Selected Coursework

Program 

1. Review of theory of ODEs: examples, classification, existence and uniqueness of solutions.

2. Simple methods: explicit and implicit Euler, convergence, stability analysis, properties.

3. High order classical methods:

  • Runge–Kutta methods: construction, explicit, implicit, IMEX, collocation methods, error and stability analysis, properties, the Butcher tableau.

  • Multistep methods: explicit, implicit methods, error and stability analysis, convergence.

4. Iterative explicit high order methods: Deferred Correction (DeC), Arbitrary Derivative (ADER) methods, properties, stability and convergence analysis.

5. Unconditionally positivity preserving schemes: implicit Euler, high order schemes, modified Patankar schemes and their stability and convergence analysis.

6. Entropy conservative high order schemes: relaxation Runge–Kutta methods.

• Introduction to RB methods, offline-online computing, elliptic coercive affine problems

• Parameters space exploration, sampling, Greedy algorithm, POD 

• Residual based a posteriori error bounds and stability factors

• Primal-Dual Approximation

• Time dependent problems: POD-greedy sampling

• Non-coercive problems

• Approximation of coercivity and inf-sup parametrized constants

• Geometrical parametrization

• Reference worked problems

• Examples of Applications in CFD and flow control

• Tutorials (5 worked problems)

This course is meant to be complementary to the courses Numerical Solution of PDEs,  Numerical Solution of PDEs Using the Finite Element Method, and Advanced FEM Techniques.

We discuss topics on advanced Finite Element Analysis, complemented with numerical implementations and examples provided via the deal.II C++ library (www.dealii.org) or the fenics library (www.fenicsproject.org).

The point of departure of the course is a generalization of the Lax Milgram Lemma, which is only a sufficient condition for well-posedness of elliptic problems in Hilbert spaces. We start by discussing necessary and sufficient conditions for well-posedness of linear problems posed in Banach spaces, and show the numerical implications of such results.  Such conditions are usually referred to as inf-sup conditions (Babuska 1972, Necas 1962). From the functional analysis point of view, the inf-sup conditions are a rephrasing of two fundamental theorems by Banach: the Closed Range Theorem and the Open Mapping Theorem, and are sometimes referred to as BNB (Banach-Necas-Babuska) conditions in the general Banach settings. They become essentials when tackling saddle point problems, and in this settings their formalization is usually known as the Brezzi theory. 

We will discuss in details well posedeness, stability, and convergence for saddle point problems, and provide some extension to non-conforming approximations.

A schematic of the syllabus follows:

• Banach-Necas-Babuska conditions (inf-sup conditions for linear problems in Banach spaces)

• Ladyzhenskaya–Babuška–Brezzi condition (inf-sup conditions for saddle point problems)

• Conforming numerical approximations of mixed problems (Stokes problem, Mixed Poisson (or Darcy problem))

• Non-conforming numerical approximations of mixed problems

• Applications and examples

• Well possessedness analysis and stability analysis for both spectral projection and interpolation.

• Investigated Gibbs phenomenon and filtering.

• Numerical implementation of Pseudo-spectral of non-linear advection - diffusion equation

Well possessedness analysis, Stability analysis – deriving CFL condition and numerical implementation.

Numerical implementation and Into. Calculus of Variations

• Finite Difference implementation over non-uniform grids (initial value problem).

• Finite Elements implementation for 1D Helmholtz equation: Reported error in H0 and H1 norm as a function element size

• Spectral (Legendre) method implementation for 1D kDV equation