Partial Differential Equations
This is the first course on partial differential equations (PDE) with applications in sciences and engineering. The objective is two-fold: To cover an up-to-date basic theory and to introduce some modern approximation tools. In the theoretical part we study existence, uniqueness and stability concepts for the basic PDEs: Poisson, heat, and wave equations. As for the approximation we focus on constructing and analysing Galerkin methods from two point of views. On one hand we consider the numerical analysis aspects of the approximation procedure: such as, variational principle, minimization problem and representation theorems, on the other hand we deal with the important implementation aspects of a priori and a posteriori error estimates, and construction of numerical algorithms deriving, e.g., stiffness-, mass- and convection matrices.