B9077 - METODI AVANZATI DI ANALISI NUMERICA

Learning outcomes

At the end of the course students have advanced knowledge of numerical analysis. In particular, for a wide class of problems that emerge in applications, they are able to derive numerical methods and analyze their theoretical and computational properties.

Course contents

Partial differential equations (PDEs) are the language through which many continuous phenomena—such as diffusion, waves, elasticity, fluids, etc.—are represented using mathematical models. However, transitioning from a mathematical formulation to a numerical simulation requires rigorous tools that combine functional analysis, approximation theory, and numerical linear algebra.

This course focuses on the theoretical foundations of the numerical analysis of PDEs: from variational formulations to finite element methods, from the study of Sobolev spaces to error analysis, from the representation of the discrete problem to the theoretical and computational reliability in solving the associated algebraic problem. The course is designed for students interested in a theoretical and structured understanding of numerical methods for PDEs, with the goal of providing a solid foundation in topics that are both relevant and cutting-edge in the field of applied mathematics.

Topics

Module 1 (M. Ruggeri): Numerical Analysis of PDEs.

• Review of elliptic PDEs (Poisson problem and minimal generalizations), and review of functional analysis (Sobolev spaces, Lax-Milgram theorem, etc.) aimed at the variational formulation of elliptic problems.
• Approximation theory in Sobolev spaces: Deny–Lions lemma and Bramble–Hilbert lemma, Lagrange interpolation and interpolation error in Sobolev spaces.
•  Finite element method: Galerkin method for elliptic problems and error estimates (Céa's lemma), mixed formulation of elliptic problems and related Galerkin discretization.

Module 2 (V. Simoncini): Numerical Linear Algebra.

• Iterative methods for large-scale linear systems: Krylov subspaces and general principles of projection-type methods; CG, MINRES, GMRES: algorithmic derivation and convergence properties depending on discretization.
• Acceleration methods: preconditioning strategies (incomplete factorizations, Algebraic Multigrid (AMG), operator preconditioning); inexact variants and sketching strategies (randomized numerical linear algebra).
• Matrix equations for differential equations.

Prerequisites

Fundamental concepts acquired during undergraduate courses in mathematical analysis and numerical calculus. Knowledge of PDEs and functional analysis is helpful, but the necessary material will be provided during the course. Familiarity with the MATLAB computational environment.

Readings/Bibliography

Useful references:

• D. Boffi, F. Brezzi, M. Fortin: Mixed Finite Element Methods and Applications. Springer, 2013.
• Leszek F. Demkowicz: Mathematical Theory of Finite Elements. SIAM, 2023.
• Alfio Quarteroni, Alberto Valli: Numerical Approximation of Partial Differential Equations. Springer, 1994.
• Y. Saad: Iterative methods for sparse linear systems. SIAM, 2003.
• G. Strang, G. J. Fix: An analysis of the Finite Element Method. Prentice-Hall Inc., 1973.
• N. Trefethen: Finite difference and Spectral Methods for ordinary and partial differential equations. Available online, 1996.

Teaching methods

The teaching activity will alternate between lectures (using slides/graphic tablet/blackboard) delivered by both instructors, and hands-on computer sessions in a MATLAB environment under the instructor's supervision. During these sessions, students will be encouraged to implement the concepts just presented in the lecture.

Assessment methods

The exam consists of two parts:

1. An oral exam covering the course content.
2. A presentation of a scientific paper that deepens or extends the topics covered in the course, to be agreed upon with the instructors. The presentation may include computational aspects, but this is not mandatory.

Links to further information

https://www.dm.unibo.it/~simoncin/AN2.html

Office hours

See the website of Michele Ruggeri

See the website of Valeria Simoncini