A velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer problem

TL;DR

This paper introduces a novel velocity-vorticity-pressure formulation for the steady Navier--Stokes--Brinkman--Forchheimer equations, providing existence results, error estimates, and efficient adaptive algorithms for porous media flow simulations.

Contribution

It develops a new formulation and analysis for the Navier--Stokes--Brinkman--Forchheimer problem, including error estimates and adaptive methods for improved numerical accuracy.

Findings

Robust a posteriori error estimates demonstrated in simulations

Enhanced convergence rates with adaptive mesh refinement

Existence of solutions established for small sources

Abstract

The flow of incompressible fluid in highly permeable porous media in vorticity - velocity - Bernoulli pressure form leads to a double saddle-point problem in the Navier--Stokes--Brinkman--Forchheimer equations. The paper establishes, for small sources, the existence of solutions on the continuous and discrete level of lowest-order piecewise divergence-free Crouzeix--Raviart finite elements. The vorticity employs a vector version of the pressure space with normal and tangential velocity jump penalisation terms. A simple Raviart--Thomas interpolant leads to pressure-robust a priori error estimates. An explicit residual-based a posteriori error estimate allows for efficient and reliable a posteriori error control. The efficiency for the Forchheimer nonlinearity requires a novel discrete inequality of independent interest. The implementation is based upon a light-weight forest-of-trees data structure handled by a highly parallel set of adaptive mesh refining algorithms. Numerical simulations reveal robustness of the a posteriori error estimates and improved convergence rates by adaptive mesh-refining.