# Spring School 2011 - Lectures

**Lectures**

**Raphaèle Herbin** (Université de Provence, Marseille, France)

Low order discretization methods for viscous flows (lectures notes in pdf Part 1) (lectures notes in pdf Part 2)

**Ulrich Langer** (RICAM Linz, Austria)

Multiharmonic methods for solving initial-boundary value problems

**Michel Pierre** (ENS Cachan, Rennes, France)

Global solutions in time for reaction-diffusion systems (lectures notes in pdf)

**Julio D. Rossi** (Universidad de Alicante, Spain)

Asymptotic behaviour for nonlocal diffusion problems (lectures notes in pdf)

**Abstracts**

**Raphaèle Herbin: Low order discretization methods for viscous flows**

I shall start with the finite volume scheme for convection diffusion problems, first on "admissible" finite volume meshes, then on general grids. In particular, the issue of heterogeneous anisotropic media will be addressed.

Then I shall turn to the incompressible Navier-Stokes equations, for which I shall consider both collocated and staggered schemes.

I will then present some recent convergence analysis of staggered schemes for the compressible Navier-Stokes equations.

**Ulrich Langer: Multiharmonic methods for solving initial-boundary value problems**

In many practical applications, for instance, in computational electromagnetics, the excitation is time-harmonic. Due to the time-harmonic excitation, we can switch from the time domain to the frequency domain. At least in the case of linear problems, this allows us to replace the expensive time-integration procedure by the solution of a simple linear elliptic system for the amplitudes belonging to the sine- and to the cosine-excitations. The fast solution of the corresponding linear system of finite element equations is crucial for the competitiveness of this method. We will present a robust and almost optimal-order multigrid-preconditioned MinRes solver for systems arising from time-harmonic scalar parabolic initial-boundary value problems and eddy current problems.

In case of non-linear time-harmonic initial-boundary value problems, the solution is not time-harmonic anymore but still periodic. Thus, the solution can be expanded in a Fourier series resulting in a special time discretization technique that is called multiharmonic approach. Truncating this Fourier series and approximating the Fourier coefficients by finite elements, we arrive at a large-scale coupled non-linear system for determining the finite element approximation to the Fourier coefficients. The construction of fast solvers for such systems is very crucial for the efficiency of this multiharmonic approach.

The multiharmonic approach also allows an efficient treatment of linear and non-linear initial-boundary value problems with non-harmonic excitations. In the linear case, we can benefit from the orthogonality of the sine- and cosine-functions. The robust and almost optimal-order multigrid-preconditioned MinRes solver constructed for the linear time-harmonic case leads to an efficient parallel solver for linear initial-boundary value problems with non-harmonic excitations.

In this course, we consider linear and non-linear parabolic initial-boundary value problems as well as linear and non-linear eddy current problems as typical model problems in electromagnetics. However, the multiharmonic technique can be applied in many other areas as well.

The author would like to thank his PhD students Michael Kolmbauer and Monika Kowalska for their contribution to the preparation of these lectures, and the Austrian Science Fund (FWF) for supporting this research work under the grant P19255.

**Michel Pierre: Global solutions in time for reaction-diffusion systems**

This series of lectures will address the question of global existence in time (or blow-up in finite time) for the so-called "reaction-diffusion systems", which are mathematical models for evolution phenomena undergoing at the same time spatial diffusion and (bio-)chemical type of reactions. Interest has increased recently for these models, in particular for applications in biology, environment and population dynamics.

Two natural properties appear in most models: the nonnegativity of the solutions is preserved for all time; the total mass of the components is controlled for all time (sometimes even exactly preserved). The fact that the total mass of the components does not blow-up in finite time suggests that solutions should exist for all time (mathematically speaking, solutions are actually bounded in L^{1} uniformly in time). But, it turns out that the answer is not so simple. In particular, it is necessary to give up looking for bounded classical solutions and rather consider weak solutions.

We will recall the main results for the 'good' situations where global existence of classical solutions holds. After showing how "incomplete blow-up" may occur, we will explain how far the notion of global weak solutions gives a satisfactory answer. Many problems are left open yet and we will indicate several of them.

These systems offer a good "L^{1}-structure", but they surprisingly satisfy an a priori L^{2}-estimate which turns out to be useful in many other questions: for instance, for the limit of (bio-)chemical systems where some rate constants tend to infinity. This will lead us to the very active and open area of "cross-diffusion systems" which happen to model more and more concrete situations and which raise new and challenging mathematical questions.

Prerequisite for these lectures: only a good knowledge of systems of ordinary differential equations together with the main properties of the heat equation.

**Julio D. Rossi: Asymptotic behaviour for nonlocal diffusion problems**

We review recent results concerning solutions to nonlocal nonlinear evolution equations with different boundary conditions, Dirichlet or Neumann and even for the Cauchy problem. We deal with existence/uniqueness of solutions and their asymptotic behaviour. We also review some results concerning limits of solutions to nonlocal equations when a rescaling parameter goes to zero. We recover in these limits some of the most frequently used diffusion models: the heat equation or the p-Laplacian evolution problem with Neumann or Dirichlet boundary conditions.