Petrov-Galerkin Enrichment Methods for Porous Media

本提案旨在发展多孔介质的新型数值计算方法。首先在aPetrov-Galerkin富化方法（PGEM）框架下对Darcy模型进行了检验。该方法包括丰富不同的试验和测试函数。测试功能富集类似于无残余气泡，对速度的正常分量具有零条件。这允许静态凝聚和浓缩计算试验函数。事实证明，这种富集是拉维亚特-托马斯的一种变体。这在单元级别上消除了，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所跨越的原始空间的计算公式。P1/P0元素和其他高阶元素的详细研究将与它们的误差分析一起进行。本工作中开发的方法是模拟复杂的大型非均质域的关键，这是多孔介质领域的主要挑战，如油藏，污染物输送和水资源问题，仅举其中的几个。本文提出的方法有助于减少大规模的计算，因为矩阵不再是半确定的。由于未知量的支持度与原始伽辽金方法相等，因此该方法也具有竞争性。总之，我们正在寻求更有效的方法，以克服几十年来一直主导该领域的复杂混合方法。我们正在解决的问题对环境和能源都有影响，对美国能源部也有帮助。

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.