Solving the Incompressible Surface Stokes Equation by Standard Velocity-Correction Projection Methods.

Area of Science:

• Computational fluid dynamics
• Numerical analysis
• Partial differential equations

Background:

• Stokes equations model slow, viscous fluid flow.
• Solving these equations on curved surfaces presents unique challenges.
• Existing numerical methods may lack efficiency or accuracy for complex geometries.

Purpose of the Study:

• To develop and analyze an effective numerical algorithm for the Stokes equation on curved surfaces.
• To ensure accurate modeling of fluid flow in complex, non-planar domains.
• To provide a robust computational tool for fluid dynamics research.

Main Methods:

• Velocity-pressure decoupling using a standard velocity correction projection method.
• Introduction of a penalty term to enforce tangential velocity conditions.
• Discretization in time using first-order backward Euler and second-order BDF schemes.
• Spatial discretization employing the mixed finite element pair (P2,P1).

Main Results:

• The proposed numerical algorithm demonstrates accuracy and effectiveness.
• Stability analysis of both backward Euler and BDF time discretization schemes was performed.
• Numerical examples validated the method's performance on curved surfaces.

Conclusions:

• The presented numerical algorithm is effective for solving Stokes equations on curved surfaces.
• The combination of projection method, penalty term, and finite elements provides a robust solution.
• The study offers a reliable computational approach for fluid dynamics problems involving curved geometries.