# On high-order conservative finite element methods

## TL;DR

This paper introduces a high-order conservative finite element method for Darcy flow that ensures mass conservation without mesh-dependent parameters, suitable for complex and heterogeneous problems.

## Contribution

The paper presents a novel residual-based, high-order finite element method using Lagrange multipliers for mass conservation, extendable to 3D and heterogeneous media.

## Key findings

- Achieves high-order convergence
- Provides locally conservative fluxes
- Applicable to 3D and heterogeneous problems

## Abstract

A new high-order conservative finite element method for Darcy flow is presented. The key ingredient in the formulation is a volumetric, residual-based, based on Lagrange multipliers in order to impose conservation of mass that does not involve any mesh dependent parameters. We obtain a method with high-order convergence properties with locally conservative fluxes. Furthermore, our approach can be straightforwardly extended to three dimensions. It is also applicable to highly heterogeneous problems where high-order approximation is preferred.