Mean field type models describing the limiting behavior of stochastic differential game problems, as the number of players tends to +∞, have recently been introduced by J-M. Lasry and P-L. Lions. Numerical methods for the approximation of stationary and nonstationary such models are proposed. In particular, existence and uniqueness are investigated, as well as bounds on the solutions. Numerical...

After revisiting some practical problems of numerical methods by playing with models introduced in Chapter 2, we will investigate the two most important properties of numerical algorithms—stability and accuracy. This is done most easily in the context of the simplest numerical method, i.e., the explicit Euler method (Section 5.2). The problem of solving stiff differential equations is addressed...

Dedicated with friendship to Saul Abarbanel on his 60th birthday

for our fruitful discussions and for their good company.

In many physical problems, a key aspect is the motion of a propagating front separating two components. As fundamental as this may be, the development of a numerical algorithm to accurately track the moving front is difficult. In this report, we describe some previous theoretical and numerical work. We begin with two examples to motivate the problem, followed by some analytical results. These t...

We analyze semidiscrete and fully discrete finite element approximations to the solution of an initial boundary value problem for a model ultraparabolic equation.

The Maxwell equations describe the interaction of electric and magnetic fields. Important applications are electric machines such as transformers or motors, or electromagnetic waves radiated from antennas or transmitted in optical fibres. To compute the solutions of real life problems on complicated geometries, numerical methods are required. In this lecture we formulate the Maxwell equations, ...