2025/12/12 by Elliott, Charles M., Mavrakis, Achilleas
#65N12 (Primary) 65N15 #65N30 (Secondary) #FOS: Mathematics #Numerical Analysis (math.NA)
paper · doi:10.48550/arxiv.2512.11737
We analyze two fully time-discrete numerical schemes for the incompressible Navier-Stokes equations posed on evolving surfaces in ℝ3 with prescribed normal velocity using the evolving surface finite element method (ESFEM). We employ generalized Taylor-Hood finite elements Pku-- P_kpr-- Pkλ, ku=kpr+1 ≥ 2, kλ≥ 1, for the spatial discretization, where the normal velocity constraint is enforced weakly via a Lagrange multiplier λ, and a backward Euler discretization for the time-stepping procedure. Depending on the approximation order of λ 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 L2ah-norm error bounds (ah an energy norm) for both schemes when kλ=ku, but only for the more information intensive one when kλ=ku-1, 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 L2L2× L2Hh-1-norm (Hh-1 a discrete dual space) when kλ=ku, 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 L2L2× L2L2-norm pressure error bounds when kλ=ku-1, using super-parametric approximation. Simulations verifying our results are provided, along with a comparison test against a penalty approach.