Petrov-Galerkin Enrichment Methods for Porous Media

多孔介质的 Petrov-Galerkin 富集方法

• 批准号: 0610039
• 负责人: Leopoldo Franca
• 金额: --
• 依托单位: University of Colorado at Denver-Downtown Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0610039&HistoricalAwards=false
• 关键词: Petrov; Galerkin; Enrichment; Methods; Porous

This proposal aims to develop novel numerical methods in porous media.Initially the Darcy model is examined under the framework of aPetrov-Galerkin enriched method (PGEM). The method consists inenriching differently trial and test funtions. The test functionenrichments resembles residual-free bubbles, with zero conditionfor the normal compoenent of the velocity. This allows forstatic condensation and computation of the enrichment for thetrial function. It turns out that this enrichment is a variationof the Raviart-Thomas. This is eliminated at the element level,so that the computational formulation of the current method isin fact that of the original spaces span by continuous piecewise linearsand piecewise constants. A detailed study for the P1/P0 elementand other higher order elements will be pursued along with theirerror analyses.The methods developed in this work are key to simulate intricatelarge heterogeneuous domains which is a major challenge in porousmedia field, as in oil reservoir, contaminant transport and waterresource problems, just to cite a few of them. The methods proposed hereincontribute to decrease large scale computations since the matricesare no longer semi-definite. The methods are also competitive sincethe support of the unknows is equal to the original Galerkin method.In summary, we are pursuing more efficient methods that can overcome complex mixed methods which has dominated the field over many decades.The problems we are addressing have impact on environmental, energyand are of use to DOE.

该提案旨在开发多孔介质中的新型数值方法。首先，在佩特罗夫-伽辽金富集方法（PGEM）的框架下检查达西模型。该方法包括丰富不同的试验和测试功能。测试函数富集类似于无残留气泡，速度的法向分量为零条件。这允许静态压缩和计算试验功能的丰富性。事实证明，这种浓缩是 Raviart-Thomas 的一种变体。这在单元层面被消除，因此当前方法的计算公式实际上是由连续分段线性和分段常数组成的原始空间跨度。将对 P1/P0 元素和其他高阶元素进行详细研究及其误差分析。这项工作中开发的方法是模拟复杂的大型异质域的关键，这是多孔介质领域的主要挑战，如油藏、污染物传输以及水资源问题，仅举其中几个。由于矩阵不再是半定的，本文提出的方法有助于减少大规模计算。这些方法也具有竞争力，因为未知数的支持等于原始伽辽金方法。总之，我们正在追求更有效的方法，可以克服几十年来主导该领域的复杂混合方法。我们正在解决的问题对环境、能源并对能源部有用。