Numerical Methods for Partial Differential Equations 1
A.Y. 2026/2027
Learning objectives
Presentation of the finite element method for elliptic boundary value problems and analysis of the error of its approximate solution.
Expected learning outcomes
The understanding of the foundations of the finite element method. The ability to apply and implement the finite element method for stationary problems and to interpret the obtained numerical results.
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
Theory:
A first finite element method. Elliptic problems. Meshes and mesh refinement. Abstract boundary value problems and Petrov-Galerkin methods. Sobolev spaces. Finite element bases. Approximation in H1 with piecewise polynomials. Numerical solution of an elliptic problem.
Practical part:
Implementation of the one-dimensional finite element model. Implementation of multidimensional boundary value problems within the ALBERTA library.
A first finite element method. Elliptic problems. Meshes and mesh refinement. Abstract boundary value problems and Petrov-Galerkin methods. Sobolev spaces. Finite element bases. Approximation in H1 with piecewise polynomials. Numerical solution of an elliptic problem.
Practical part:
Implementation of the one-dimensional finite element model. Implementation of multidimensional boundary value problems within the ALBERTA library.
Prerequisites for admission
Essential: Mathematical Analysis and Linear Algebra, and programming practice in C.
Useful: Basic knowledge of the Lebesgue integral and Lebesgue spaces, Numerical Linear Algebra, and Constructive Approximation.
Useful: Basic knowledge of the Lebesgue integral and Lebesgue spaces, Numerical Linear Algebra, and Constructive Approximation.
Teaching methods
Lectures, exercise classes, and lab sessions held in person, except in special circumstances (e.g., strikes).
Teaching Resources
· A. Bonito, C. Canuto, R. H. Nochetto, A. Veeser, Adaptive Finite Element Methods, Acta Numerica 33 (2024), 163-485. https://doi.org/10.1017/S0962492924000011
· D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, 2007
· S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, 3rd edition, Springer, 2007
· A. Ern, J.-L. Guermond, Finite Elements I-III, Texts in Applied Mathematics 72-74, Springer, 2021
· W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, Springer, 1987
· C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987
· S. Larsson, V. Thomée, Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics 45, 2nd edition, Springer, 2008
· R. H. Nochetto, A. Veeser, Primer of Adaptive Finite Element Methods, in: Multiscale and Adaptivity: Modeling, Numerics and Applications, G. Naldi, G. Russo (eds.), Lecture Notes in Mathematics 2040, Springer, 2012
· A. Quarteroni, Modellistica numerica per problemi differenziali, Springer Italia, 2000
· A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1991
· A. Schmidt, K. G. Siebert, Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42, Springer, 2005
· D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edition, Cambridge University Press, 2007
· S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15, 3rd edition, Springer, 2007
· A. Ern, J.-L. Guermond, Finite Elements I-III, Texts in Applied Mathematics 72-74, Springer, 2021
· W. Hackbusch, Elliptic Differential Equations: Theory and Numerical Treatment, Springer Series in Computational Mathematics 18, Springer, 1987
· C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Cambridge University Press, 1987
· S. Larsson, V. Thomée, Partial Differential Equations with Numerical Methods, Texts in Applied Mathematics 45, 2nd edition, Springer, 2008
· R. H. Nochetto, A. Veeser, Primer of Adaptive Finite Element Methods, in: Multiscale and Adaptivity: Modeling, Numerics and Applications, G. Naldi, G. Russo (eds.), Lecture Notes in Mathematics 2040, Springer, 2012
· A. Quarteroni, Modellistica numerica per problemi differenziali, Springer Italia, 2000
· A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1991
· A. Schmidt, K. G. Siebert, Design of Adaptive Finite Element Software: The Finite Element Toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42, Springer, 2005
Assessment methods and Criteria
The examination consists of two parts:
1)the evaluation of a small project chosen from a given list; and
2) a final oral examination, held by individual appointment after enrollment in an appello (official examination session).
The project must be chosen from a list that will be published at the beginning of each examination session. The project may be completed in collaboration with one other student; however, all members of the group must complete the examination within the period of validity of the same project list.
The project must be submitted by email as a ZIP archive containing the source code (but no executable files, to avoid issues with antivirus software) and a PDF report describing the results in no more than five pages. Although the project may be completed collaboratively, student are advied to write their own report independently.
The ZIP archive, together with the name of the collaborator (if any), must be submitted by email at least two working days before the scheduled oral examination.
To arrange the date of the oral examination, the student must be enrolled in a current appello. Students are advised to contact the instructor at least one week before their preferred examination date. The oral examination usually begins with a brief discussion of the report and lasts approximately 45 minutes. Students should bring a copy of their report and be prepared to answer questions both related and unrelated to the chosen project. The examination cannot be repeated using the same project.
The examination is considered passed if both the report (including its discussion) and the oral examination are evaluated positively. Final grades are expressed on a scale from 0 to 30 and will be communicated at the end of the oral examination.
1)the evaluation of a small project chosen from a given list; and
2) a final oral examination, held by individual appointment after enrollment in an appello (official examination session).
The project must be chosen from a list that will be published at the beginning of each examination session. The project may be completed in collaboration with one other student; however, all members of the group must complete the examination within the period of validity of the same project list.
The project must be submitted by email as a ZIP archive containing the source code (but no executable files, to avoid issues with antivirus software) and a PDF report describing the results in no more than five pages. Although the project may be completed collaboratively, student are advied to write their own report independently.
The ZIP archive, together with the name of the collaborator (if any), must be submitted by email at least two working days before the scheduled oral examination.
To arrange the date of the oral examination, the student must be enrolled in a current appello. Students are advised to contact the instructor at least one week before their preferred examination date. The oral examination usually begins with a brief discussion of the report and lasts approximately 45 minutes. Students should bring a copy of their report and be prepared to answer questions both related and unrelated to the chosen project. The examination cannot be repeated using the same project.
The examination is considered passed if both the report (including its discussion) and the oral examination are evaluated positively. Final grades are expressed on a scale from 0 to 30 and will be communicated at the end of the oral examination.
MATH-05/A - Numerical Analysis - University credits: 9
Laboratories: 36 hours
Lessons: 42 hours
Lessons: 42 hours
Professors:
Fierro Francesca, Veeser Andreas
Shifts:
Professor(s)