Trace finite element method for material surface flows

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.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2022

Author dc:creator

• Zhiliakov, Alexander

Advisor dc:contributor.advisor

• Olshanskii, Maxim A.

Committee members dc:contributor.committeemember

• Mamonov, Alexander V.
• Mang, Andreas
• Demlow, Alan

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

Statement dc:rights

• The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/10657/13640

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/13640