Mathematics

This course provides a high level overview on the numerical solution of Partial Differential Equations (PDEs). The course focuses on the Finite Element Methods (FEMs) but insights on all major classes of numerical methods will be discusses. Numerical methods will be presented and analysed within a rigorous mathematical setting. Key aspects such as consistency, stability, and convergence will be thoroughly investigated, providing the guidelines for the correct choice and implementation of numerical methods for a range of problems. The lectures will be completed by computer classes based on the Python language. These are hands-on sessions where codes are produced to test the properties of the numerical algorithms seen in the lectures, helping everyone to build their own suits of numerical codes.Topics.1. Elliptic problems. Finite Differences discretisation; Discrete Maximum principle, consistency, stability, and convergence.2. Basic notions on functional spaces. Weak formulations; Lax-Milgram lemma.3. The method of Galerkin; Lemma of Cea.4. Finite Element Methods (FEM).5. Interpolation; Bramble-Hilbert lemma. Nitsche duality trick.6. Conditioning.7. A posteriori error bounds.8. Generalised Galerkin method; the lemma of Strang.9. Convection-reaction-diffusion problems; the streamline diffusion method.10. Parabolic problems. Weak formulations; well-posedness, energy estimates.11. Finite Differences discretisation; consistency, stability, and convergence.12. FEM for parabolic problems; elliptic projection, convergence analysis.13. Discontinuous Galerkin time-stepping.14. Other methods. This course is followed up by the course Advanced topics on the analysis of Finite Element Methods and Advanced FEM Techniques. DSSC students: the Advanced Numerical Analysis course is composed of this course together with the first half of Advanced topics on the analysis of Finite Element Methods All material of the course can be found here: NSPDE-ANA github