Applied Mathematics

Some partial differential equations of interest in the applications and the main numerical methods used to solve them. After a short recall of the main theoretical results (students are supposed to have already attended a basis course on PDE), the approximation techniques will be discussed: standard finite difference schemes and mainly the finite element method for the numerical study of linear elliptic, parabolic and hyperbolic problems in two dimensions. The course also includes computer sessions. Hyperbolic problems Basics of transport problems in one and two dimensions. Main finite difference schemes. Convergence analysis and study of dispersion and diffusion properties. Computer implementation of the methods. Elliptic problems Recall of boundary problems for second order linear equations: classical solutions, maximum principle, variational formulation in Sobolev spaces. Finite difference schemes for Poisson equation, discrete maximum principle and convergence analysis. The Galerkin method for the approximation of variational problems. Lagrange finite elements. Interpolation theory in Sobolev spaces, convergence theorems and error estimates for the finite element approximation method, computational aspects and comparison with the finite difference approach. Numerical analysis of elliptic problems with a dominant transport (or reaction) term and their resolution with finite difference or finite element techniques. Up-wind type schemes and artificial diffusion. Some notes on stabilization methods for finite element schemes in advection-diffusion problems. Computer implementation of the methods. Parabolic problems Recall of classical results and variational formulation for linear parabolic problems. Finite difference schemes for the heat equation, consistency error and stability estimate.Schemes base on finite elements in space and finite differences in time (theta method), stability and convergence theorems, remarks on implementation.

The course requires familiarity with the basic instruments of mathematical analysis, and the knowledge of main theoretical aspects of Ordinary Differential and Linear Partial Differential Equations introduced in the parallel MAT/05 module, as well as the ability of writing simple Matlab programs.

A. Quarteroni, Numerical Models for Differential Problems, Springer

The evaluation will be based on a written exam and an oral exam. The written exam will mainly aim to verify the most theoretical knowledge. Instead, during the oral examination practical knowledge will be verified and the programs, developed in C or MatLab, will be evaluated.

For further information, we refer also to: A. Quarteroni - A. Valli, Numerical Approximation of Partial Differential Equations, Springer. L. Formaggia - F. Saleri - A. Veneziani, Applicazioni ed esercizi di modellistica numerica per problemi differenziali, Springer. J.C. Strickwerda, Finite Difference Schemes and PDE, Wadsworth \& Brooks Cole.

The evaluation will be based on a written exam and an oral exam. The written exam will mainly aim to verify the most theoretical knowledge. Instead, during the oral examination practical knowledge will be verified and the programs, developed in C or Matlab, will be evaluated.