ECTS credits ECTS credits: 6
ECTS Hours Rules/Memories Student's work ECTS: 99 Hours of tutorials: 3 Expository Class: 24 Interactive Classroom: 24 Total: 150
Use languages Spanish, Galician
Type: Ordinary Degree Subject RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Faculty of Mathematics
Call: First Semester
Teaching: With teaching
Enrolment: Enrollable
1. Complete training in the methods of finite difference and introduce the finite element method for the numerical solution of partial differential equations.
2. Check the properties and operation of the methods through computer programming.
1. Finite differences (FROM THE BEGINNING OF THE COURSE UNTIL THE PRINCIPLES OF NOVEMBER, APPROXIMATELY 14 HOURS OF BIG GROUP LECTURES).
Design and implementation of finite difference methods for partial differential equations (PDEs). Basic concepts in their analysis: consistency, order, stability and convergence.
- PARABOLIC AND HYPERBOLIC PDEs, ONE-DIMENSIONAL IN SPACE: heat equation (4 HOURS: explicit method, implicit method, theta-methods, Crank-Nicolson), transport equation (4 HOURS: explicit schemes: FTFS, FTBS, Lax-Wendroff; implicit schemes: BTFS, BTBS, BTCS), wave equation (4 HOURS: standard explicit, schemes of order O(k^2) + O(h^4), theta-methods, Crank-Nicolson).
- ELLIPTIC PDEs IN TWO SPATIAL DIMENSIONS (2 HOURS): Dirichlet problem for the Poisson equation (standard scheme with computational molecule of 5 points).
Lab classes are devoted to code some of these methods.
2. Finite elements (FROM THE PRINCIPLES OF NOVEMBER UNTIL THE END OF THE COURSE, APPROXIMATELY 14 HOURS OF BIG GROUP LECTURES).
- Concept of distributional derivative. Spaces H^1 (a,b) and H_0^1 (a,b). The Lax-Milgram lemma. (2 HOURS.)
- Finite element method (FEM) in one spatial dimension (resolution of the Sturm-Liouville problem with diffferent types of boundary conditions by means of the FEM Lagrange P_k): variational formulation, discretization with the FEM Lagrange P_k, matrix formulation and assembly in the case k=1. (10 HOURS.)
- Variational formulation of a two-dimensional elliptic problem. (2 HOURS.)
As in the first part of the course, lab classes are devoted to code some of these methods.
Basic bibliography:
1. ISERLES, A. (2008, second edition) A first course in the numerical analysis of differential equations. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge. [First edition: 1997.] Available online.
2. JOHNSON, C. (1987) Numerical solution of partial differential equations by the finite element method. Cambridge University Press, Cambridge.
3. KRIZEK, M.; NEITTAANMÄKI, P. (1990) Finite element approximation of variational problems and applications. Longman Scientific and Technical, Harlow (UK).
4. RAVIART, P.-A.; THOMAS, J.-M. (1983) Introduction à l'ánalyse numérique des équations aux dérivées partielles. Masson, Paris.
5. STRIKWERDA, J. CH. (2004, second edition) Finite difference schemes and partial differential equations. SIAM, Philadelphia, PA. [First edition: 1989, Wadsworth & Brooks/Cole, Pacific Grove, CA.]
6. VIAÑO REY, J. M.; FIGUEIREDO, J. (2000) Implementação do método de elementos finitos. Notas.
Complementary bibliography:
1. CIARLET, PH. G. (1991) Basic error estimates for elliptic problems. In Handbook of Numerical Analysis, vol. II, pp. 17—351. Editors: J. L. Lions and Ph. G. Ciarlet. North-Holland, Amsterdam.
2. GODUNOV, S. K.; RYABENKII, V. S. (1987) Difference schemes: an introduction to the underlying theory. North-Holland, Amsterdam.
3. LEVEQUE, R. J. (2007) Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems. SIAM, Philadelphia, PA.
4. THOMAS, J. W. (1995) Numerical partial differential equations: finite difference methods. Springer, New York, NY. Available online.
5. THOMAS, J. W. (1999) Numerical partial differential equations: conservation laws and elliptic equations. Springer, New York, NY. Available online.
Subject-specific competences:
1. To know the basic techniques for obtaining finite difference schemes for partial differential equations (PDEs).
2. To know the more usual finite difference schemes for PDEs.
3. To assimilate the fundamental concepts for analyzing the numerical schemes for PDEs: consistency, order, stability, and convergence.
4. To know the theoretical-practical foundations of the finite element method for boundary value problems for PDEs: weak formulations, variational equalities, analysis of the existence of solution, discretization, meshing, implementation, and error.
5. To be able to implement the studied methods by employing some programming language.
6. To use commercial/academic software to solve some problems by the methods studied.
7. To put into practice, to validate, and to evaluate with criticism the results obtained with the methods studied.
The competences above, as well as the ones described on page 5 of the degree memory at the link
https://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/me…,
are worked during clases, and they are assessed according to the system described in the section "Assessment system".
Blackboard classes in big groups (lectures).
Lab seminars.
Tutorials.
All grades (CA, PA, OT, FE, EX, PAnew and FG) must be understood on the 0-10 scale.
The evaluation system contemplates, on the one hand, a continuous assessment and, on the other, a final evaluation.
The continuous assessment (CA) consists of the control of the programming assignments (PA) and, if applicable, of other knowledge tests (OT), two maximum, which would be carried out within the time reserved for the subject. The value of CA is calculated using the following formula:
CA = 0.80*PA + 0.20*OT if tests other than programming assignments are done;
CA = PA in another case.
The number of activities that lead to the calculation of CA is not greater than 3.
The CA grade can be kept for the second evaluation opportunity.
The final evaluation (FE) is made by means of a written exam (EX), which is carried out on the officially scheduled dates. The value of FE is calculated as follows:
** If PA >= 3, FE = EX.
** If PA is less than 3, some questions related to programming assignments (PAnew) are added to the written exam, and FE = max{EX,0.70*EX+0.30*PAnew}, except in the case specified next: since this subject has to provide programming skills, the value of FE is limited to a maximum of 4 points if PAnew is less than 3.
The final grade (FG) is calculated using the following formula: FG = max{FE,0.70*FE+0.30*CA}.
The second evaluation opportunity will be governed by the same rules as the first one.
+ On-site work at the classroom (attendance to classes and participation on them) = 58 hours.
Classes on blackboard to big groups: 28.
Classes in computer / laboratory in small group: 28.
Tutorials: 2.
+ Personal work (autonomous study, doing exercises, programming, recommended readings) = 92 hours.
- Maintain current knowledge of the contents explained in class.
- Do the exercises and programs proposed.
- Start making practices from the first session.
- Check all doubts with the teacher.
The programming assignments will be done in MATLAB®.
In cases of fraudulent performance of exercises or tests, the USC regulations contained in the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións" will apply.
Óscar López Pouso
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813228
- oscar.lopez [at] usc.es
- Category
- Professor: University Lecturer
Rafael Vazquez Hernandez
- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813134
- rafael.vazquez [at] usc.es
- Category
- Investigador/a Distinguido/a
| Monday | |||
|---|---|---|---|
| 13:00-14:00 | Grupo /CLE_01 | Spanish | Ramón María Aller Ulloa Main Hall |
| Tuesday | |||
| 11:00-12:00 | Grupo /CLIL_02 | Spanish, Galician | Computer room 4 |
| 12:00-13:00 | Grupo /CLIL_01 | Spanish | Computer room 4 |
| 13:00-14:00 | CLIL_03 | Spanish | Computer room 4 |
| Wednesday | |||
| 09:00-10:00 | Grupo /CLIL_01 | Spanish | Computer room 4 |
| 10:00-11:00 | Grupo /CLIL_02 | Galician, Spanish | Computer room 4 |
| 13:00-14:00 | Grupo /CLE_01 | Spanish | Classroom 03 |
| Thursday | |||
| 09:00-10:00 | CLIL_03 | Spanish | Computer room 4 |
| 01.24.2025 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.19.2025 16:00-20:00 | Grupo /CLE_01 | Classroom 06 |