We consider a model convection-diffusion problem and present useful connections between the finite differences and finite element discretization methods. We introduce a general upwinding Petrov-Galerkin discretization based on bubble modification of the test space and connect the method with the general upwinding approach used in finite difference discretization. We write the finite difference and the finite element systems such that the two corresponding linear systems have the same stiffness matrices, and compare the right hand side load vectors for the two methods. This new approach allows for improving well known upwinding finite difference methods and for obtaining new error estimates. We prove that the exponential bubble Petrov-Galerkin discretization can recover the interpolant of the exact solution. As a consequence, we estimate the closeness of the related finite difference solutions to the interpolant. The ideas we present in this work, can lead to building efficient new discretization methods for multidimensional convection dominated problems.

翻译：我们考虑一个模型对流扩散问题，并揭示有限差分与有限元离散方法之间的有用联系。我们引入一种基于测试空间气泡修正的广义迎风格式Petrov-Galerkin离散化方法，并将其与有限差分离散中使用的广义迎风方法联系起来。我们构建了有限差分与有限元系统，使得两个对应的线性系统具有相同的刚度矩阵，并比较两种方法的右端荷载向量。这一新方法有助于改进经典迎风有限差分方法并获得新的误差估计。我们证明指数气泡Petrov-Galerkin离散化能够恢复精确解的插值函数。由此，我们估算了相关有限差分解与插值函数的接近程度。本文提出的思想可为构建多维对流主导问题的高效新离散方法提供基础。