Speaker
Description
We present residual-based stabilization techniques for finite element methods to simulate Navier-Stokes flows in stress-divergence form on curved surfaces. The mixed formulation in velocity and pressure variables leads to the inf-sup condition. One may use Taylor-Hood elements, e.g., [1,2], or equal-order element pairs for velocity and pressure together with a stabilization scheme, e.g., the Brezzi-Pitkäranta stabilization in [3], to achieve stable results. In this work, we extend the PSPG method [4] for classical Euclidean geometries to Navier-Stokes flows on surfaces to enable consistent equal-order approximations. Furthermore, for Navier-Stokes flows at high Reynolds numbers and transport-dominated advection-diffusion problems, the SUPG stabilization technique [5] is applied to get stable solutions for stability, herein extended to convection-dominated applications on curved surfaces. A crucial aspect for flow problems on surfaces is the consideration of the tangential velocity in the governing equations, e.g., [1–3,6], which also significantly influences the formulation of the residual-based stabilization techniques. A model with a Lagrange multiplier [1,2,6] for the tangentiality constraint is compared with a projection of the velocities onto the tangent space [3,6]. Results obtained with PSPG and equal-order element pairs for velocity and pressure are compared to stable solutions computed with Taylor-Hood elements to verify the stabilized finite elements.
REFERENCES
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[5] A.N. Brooks and T.J.R. Hughes, Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comp. Methods Appl. Mech. Engrg., 32, 199-259, 1982.
[6] T. Jankuhn, M. Olshanskii, and A. Reusken, Incompressible fluid problems on embedded surfaces: Modeling and variational formulations, Interface Free Bound., 20, 353-377, 2018.
