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

9707015托德阿伯加斯特 混合有限元法的后验误差估计及尺度放大 本计画系关于在逻辑矩形网格上，用混合有限元素法来逼近二阶椭圆型与抛物型偏微分方程。 第一个目标是开发一个本地后验误差估计或指标，使空间误差可以本地化。 我们建议利用矩形元的混合和非协调方法之间的等价性来估计误差。 第二个目标是为高度可变系数和点状源开发放大或均匀化技术，即，用于解析模型系统中低于实际计算网格尺寸的精细长度尺度。 与使用这些放大技术相关的误差必须是可量化的。 我们建议我们的技术的基础上的离散方程，并使用粗规模的Raviart-Thomas投影算子，保留通量在任何元素的脸。 第三个目标是证明在实际环境中的技术的适用性，我们认为地下水流的模拟。 前两个目标是相辅相成的。 误差估计将允许我们在解表现不佳的地方细化网格，如在尖锐前沿附近、局部异质性或源（即，威尔斯），从而可以集中计算工作来解析数据和解中的主要长度尺度。 放大将使我们能够进一步解决网格尺寸以下的一些尺度。 我们对地下流体流动的理解对一系列活动都很重要，包括地下水污染的清理和石油天然气生产。 地下水供应日益受到因处置不当或意外排放而进入环境的污染物的威胁。 近年来，美国石油产量大幅下降。 这些问题可以通过复杂的工程过程来改善，这些过程需要仔细的设计和监控，而这又取决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，以便我们能够进一步预测物理，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。 由于多种原因，很难做到这一点，但最基本的是缺乏分辨率：我们只能使用数据并计算空间中少量网格点处的流体速度。 到目前为止，还没有有效的方法来估计流体速度近似的误差。 如果我们知道我们的误差很大，我们可以通过增加网格分辨率来采取纠正措施，直到可用计算机资源的极限。 超过这个极限，就有必要用一些适当定义的平均量来代替网格尺度下某些非常小尺度的量（如岩石性质的局部变化）。 我们解决这些问题，在这个建议，并证明我们的技术在实际环境中的适用性，以获得工程师的认可。

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.