TOPICS:
- A quick introduction to Sobolev spaces, Lax-Milgram Lemma; Review of Elliptic PDE: existence of unique weak solution, regularity.
- Quick review of FEM: Cea's Lemma Mixed FEM, Aubin-Nitsche deality technique, convergence rates.
- A posteriori error analysis, reliability, efficiency, adaptive mesh refinement, contraction property, discrete reliability, optimality of closure algorithm, approximation classes, optimality of adaptive finite element algorithms. Adaptive finite element methods for elliptic PDEs: basic concepts and aposteriori estimators, Adaptive Galerkin Discretion of mixed finite element methods.
- Discontinous Petrov Galerkin Methods (DPG): Introduction of the DPG method as a Petrov-Galerkin method, connection to least squares methods, a mixed method reformulation, theory of a priori error estimates for the DPG method, Application to the Laplace operator.
- Scientific Labs: MATLAB implementation of the FEM and Adaptive FEM.