The course aims at studying in detail some modern methods for the numerical approximation of partial differential equation that are relevant for applications. The methods under consideration will be analysed theoretically and implemented numerically.
Learning Objectives: To be able to derive partial differential and integral equation from simple physical phenomena; to know and to be able to use basic analytical and mathematical tools for boundary value problems; to be able to implement advanced numerical methods and to study their properties, both starting from theory and from numerical experiments.
Course Prerequisites
Basic knowledge of numerical analysis, mathematical analysis, partial differential equations. A basic knowledge of Matlab, python or similar languages is advisable. It is preferable to have attended, or to attend during the same term, the Finite Elements class.
Teaching Methods
Classroom lectures, tutorials in the computer lab, study of research papers, seminars. The topics presented may vary according to the students' preferences.
Assessment Methods
Oral exam and report. Every student will be able to implement the numerical methods presented during the course, focusing on some extensions or applications, or studying in details some theoretical aspects, also using the most recent scientific literature suggested by the lecturers.
Texts
Notes prepared by the lecturer, available on the course web page. Scientific papers provided by the lecturer. A list of references to study in more depth the different course subjects is available in the lecture notes.
Contents
The course will focus on some advanced techniques for the solution of partial differential equations (PDEs) that complement and extend the programme of the Finite Element course. In particular, the course will focus on the Boundary Element Method (BEM) for the approximation of the Helmholtz equation. In details: Derivation of the Helmholtz equation from the acoustic, electromagnetic, and elastic wave propagation models. Particular solutions of the Helmholtz equation. Exterior Dirichlet problems and scattering problems. Analytical tools: Sobolev spaces, Green identity, variational problems, Fredholm theory. Single-layer potential and operator, corresponding boundary integral equation. Boundary element method (BEM), properties and implementation. Well-posedness of Helmholtz problems in bounded and unbounded domains, resonances and eigenfunctions. Green representation formula Well-posedness analysis for the single-layer integral equation, spurious resonances. Other integral equation. Galerkin method analysis for problems satisfying a Garding inequality, Schatz argument. Transmission problem: derivation, refraction, first- and second-kind boundary integral equations. Acoustic waves in heterogeneous media, volume integral equations.
Course Language
Italian
More information
Office hours, in presence or online, can be arranged by appointment. The lecturer's notes are available on the course page.
