Trace finite element method for material surface flows

Abstract

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.