 ## Finite Elements and Calculus of Variations, 2003

Below find a list of the topics to be discussed in class. The references refer to the lecture notes to be handed out. During the semester the order of presentation may change, the changes should show on this page.

### Main goal of this class:

Understand the basic mathematical background of the Finite Element Method (FEM) and some aspects of an efficent implementation on computers

### Main topics to be discussed

• formulate a physical or mechanical problem as a Partial Differential Equation (PDE) with boundary conditions
• determine the FEM formulation of the PDE, create the system of linear equations to be solved
• analyze and visualize the computed solution
• how to solve large systems of linear equations
• how to estimate and control the unavoidable numerical approximation errors

### Lecture notes and a tentative program

The lectures will be based on two sets of lecture notes
•  Calculus of Variations and Finite Elements, Andreas Stahel
•  Applications of Partial Differential Equations, Andreas Stahel
A possible sequence of topics to be treated is
1. The PDE for a heat equation 
2. solving the heat equation, using FEMLAB
3. From a minimization problem to a system of linear equations 
4. a system of simple bars 
5. 1-d Calculus of variations
6. 1-d FEM for second order problems, a first attempt and an efficient algorithm
7. 2-d calculus of variations
8. 2-d FEM for second order problems
9. coding in Octave, FEMoctave
10. fast solvers for linear systems, Gauss, Cholesky, a sparse solver
11. error estimates
12. dynamic problems, heat and wave equation
13. consistency, stability, convergence
Software to be used
• FEMLAB, based on Matlab, fully commercial
• FEMoctave, based on Octave, open source, free
• BVP2, based on Mathematica, free Go back to Home Page of Andreas Stahel

October, 2003 by Andreas.Stahel@hta-bi.bfh.ch
I take neither credit nor responsibility for the above e-mail address