Numerical solution of PDEs with FreeFem++

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Abstract:

The aim of this series of lectures is to introduce the audience to the numerical solution of PDEs with finite element techniques.

We will consider both stationary and time-dependent problems. The basic theoretical results will be briefly recalled. Then, the fundamentals of finite element methods will be presented, together with the related convergence properties.

The FreeFem++ package will be used to implement these methods under several circumstances. FreeFem++ is a partial differential equation solver by F. Hecht and collaborators. It has its own language. The scripts can solve linear and nonlinear problems in 2D and 3D.

In particular, we will see how meshes can be constructed, how linear and nonlinear PDEs can be (re)formulated in a weak sense, appropriate for numerical approximation, how large systems of linear algebraic equations can be solved, how solutions can be visualized, etc.

Contents:

1 - Elliptic PDEs and stationary models.

1.1 - Weak solutions and Lax-Milgram Theorem.

1.2 - Numerical approximations and finite elements. Description and results.

1.3 - The P_k - Lagrange finite elements. Properties.

1.4 - Implementation: FreeFem, 2D and 3D meshes, weak formulations, computations and visualizations.

1.5 - Examples

2 - Parabolic and hyperbolic PDEs and time-dependent models.

2.1 - Weak solutions and Lions Theorem.

2.2 - Numerical approximations with finite differences in time and finite elements in space.

2.3 - Euler and Gear implicit and semi-implicit schemes. Properties.

2.4 - Implementation and examples.

3 - A collection of interesting problems and their numerical solution.

3.1 - Population models, Lotka-Volterra and competition systems, etc.

3.2 - Systems of mixed kinds (parabolic-elliptic, hyperbolic-parabolic, etc.)

3.3 - Linear elasticity and Lamé systems.

3.4 - Semilinear elliptic and parabolic PDEs.

3.5 - PDEs of the Stokes and Navier-Stokes kinds.

Important: Download the FreeFem package from

http://www.freefem.org

and follow the instructions to install in your computer.