Petrov-Galerkin Enrichment Methods for Porous Media

本研究的目的是发展新的数值方法，首先在一个Petrov-Galerkin富集方法（PGEM）的框架下研究达西模型。该方法包括丰富不同的试验和测试功能。试验函数的富集类似于无残余气泡，其正常速度分量为零。这允许静态浓缩和计算的丰富的尝试功能。事实证明，这种富集是拉维亚-托马斯的一种变体。这是消除在元素级，使目前的方法的计算公式，实际上是由连续的分段线性和分段常数的原始空间跨度。沿着对P1/P0单元和其他高阶单元的详细研究以及它们的误差分析，在这项工作中开发的方法是模拟复杂的大非均匀域的关键，这是多孔介质领域的一个主要挑战，如油藏，污染物运移和水资源问题，仅举几例。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。这些方法也是有竞争力的，因为未知数的支持等于原始的Galerkin方法。总之，我们正在寻求更有效的方法，可以克服复杂的混合方法，这些方法已经主导了该领域几十年。我们正在解决的问题对环境，能源和能源都有影响。

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.