CME358: The Finite Element Method for Fluid Mechanics

Course Description

This course presents the basic mathematical theory of the finite element method for incompressible flows. It also covers related computational algorithms and computer implementation details. It is intended primarily for graduate students interested either in developing modern and rigorous skills in the numerical solution of fluid mechanics problems, or in developing further their basic skills in the finite element methodology. Using the Poisson equation as a background problem, the course begins with a fast review of the basic finite element method for simple elliptic problems. Next, it explains why this basic theory is insufficient for problems such as the mixed formulation of elliptic equations, incompressible flows, the advection-diffusion problem, and the Navier-Stokes equations. To address such problems, the course continues with notions of the mathematical analysis of non coercive partial differential equations, the inf-sup (or Babuska-Brezzi) condition and its application to the Stokes and Darcy problems, and a presentation of stable mixed finite element methods and corresponding algebraic solvers. Stabilization approaches are then discussed in the context of the advection-diffusion equation. Finally, the numerical solution of the incompressible Navier-Stokes equations by a suitable finite element method is covered. The course material described above is complemented by a balanced set of theoretical, computational, and Matlab computer programming homeworks.

Contact Information


Professor Jean-Frederic Gerbeau
Email: jean-frederic.gerbeau AT inria DOT fr
Office: Durand 028
Office Hours: Wednesdays 4:00 to 5:00 PM

Course Assistants:

Paolo Massimi
Email: pmassimi AT stanford DOT edu
Office: Durand 023C
Office Hours: Wednesdays 4:00 to 5:00 PM
David Amsallem
Email: amsallem AT stanford DOT edu
Office: Durand 028
Office Hours: Wednesdays and Thursdays 4:00 to 5:00 PM


Time: Tuesdays and Thursdays 1:15 to 3:05 PM
Location: Rm 240-110


  • Solid foundations in mutivariable calculus.
  • ME335A or equivalent.
  • Also, graduate students interested in this course are advised to take ME412/CME356 during the previous quarter or earlier.


    There is no required textbook for this class.


    There will be assignments due roughly every two weeks. Collaboration among students is encouraged. You should feel free to discuss problems with your fellow students (please document on each assignment with whom you worked). However, you must write your own solutions, and copying homework from another student (past or present) is forbidden. The Stanford Honor Code will apply to all assignments, both in and out of class.


    There will be no midterm. There will be a final exam. It is open notes and open book.


    The course grade will be based on assignments (50%) and the final exam (50%).