Linear algebra (eigenvectors, eigenvalues, matrix algebra, Gershgorin's circle theorem, inner products, projections)
Calculus (differentiation, integration, integration over lines, surfaces and domains, integral theorems (Gauss, Green))
Partial differential equations (definition, heat, Laplace, Poisson, wave equation))
Introductory numerical analysis (numerical time integration, interpolation, finite differences, quadrature, approximation methods for nonlinear equations)
Aim of the course
After completion of the course, the student will be able to construct and to use finite-element methods to solve partial differential equations. Furthermore, the student will be able to assess the quality of the obtained numerical approximations.
The course aims at learning how to apply and construct finite-element methods to various kinds of partial differential equations. The emphasis will be on the application and implementation of the finite-element methods. Theoretical issues will be assessed superficially and only if needed. The finite-element formalisms due to Ritz and Galerkin will be treated. The course will include linear, quadratic, bilinear elements for time-independent and time dependent problems. Several lab assignments will be helpful in gaining understanding in the development of finite-element methods.
Rules about Homework / Exam
There will be two series of obligatory take-home assignments, a lab assignment report and a written test. The written test counts for 60% and the lab report for 40% of the total grade. The grade of the written exam should be at least 5 (five). If the two take-home assignments are satisfactory, then at most one bonus point is obtained for the written exam. You may hand in the take-home assignments of couples of two participants.
The lab assignments consist of two 1D-programming exercises and one 2D-programming task. The 2D-programming task has to be finalized with a written report of format: Introduction - Mathematical Model - Finite Element Strategy - Numerical Results - Conclusion
The two 1D lab assignments can be handed in as two short appendixes to the final report of the 2D-assignment.
Lecture Notes / Literature
J. van Kan, A. Segal, F. Vermolen. Numerical methods in scientific computing, Delft Academic Press, second edition, 2014