A nonconforming method for a generalized Darcy-Forchheimer model

TL;DR

This paper introduces a new dual mixed nonconforming discretization for the generalized Darcy-Forchheimer model, accommodating complex nonlinearities, boundary conditions, and permeability tensors, with proven convergence and error estimates.

Contribution

It extends previous schemes by handling nonquadratic nonlinearities, inhomogeneous boundary conditions, and lower regularity permeability tensors, with general-order schemes and convergence proofs.

Findings

Convergence to the exact solution under low regularity assumptions.

Error estimates for schemes assuming extra regularity.

Numerical results demonstrating scheme performance for various nonlinearities.

Abstract

We analyze a dual mixed nonconforming discretization of a generalized Darcy-Forchheimer model. Compared to the analogous scheme proposed by Girault and Wheeler, we consider general, i.e., nonquadratic, Forchheimer nonlinearities; we admit mixed, inhomogeneous boundary conditions; we allow for more general, i.e., with lower Lebesgue regularity, permeability tensors; we construct general-order schemes; we prove convergence to the exact solution under low regularity assumptions, based on novel Sobolev-trace inequalities for broken spaces; we derive error estimates of general-order assuming extra regularity of the exact solution and data; we present numerical results assessing the performance of the proposed schemes for different types of nonlinearity and nonlinear solvers.