A penalty finite element method for a fluid system posed on embedded surface

Abstract

The paper introduces a finite element method for the incompressible Navier--Stokes equations posed on a closed surface $\Gamma \subset {\mathbb{R}}^3$. The method needs a shape regular tetrahedra mesh in ${\mathbb{R}}^3$ to discretize equations on the surface, which can cut through this mesh in a fairly arbitrary way. Stability and error analysis of the fully discrete (in space and in time) scheme is given. The tangentiality condition for the velocity field on $\Gamma$ is enforced weakly by a penalty term. The paper studies both theoretically and numerically the dependence of the error on the penalty parameter. Several numerical examples demonstrate convergence and conservation properties of the finite element method.

References in corpus (2)

• Higher-order surface FEM for incompressible Navier-Stokes flows on manifolds
• Solving the incompressible surface Navier-Stokes equation by surface finite elements

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• Parametric Finite Element Discretization of the Surface Stokes Equations
• Simultaneous solution of incompressible Navier-Stokes flows on multiple surfaces
• The Ramshaw-Mesina Hybrid Algorithm applied to the Navier Stokes Equations