A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation Journal Article


Author(s): Fischer, Julian
Article Title: A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation
Affiliation
Abstract: We derive a posteriori estimates for the modeling error caused by the assumption of perfect incompressibility in the incompressible Navier-Stokes equation: Real fluids are never perfectly incompressible but always feature at least some low amount of compressibility. Thus, their behavior is described by the compressible Navier-Stokes equation, the pressure being a steep function of the density. We rigorously estimate the difference between an approximate solution to the incompressible Navier-Stokes equation and any weak solution to the compressible Navier-Stokes equation in the sense of Lions (without assuming any additional regularity of solutions). Heuristics and numerical results suggest that our error estimates are of optimal order in the case of "well-behaved" flows and divergence-free approximations of the velocity field. Thus, we expect our estimates to justify the idealization of fluids as perfectly incompressible also in practical situations.
Keywords: A posteriori estimate; Compressible fluid; Incompressible limit; Modeling error; Navier-Stokes equation
Journal Title: SIAM Journal on Numerical Analysis
Volume: 53
Issue 5
ISSN: 1095-7170
Publisher: Society for Industrial and Applied Mathematics  
Date Published: 2015-01-01
Start Page: 2178
End Page: 2205
Sponsor: The research of the author was supported by the Lithuanian-Swiss cooperation program under the project agreement CH-SMM-01/0.
DOI: 10.1137/140966654
Open access: no