CME358: The Finite Element Method for Fluid Mechanics

Course Outline

Introduction

• Incompressible fluid mechanics.
• Reminder on finite element in the coercive frameworks (Lax- Milgram, Sobolev spaces, Lagrangian finite elements).
• Why the coercive framework is not sufficient in many applications.

Theory for mixed problems

• Continuous case, inf-sup condition.
• Application to Stokes problem.
• Link with optimization under constraints. Saddle point problems.

Stable finite elements for mixed problems

• Convergence analysis.
• Algebraic aspects. Uzawa algorithm. Conditioning.

Stable finite elements for mixed problems

• Fortin lemma. Analysis of P1-bubble-P1 finite element.
• Other examples of finite elements.
• Other examples of finite elements.

Stabilized finite element

• The advection diffusion case.

Introduction to Navier-Stokes: projection algorithms