MIGSAA What is Numerical Analysis (Semester 2, 2018)

Weeklies

Week 1: We discuss the Laplace equation, its numerical solution using finite difference methods as well as the formulation of the Laplace equation as a minimization problem for the energy. See e.g. here, Chapter 1, and Chapter I.3 in Braess' book.

Week 2: We discussed finite element methods for the Laplace equation, following the finite element primer. They are based on the formulation of the Laplace equation as a minimization problem. To do so, in particular we needed the Poincare inequality to prove the existence of a unique finite element solution (and also finite difference solution). We concluded Cea's Lemma / the best approximation property. The practical implementation is in Chapter 3. Braess gives a more thorough introduction in Chapter II.

Week 3: We discuss the approximation properties of piecewise polynomial functions, which allow to determine the convergence rate in L2 and Sobolev norms (Finite element primer, also Braess, Chapter II.6, in particular Thm. 6.4). We then obtained a computable (a posteriori) error estimate and discussed adaptive mesh refinements based on the strategy Solve-Estimate-Mark-Refine. The resulting mesh refinements are illustrated in examples with nonsmooth solutions, for Laplace and time-dependent equations.

Week 4: We reviewed a posteriori error estimates and saw examples of the resulting adaptive mesh refinements on the computer. Following Braess, we then introduced an abstract framework for finite element methods: coercive bilinear forms on Hilbert spaces.

Week 5: Jakub discussed the implementation from scratch of piecewise linear finite elements for problems in two dimensions. If time permits, we also review the convergence of finite difference methods (Braess, Chapter I.4), based on the discrete maximum principle 3.5, in I.3 (see here).

Week 6: We discussed the best approximation property of finite elements / Cea's Lemma in the abstract framework of coercive, continuous bilinear forms on Hilbert spaces. The general approach allows to use a wide variety of finite elements, and we surveyed some important ones: higher order polynomials, rectangular meshes, C1-continuous elements, etc. Also virtual element methods fall in this frame work, and we illustrated them with a discretization by Pegasus meshes. Slides here and here. We finally reviewed the convergence of finite difference methods, following Braess, Chapter I.4, based on the discrete maximum principle 3.5 in I.3 there (see these slides).

Week 7: We finished the discussion of finite difference methods, following Braess and these slides. We then started the discussion of weak formulations of PDEs and spectral methods, specifically spectral collocation and spectral Galerkin methods. We mostly followed the first section in the book on spectral methods by Shen, Tang, Wang, but see also Trefethen.

Week 8: We continued the discussion of spectral methods, following the early sections in the books by Shen, Tang, Wang and Trefethen. In particular, we discussed the fast approximation of smooth functions by trigonometric functions and saw a generalization of Lax-Milgram's and Cea's Lemmas for methods based on possibly different spaces of test and ansatz functions (see the notes by Babuska and Aziz for a thorough treatment of the classical theory).

Week 9: We finished the discussion of generalizations of Lax-Milgram's and Cea's Lemmas and illustrated the use of spectral methods for the Allen-Cahn equation.

Week 10: We conclude the discussion of spectral methods with methods based on orthogonal polynomials, as in Chapter 4 in the book by Shen, Tang, Wang and in the book by Trefethen. We then conclude the course with applications to continuum mechanics.