A finite element method for the surface Stokes problem (1801.06589v1)

Abstract: We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a Trace finite element method (TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element spaces defined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on $P_1$ bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface we introduce a penalty term. The method is straightforward to implement and has an $O(h^2)$ geometric consistency error, which is of the same order as the approximation error due to the $P_1$--$P_1$ pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface $H^1$ and $L^2$ norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.

• The paper introduces the Trace Finite Element Method (TraceFEM), a novel approach for solving the surface Stokes problem on 2D surfaces embedded in 3D using a fixed background mesh and bulk finite elements.
• TraceFEM utilizes P1 elements for velocity and pressure approximation and a penalty term to enforce tangential flow, avoiding complex surface parametrization and mesh fitting.
• Numerical analysis demonstrates the method's stability and optimal error bounds (O(h^2) geometric error), validated by experiments and stabilization techniques for pressure approximation.

Overview of "A finite element method for the surface Stokes problem"

The paper by Olshanskii et al. presents a finite element method tailored for the surface Stokes problem, a mathematical formulation describing a fluid flow constrained on two-dimensional surfaces embedded within a three-dimensional domain. This research addresses both practical aspects of numerical simulation and theoretical challenges related to the Stokes equations adapted to curved surfaces, which are crucial in the modeling of complex fluidic phenomena such as those occurring in emulsions, foams, and biological membranes.

Methodological Framework

The authors propose the Trace Finite Element Method (TraceFEM), which operates on a fixed, tetrahedral background mesh that is independent of the surface. The key innovation here is the method's reliance on bulk finite elements, specifically the $P_1$ elements, to approximate both velocity and pressure. This approach circumvents the need for mesh fitting or surface parametrization, which are often cumbersome in fluid simulations on complex geometries. A penalty term is introduced to ensure that the computed velocity field adheres to the surface tangentially, effectively enforcing the Stokes equations' constraints on the surface.

Numerical Analysis and Error Evaluation

The researchers provide a comprehensive mathematical analysis of TraceFEM, demonstrating its stability and optimal order discretization error bounds in both the surface $H^1$ and $L^2$ norms. They provide numerical experiments to validate their theoretical findings, showing the method's auspicious performance on various test problems. Significantly, they highlight that the geometric error stemming from surface approximation is $O(h^2)$, aligning with the approximation error's order for the employed finite element pair.

A crucial aspect discussed is the stabilization mechanism for the $P_1$--$P_1$ finite element setup, achieved through a Brezzi--Pitkäranta stabilization technique. This ensures the inf-sup condition necessary for stable and reliable pressure approximation.

Implications and Future Directions

The introduction of TraceFEM represents a notable advancement in computational fluid dynamics for manifold-based problems. By sidestepping the challenges involved in surface parametrization, the proposed method broadens the scope of potential applications, extending to dynamic interfaces with complex evolution patterns—a frequent requirement in both natural and industrial scenarios.

The authors hint at future explorations involving the extension of their method to evolving surfaces, which will necessitate further research into the interplay between the dynamic changes of the manifold geometries and the underlying computational mesh. Moreover, investigating higher order finite element methods and alternative stabilization techniques stands to enhance the robustness and applicability of TraceFEM in diverse and challenging fluid flow contexts.

Conclusion

The paper successfully demonstrates the efficacy of a systematically derived finite element method tailored for surface flows, opening avenues for robust simulations of fluid dynamics on complex geometries without direct manipulation of the computational mesh relative to the manifold. Future research could potentially expand TraceFEM's capabilities, offering even more refined tools for tackling intricate interfaces in both static and dynamic environments.