MIGSAA What is Numerical Analysis (Semester 2, 2018)

Weeklies

Week 1: We discussed 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 to learn about weak derivatives and Sobolev spaces (Def. x.7), the Poincare inequality (Lemma 1.3) and concluded the best approximation property (Thm. 2.2). The practical implementation is in Chapter 3. Braess gives a more thorough introduction in Chapter II.

Week 3: We discussed the convergence of finite element and finite difference methods for the Laplace equation, following the finite element primer and Braess' book. In particular, we prove the existence of a unique finite element solution and the best approximation property (Thm. 2.2). If time permits, we discuss more about the practical implementation from Chapter 3. Braess gives a more thorough introduction in Chapter II.

Week 4: 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 were illustrated in examples with nonsmooth solutions, for Laplace and time-dependent equations.

Week 5: Following Braess, we introduced an abstract framework for finite element methods: coercive bilinear forms on Hilbert spaces. We also reviewed the convergence of finite difference methods (Chapter I.4), based on the discrete maximum principle 3.5, in I.3.

Week 6: The lecture started with a survey of different finite elements: higher order polynomials, rectangular meshes, C1-continuous elements, etc. We then introduced gradient descent as a method to solve the linear systems arising from finite element or finite difference discretizations. Its convergence and shortcomings were explained (Braess, Chapter IV.2). In the remaining time we discuss time stepping schemes for time dependent problems, following this.

Week 7: The lecture discussed the conjugate gradient method as a Fourier analytic improvement of the gradient descent method, and introduced the related MINRES and BICGSTAB (Braess, Chapter IV.3).

Week 8: We discussed time stepping schemes for ODEs, following this. In particular, we looked at the theorems about zero-stability and convergence.

Week 9: Following this, we briefly discuss absolute stability and then move on to parabolic PDEs. We consider both the time stepping methods for we have seen for ODEs and variational Galerkin methods. In the remaining time we start with the numerical approximation of the Stokes problem from fluid mechanics (= the linearized Navier-Stokes equations), based on Chapter 3 of Braess. This will involve both the relevant finite element spaces to approximate divergence free functions and some functional analysis of saddle-point problems.

Week 10: We discuss the numerical approximation of the Stokes problem from fluid mechanics (= the linearized Navier-Stokes equations), based on Chapter 3 of Braess. This will involve both the relevant finite element spaces to approximate divergence free functions and some functional analysis of saddle-point problems.