Robust stability and preconditioning of Darcy-Forchheimer equations

TL;DR

This paper develops robust error estimates and preconditioners for mixed finite element methods applied to nonlinear Darcy-Forchheimer equations, ensuring efficiency and stability across varying parameters.

Contribution

It introduces parameter-robust error estimates and designs efficient block preconditioners for the nonlinear Darcy-Forchheimer equations, enhancing computational stability and efficiency.

Findings

Error estimates are robust with respect to parameters and mesh size.

Preconditioners show stability across permeability and inertia variations.

Numerical examples confirm theoretical properties.

Abstract

We derive parameter-robust quasi-optimal error estimates for mixed finite element methods for the nonlinear Darcy--Forchheimer equations with mixed boundary conditions. Using the framework of operator preconditioning, we also design efficient block preconditioners for the linearised system, that exhibit robustness with respect to the coefficients that modulate permeability and inertia of the system. The properties of the formulation (parameter and mesh-size independence of the convergence rates) are illustrated by means of several numerical examples.