A decoupled, stable, and linear FEM for a phase-field model of variable density two-phase incompressible surface flow

The paper considers a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids. The model allows for a non-linear dependence of fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier-Stokes-Cahn-Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. Numerical examples are provided to demonstrate the stability, accuracy, and overall efficiency of the approach. Our computational study of several two-phase surface flows reveals some interesting dependencies of flow statistics on the geometry.Comment: 22 pages, 5 figures, 1 tabl

Trace finite element method for material surface flows

This dissertation studies a geometrically unfitted finite element method (FEM), known as trace FEM, for the numerical solution of the Navier-Stokes problem posed on a closed smooth material surface. The key result proved is an inf-sup stability of the discrete formulation based on standard Taylor-Hood bulk elements, with the stability constant uniformly bounded w.r.t. the mesh parameter and position of the surface in the bulk mesh. Optimal order convergence rates follow from this new stability result and interpolation properties of the trace FEM. An augmented Lagrangian preconditioner which is robust w.r.t. variation of the Reynolds number is proposed, along with an efficient recycling strategy of the velocity matrix factorization. Eigenvalue bounds for the preconditioned Schur complement are derived. Properties of the proposed method are illustrated with numerical examples which include simulation of Kelvin--Helmholtz instability at different Reynolds numbers on a sphere and torus, as well as tangential flow induced by inextensible radial deformations of a surface