In this work we present an enriched Petrov-Galerkin (EPG) method for the simulation of the Darcy flow in porous media. The new method enriches the approximation trial space of the conforming continuous Galerkin (CG) method with bubble functions and enriches the approximation test space of the CG method with piecewise constant functions, and it does not require any penalty term in the weak formulation. Moreover, we propose a framework for constructing the bubble functions and consider a decoupled algorithm for the EPG method based on this framework, which enables the process of solving pressure to be decoupled into two steps. The first step is to solve the pressure by the standard CG method, and the second step is a post-processing correction of the first step. Compared with the CG method, the proposed EPG method is locally mass-conservative, while keeping fewer degrees of freedom than the discontinuous Galerkin (DG) method. In addition, this method is more concise in the error analysis than the enriched Galerkin (EG) method. The coupled flow and transport in porous media is considered to illustrate the advantages of locally mass-conservative properties of the EPG method. We establish the optimal convergence of numerical solutions and present several numerical examples to illustrate the performance of the proposed method.

翻译：本文提出了一种增强型Petrov-Galerkin (EPG) 方法，用于模拟多孔介质中的达西流动。新方法通过气泡函数增强相容连续Galerkin (CG) 方法的逼近试验空间，并通过分片常数函数增强CG方法的逼近检验空间，且其弱形式无需任何惩罚项。此外，我们提出了一种构建气泡函数的框架，并基于该框架设计了EPG方法的解耦算法，使压力求解过程可分解为两个步骤：第一步采用标准CG方法求解压力，第二步为第一步的后处理校正。与CG方法相比，所提出的EPG方法具有局部质量守恒特性，同时相比间断Galerkin (DG) 方法保持更少的自由度。此外，该方法在误差分析上比增强型Galerkin (EG) 方法更简洁。我们考虑了多孔介质中耦合流动与输运问题，以说明EPG方法局部质量守恒特性的优势。我们建立了数值解的最优收敛性，并通过多个数值算例展示了所提方法的性能。