Numerical Methods for Partial Differential Equations 3
A.Y. 2025/2026
Learning objectives
The course aims at providing the basic techniques concerning parallel computing for the numerical treatment of problems arising from the approximation of PDEs, and, more generally, from numerical linear algebra.
Expected learning outcomes
At the end of the course students wil have acquired the basic ideas of parallel programming, as well as the ability to implement some parallel algorithms for the solution of partial differential equations, and, more genearlly, for linear algebra problems.
Lesson period: First semester
Assessment methods: Esame
Assessment result: voto verbalizzato in trentesimi
Single course
This course cannot be attended as a single course. Please check our list of single courses to find the ones available for enrolment.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
1. Numerical Approximation of Parabolic Equations
- 1.1 Semi-discretization in space using finite elements
- 1.2 Stability and convergence of the semi-discrete problem
- 1.3 Time discretization using finite differences
- 1.4 Stability and convergence of the discrete problem
2. Numerical Approximation of Parabolic Equations
- 2.1 Introduction to finite difference methods
- 2.2 Consistency, stability, and convergence
- 2.3 Overview of the wave equation
3. Domain Decomposition Methods and Parallel Computing
- 3.1 Overlapping Domain Decomposition Methods: Additive Schwarz and Multiplicative Schwarz Methods
- 3.2 Non-overlapping Domain Decomposition Methods: Schur Complement, Dirichlet-Neumann and Neumann-Neumann Methods
- 3.3 Abstract Schwarz Theory
- 1.1 Semi-discretization in space using finite elements
- 1.2 Stability and convergence of the semi-discrete problem
- 1.3 Time discretization using finite differences
- 1.4 Stability and convergence of the discrete problem
2. Numerical Approximation of Parabolic Equations
- 2.1 Introduction to finite difference methods
- 2.2 Consistency, stability, and convergence
- 2.3 Overview of the wave equation
3. Domain Decomposition Methods and Parallel Computing
- 3.1 Overlapping Domain Decomposition Methods: Additive Schwarz and Multiplicative Schwarz Methods
- 3.2 Non-overlapping Domain Decomposition Methods: Schur Complement, Dirichlet-Neumann and Neumann-Neumann Methods
- 3.3 Abstract Schwarz Theory
Prerequisites for admission
Calcolo Numerico 1
Teaching methods
Lectures and laboratory exercises.
Teaching Resources
- A. Quarteroni. Numerical Models for Differential Problems. Springer, 2014.
- V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer, 1984.
- A. Toselli and O. B. Widlund. Domain Decomposition Methods - Algorithms and Theory.
- V. Thomee. Galerkin Finite Element Methods for Parabolic Problems. Springer, 1984.
- A. Toselli and O. B. Widlund. Domain Decomposition Methods - Algorithms and Theory.
Assessment methods and Criteria
The exam consists of submitting a report on the laboratory activities carried out during the course and an oral examination. During the oral exam, students will be asked to present their laboratory report and discuss some topics from the course program, in order to assess their knowledge, understanding of the subjects covered, and ability to apply them. The exam is considered passed if the report is submitted and the oral exam is successfully completed. The final grade is expressed out of thirty and will be communicated immediately at the end of the oral exam.
MAT/08 - NUMERICAL ANALYSIS - University credits: 9
Laboratories: 36 hours
Lessons: 42 hours
Lessons: 42 hours
Professor:
Scacchi Simone
Professor(s)