Paper
A Fully Discrete Surface Finite Element Method for the Navier--Stokes equations on Evolving Surfaces with prescribed Normal Velocity
Authors
Charles M. Elliott, Achilleas Mavrakis
Abstract
We analyze two fully time-discrete numerical schemes for the incompressible Navier-Stokes equations posed on evolving surfaces in with prescribed normal velocity using the evolving surface finite element method (ESFEM). We employ generalized Taylor-Hood finite elements -- -- , , , for the spatial discretization, where the normal velocity constraint is enforced weakly via a Lagrange multiplier , and a backward Euler discretization for the time-stepping procedure. Depending on the approximation order of and weak formulation of the Navier-Stokes equations, we present stability and error analysis for two different discrete schemes, whose difference lies in the geometric information needed. We establish optimal velocity -norm error bounds ( an energy norm) for both schemes when , but only for the more information intensive one when , using iso-parametric and super-parametric discretizations, respectively, with the help of a newly derived surface Ritz-Stokes projection. Similarly, stability and optimal convergence for the pressures is established in an -norm ( a discrete dual space) when , using a novel Leray time-projection to ensure weakly divergence conformity for our discrete velocity solution at two different time-steps (surfaces). Assuming further regularity conditions for the more information intensive scheme, along with an almost weak divergence conformity result at two different time-steps, we establish optimal -norm pressure error bounds when , using super-parametric approximation. Simulations verifying our results are provided, along with a comparison test against a penalty approach.