Details

Autor(en) / Beteiligte

• Diegel, Amanda E.
• Wang, Cheng
• Wang, Xiaoming
• Wise, Steven M.

Titel

Convergence analysis and error estimates for a second order accurate finite element method for the Cahn–Hilliard–Navier–Stokes system

Ist Teil von

• Numerische Mathematik, 2017-11, Vol.137 (3), p.495-534

Ort / Verlag

Berlin/Heidelberg: Springer Berlin Heidelberg

Erscheinungsjahr

2017

Link zum Volltext

Quelle

SpringerNature Journals

Beschreibungen/Notizen

• In this paper, we present a novel second order in time mixed finite element scheme for the Cahn–Hilliard–Navier–Stokes equations with matched densities. The scheme combines a standard second order Crank–Nicolson method for the Navier–Stokes equations and a modification to the Crank–Nicolson method for the Cahn–Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and a trapezoidal rule are included to help preserve the energy stability natural to the Cahn–Hilliard equation. We show that our scheme is unconditionally energy stable with respect to a modification of the continuous free energy of the PDE system. Specifically, the discrete phase variable is shown to be bounded in ℓ ∞ 0 , T ; L ∞ and the discrete chemical potential bounded in ℓ ∞ 0 , T ; L 2 , for any time and space step sizes, in two and three dimensions, and for any finite final time T . We subsequently prove that these variables along with the fluid velocity converge with optimal rates in the appropriate energy norms in both two and three dimensions.