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.

首先，在一种Petrov-Galerkin富集法(PGEM)的框架下研究了Darcy模型。该方法包括丰富不同的试验功能和测试功能。测试函数富集度类似于无残留物的气泡，速度的正常成分为零。这允许对三元函数进行静态凝聚和浓缩计算。事实证明，这种浓缩是拉维亚特-托马斯的一种变体。在单元水平上消除了这一点，从而使当前方法的计算公式实际上是由连续分段线性化和分段常数组成的原始空间跨度的计算公式。对P1/P0单元和其他高阶单元进行了详细的研究，并进行了误差分析。本文提出的方法是模拟复杂的非均质区域的关键，这是多孔介质领域的主要挑战，例如在油藏、污染物传输和水资源问题中，仅举几个例子。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的计算量。这些方法也是有竞争力的，因为未知因素的支持等同于原始的Galerkin方法。总之，我们正在寻求更有效的方法，可以克服统治该领域数十年的复杂混合方法。我们正在解决的问题对环境、能源和能源部有用。