# A second order multipoint flux mixed finite element method on hybrid meshes

## TL;DR

This paper develops a second order mixed finite element method with mass lumping for simulating single phase flow in porous media on hybrid meshes, extending previous work with new analysis and computational validation.

## Contribution

It introduces a second order approximation method on hybrid meshes with a comprehensive convergence analysis and new quadrature rules, advancing numerical techniques for porous media flow.

## Key findings

- Method achieves second order accuracy in hybrid meshes.
- Convergence analysis confirms theoretical predictions.
- Computational tests validate the method's effectiveness.

## Abstract

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.