Trace finite element method for material surface flows



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This dissertation studies a geometrically unfitted finite element method (FEM), known as trace FEM, for the numerical solution of the Navier-Stokes problem posed on a closed smooth material surface. The key result proved is an inf-sup stability of the discrete formulation based on standard Taylor-Hood bulk elements, with the stability constant uniformly bounded w.r.t. the mesh parameter and position of the surface in the bulk mesh. Optimal order convergence rates follow from this new stability result and interpolation properties of the trace FEM. An augmented Lagrangian preconditioner which is robust w.r.t. variation of the Reynolds number is proposed, along with an efficient recycling strategy of the velocity matrix factorization. Eigenvalue bounds for the preconditioned Schur complement are derived. Properties of the proposed method are illustrated with numerical examples which include simulation of Kelvin--Helmholtz instability at different Reynolds numbers on a sphere and torus, as well as tangential flow induced by inextensible radial deformations of a surface.



Material surfaces, Fluidic membranes, Surface Navier-Stokes problem, Augmented Lagrangian preconditioner, Grad-div stabilization, Trace, finite element method, Kelvin-Helmholtz instability


Portions of this document appear in: Olshanskii, Maxim, Arnold Reusken, and Alexander Zhiliakov. "Inf-sup stability of the trace 𝐏₂–𝐏₁ Taylor–Hood elements for surface PDEs." Mathematics of Computation 90, no. 330 (2021): 1527-1555; and in: Olshanskii, Maxim A., and Alexander Zhiliakov. "Recycling augmented Lagrangian preconditioner in an incompressible fluid solver." Numerical Linear Algebra with Applications 29, no. 2 (2022): e2415; and in: Olshanskii, Maxim A., Arnold Reusken, and Alexander Zhiliakov. "Tangential Navier-Stokes equations on evolving surfaces: Analysis and simulations." arXiv preprint arXiv:2203.01521 (2022).