point-like sources, i.e., for resolving fine length scales in the model system that are below the size of a practical computational mesh. The error associated with using these up-scaling techniques must be quantifiable. We propose to base our techniques on the discrete equations, and use the coarse scale Raviart-Thomas projection operator which preserves the flux across any element face. The third objective is to demonstrate the applicability of the techniques in a practical setting; we consider the simulation of subsurface flow. The first two objectives are complementary. Error estimation would allow us to refine the mesh where the solution is ill behaved, as near sharp fronts, local heterogeneity, or sources (i.e., wells), so that computational effort can be concentrated to resolve the major length scales in both the data and the solution. Up-scaling would allow us to further resolve some scales below the mesh size. Our understanding of fluid flow underground is important to a range of activities, including the clean-up of ground-water contamination and oil and gas production. Ground-water supplies are increasingly threatened by contaminants introduced into the environment by improper disposal or accidental release. U.S. petroleum production has declined markedly in recent years. These problems can be ameliorated by complex engineering processes that require careful design and monitoring, which in turn depend on our ability to simulate on a computer the movement of fluids underground. Our computer simulations must be sufficiently detailed that we can further predict physical, chemical, and biological processes and the consequences of human intervention. Such simulation requires that we approximate accurately the differential equations governing the movement and interaction of the fluids. It is difficult to do this for a number of reasons, but the most basic is a lack of resolution: we can only use data and compute fluid velocities at a small number of grid points in space. To date there has been no effective way to estimate the error in the approximation to the fluid velocity. If we knew that our errors were large, we could take corrective action by increasing the grid resolution, up to the limits of the computer resources available. Beyond that limit, it is necessary to approximate certain very small-scale quantities below the grid scale (such as local variations in rock properties) by replacing them by some appropriately defined average quantities. We address these concerns in this proposal, and demonstrate the applicability of our techniques in practical settings to gain their acceptance by engineers.