Partial Differential Equations

This is the first course on partial differential equations (PDE) with applications in science and engineering. The objective of the course is two-fold: To introduce a theoretical foundation for classical PDEs such as Poisson's equation and the heat and wave equations and to introduce some modern approximation tools. In the theoretical part we study existence, uniqueness and stability concepts of model PDEs. As for the approximation tools, we focus on constructing and analysis of Galerkin methods. On one hand we consider the numerical analysis of approximation procedures such as: the variational principle, the minimization problem and representation theorems. On the other hand we deal with important implementation aspects such as a priori and a posteriori error estimates, and construction of numerical algorithms deriving, e.g., stiffness-, mass- and convection matrices.