A modified Brinkman penalization fictitious domain method for the unsteady Navier-Stokes equations

TL;DR

This paper presents a modified fictitious domain method with a solution-dependent parameter choice for unsteady Navier-Stokes equations, achieving higher-order convergence and strong mass conservation in numerical simulations.

Contribution

It introduces a new modification enabling solution-dependent parameter selection, proves existence and convergence results, and develops a divergence-free finite element discretization for improved accuracy.

Findings

Proved global-in-time existence and convergence of weak solutions.

Established local-in-time existence and uniqueness of strong solutions.

Demonstrated higher-order convergence rates and mass conservation in numerical tests.

Abstract

This paper investigates a modification of the fictitious domain method with continuation in the lower-order coefficients for the unsteady Navier-Stokes equations governing the motion of an incompressible homogeneous fluid in a bounded 2D or 3D domain. The modification enables {a solution-dependent} choice of the critical parameter. Global-in-time existence and convergence of a weak solution to the auxiliary problem are proved, and local-in-time existence and convergence of a unique strong solution are established. For the strong solution, a new higher-order convergence rate estimate in the penalization parameter is obtained. The introduced framework allows us to apply a pointwise divergence free finite element method as a discretization technique, leading to strongly mass conservative discrete fictitious domain method. A numerical example illustrates the performance of the method.