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Divergence-free tangential finite element methods for incompressible flows on surfaces
Philip L Lederer1, Christoph Lehrenfeld2, Joachim Schöberl1
1Institute for Analysis and Scientific Computing TU Wien Vienna Austria.
International Journal for Numerical Methods in Engineering
|December 2, 2021
Summary
This study presents new finite element methods for solving incompressible fluid flows on curved surfaces. These methods ensure accurate velocity approximations on manifolds, improving numerical simulations for complex geometries.
Area of Science:
- Computational fluid dynamics
- Numerical analysis
- Differential geometry
Background:
- Solving incompressible fluid flow problems on manifolds presents unique challenges.
- Standard finite element methods struggle with velocity fields constrained to tangential spaces of curved geometries.
- Approximating divergence-free velocity fields on surfaces requires specialized techniques.
Purpose of the Study:
- To develop and analyze novel finite element discretizations for incompressible flows on two-dimensional manifolds.
- To address the challenges of approximating tangential velocity fields and ensuring pressure-velocity compatibility on curved surfaces.
- To introduce methods that yield exactly tangential and divergence-free velocity solutions.
Main Methods:
- Utilizing Piola transformations to construct exactly tangential finite elements.
- Employing (hybrid) discontinuous Galerkin techniques to enforce weak continuity.
- Incorporating Nédélec (or similar) conforming finite elements for divergence-free solutions.
- Developing new finite element discretizations tailored for manifold geometries.
Main Results:
- New finite element methods are presented for incompressible flows on manifolds.
- The proposed methods successfully handle the approximation of tangential velocity fields.
- The use of discontinuous Galerkin and Nédélec elements leads to accurate, divergence-free velocity solutions.
- Numerical examples demonstrate the qualitative properties and accuracy of the developed discretizations.
Conclusions:
- The developed finite element methods offer a robust approach for simulating incompressible flows on two-dimensional manifolds.
- Abandoning H1-conformity and using Piola transformations enables exact tangentiality.
- Discontinuous Galerkin and Nédélec elements are effective for achieving weak continuity and divergence-free solutions.
- The study provides validated numerical examples showcasing the effectiveness of the new discretizations.
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