Numerical Methods for Partial Differential Equations 3
A.Y. 2023/2024
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 methods for parabolic equations
- 1.1 Semi-discretization in space with finite elements
- 1.2 Stability and convergence of the semi-discrete problem
- 1.3 Discretization in time with finite difference
- 1.4 Stability and convergence of the discrete problem
- 2 Numerical methods for hyperbolic equations
- 2.1 Introduction to finite difference methods
- 2.2 Consistency, stability and convergence
- 2.3 Numerical methods for the wave equation
- 3 Domain decomposition methods and parallel computing
- 3.1 Overlapping Domain Decomposition methods: Additive Schwarz, Multiplicative Schwarz methods.
- 3.2 Non-overlapping Domain Decomposition methods: Schurm complement system, Dirichlet-Neumann, Neumann-Neumann Methods.
- 3.3 Abstract Schwarz theory.
- 1.1 Semi-discretization in space with finite elements
- 1.2 Stability and convergence of the semi-discrete problem
- 1.3 Discretization in time with finite difference
- 1.4 Stability and convergence of the discrete problem
- 2 Numerical methods for hyperbolic equations
- 2.1 Introduction to finite difference methods
- 2.2 Consistency, stability and convergence
- 2.3 Numerical methods for the wave equation
- 3 Domain decomposition methods and parallel computing
- 3.1 Overlapping Domain Decomposition methods: Additive Schwarz, Multiplicative Schwarz methods.
- 3.2 Non-overlapping Domain Decomposition methods: Schurm complement system, Dirichlet-Neumann, Neumann-Neumann Methods.
- 3.3 Abstract Schwarz theory.
Prerequisites for admission
Basic notions of numerical analysis
Teaching methods
Lectures and lab sessions.
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 a project and an oral exam.
- The project consists in the development of a code for the solution of a model described by partial differential equations assigned by the teacher.
- During the oral exam you will be asked to illustrate the project and some results of the teaching program, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to know how to apply them.
The exam is passed if the project is carried out and the oral exam is passed. The mark is expressed out of thirty and will be communicated immediately at the end of the oral exam.
- The project consists in the development of a code for the solution of a model described by partial differential equations assigned by the teacher.
- During the oral exam you will be asked to illustrate the project and some results of the teaching program, in order to evaluate the knowledge and understanding of the topics covered, as well as the ability to know how to apply them.
The exam is passed if the project is carried out and the oral exam is passed. The mark 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
Educational website(s)
Professor(s)