- 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 particpant 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. The finite-element formalisms due to Ritz and Galerkin will be treated. The course will include linear, quadratic, bilinear Lagrangian elements for time-independent and time dependent problems. Next to Galerkin frameworks, Petrov-Galerkin frameworks will be considered for the treatment of convection-dominated cases. Theoretical issues will be assessed in terms of convergence and error analysis, although this will not be the main focus of the course. Several lab assignments will be helpful in gaining understanding in the development of finite-element methods.
After completion of the course, the participant 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.
Fred Vermolen (TUD), Deepesh Toshniwal (TUD), Jaap van der Vegt (UT)