We consider the numerical approximation of single phase flow in porous media by a mixed finite element method with mass lumping. Our work extends previous results of Wheeler and Yotov, who showed that mass lumping together with an appropriate choice of basis allows to eliminate the flux variables locally and to reduced the mixed problem in this way to a finite volume discretization for the pressure only. Here we construct second order approximations for hybrid meshes in two and three space dimensions which, similar to the method of Wheeler and Yotov, allows the local elimination of the flux variables. A full convergence analysis of the method is given for which new arguments and, in part, also new quadrature rules and finite elements are required. Computational tests are presented for illustration of the theoretical results.

中文翻译：

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混合网格上的二阶多点通量混合有限元方法

我们通过质量集总的混合有限元方法考虑多孔介质中单相流动的数值近似。我们的工作扩展了Wheeler和Yotov的先前结果，该结果表明，与适当的基础一起占据了群众，允许在本地消除磁通变量，并以这种方式降低混合问题，仅用于压力的有限体积离散化。在这里，我们为两个和三个空间维度的混合网格构建二阶近似，类似于 Wheeler 和 Yotov 的方法，允许局部消除通量变量。给出了该方法的完全收敛分析，其中需要新的参数，部分还需要新的正交规则和有限元。计算测试用于说明理论结果。