Finite Elements and Calculus of Variations

Below you find a possible sequence of topics to be examined in this class. Lecture notes will be prepared. The notes from last year (referred to as [N]) will serve as a basis, but a major revision is ahead.

The main goal of this optional class is to understand the mathematical background of the Finite Element Method and to develop code to solve a typical problem. The code in class is given in Mathematica and it is up to each student to choose his/her favourite language for his/her implementation.

The student is expected to write code to solve a typical problem with the Finite Element Method and to solve a few homework problems.

• A typical finite element problem
  • set up a typical heat conduction problem, mathematical description as a differential equation
  • use Matlab to find a numerical solution of the problem,
    learn how to use the PDE-toolbox with this sample problem.
  • identify the essential steps
    1. description of the domain
    2. description of the differential equation and boundary conditions
    3. meshing
    4. setting up of the system of linear equations and solving the system
    5. interpretation of the results, graphically and quantitatively
  • show that the solution of the heat equation corresponds to the minimum of an integral expression
• The connection between extremal problems and systems of linear equations [N §1]
• Finite Element Method applied to a stretched bar [N §2]
  This problem is to be examined very carefully.
• Introduce the basic ideas of the Calculus of Variations (use parts of [N §3]), apply it to the heat conduction problem.
• Now go back to the heat equation of the first section and examine all steps and try to understand the mathematical background. As a test code to solve the above problem will be developed.
  1. compute energy on a single triangular element, construct the element stiffness matrix
  2. the mesh is to be generated by the code EasyMesh. This code is avaliable as C source and also as a DOS/WIN executable. It can be used to describe domains in the plane and generate a triangulation. Code to read the corresponding output will be given in Mathematica
  3. combine the above to generate a global stiffness matrix
  4. incorporate the boundary conditions
  5. solve the system of linear equations and generate illustrative graphics
• Illustrate some of the finer points of the Finite Element Method
  use [N §4]
  • numerical integration
  • higher order methods
  • Gauss integration
• Instead of the heat problem another type of problem may be chosen by each student, e.g. electrostatic problems. For details contact the instructor.