Unconditionally Stable Mixed Finite Element Methods for Darcy Flow

TL;DR

This paper introduces unconditionally stable mixed finite element methods for Darcy flow that are parameter-free, flexible in element choice, and backed by theoretical stability and convergence analysis with supporting numerical experiments.

Contribution

It presents a novel class of finite element methods for Darcy flow that are unconditionally stable, parameter-free, and compatible with classical Lagrangian spaces, with proven stability and convergence.

Findings

Methods are unconditionally stable and mesh-independent.

Theoretical stability and convergence are established.

Numerical experiments confirm predicted convergence rates.

Abstract

Unconditionally stable finite element methods for Darcy flow are derived by adding least-squares residual forms of the governing equations to the classical mixed formulations. The proposed methods are free of mesh dependent stabilization parameters and allow the use of the classical continuous Lagrangian finite element spaces of any order for the velocity and the potential. Stability, convergence and error estimates are derived and numerical experiments are presented to demonstrate the flexibility of the proposed finite element formulations and to confirm the predicted rates of convergence.