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. To complete training in finite difference methods and introduce the finite element method for the numerical resolution of partial differential equations.
2. To verify the properties and operation of the methods through computer programming.
1. Finite Differences (FROM THE BEGINNING OF THE COURSE UNTIL EARLY NOVEMBER, APPROXIMATELY 14 LECTURE HOURS)
Design and implementation of finite difference methods for partial differential equations (PDEs). Basic concepts of their analysis: consistency, order, stability, and convergence.
- PARABOLIC AND HYPERBOLIC PDES IN ONE SPATIAL DIMENSION: 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 scheme, 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 5-point stencil discretization).
Lab sessions will focus on programming some of these methods
2. Finite elements (FROM EARLY NOVEMBER UNTIL THE END OF THE COURSE, APPROXIMATELY 14 LECTURE HOURS).
- Concept of distributional derivative. Spaces H^1 (a,b) and H_0^1 (a,b). Lax-Milgram lemma. (2 HOURS.)
- Finite element method (FEM) in one spatial dimension (solving the Sturm-Liouville problem with different boundary conditions using P_k Lagrange FEM): Variational formulation, discretization via P_k Lagrange FEM, matrix formulation, and assembly for 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 sessions will focus on programming some of these methods.
BASIC BIBLIOGRAPHY:
1. Iserles, A. (2008) A first course in the numerical analysis of differential equations (2nd ed.). Cambridge: Cambridge University Press. (Series: Cambridge Texts in Applied Mathematics). (First edition: 1997). Available online.
2. Johnson, C. (1987) Numerical solution of partial differential equations by the finite element method. Cambridge: Cambridge University Press.
3. Krizek, M., and Neittaanmäki, P. (1990) Finite element approximation of variational problems and applications. Harlow, UK: Longman Scientific and Technical.
4. Raviart, P.-A., and Thomas, J.-M. (1983) Introduction à l'analyse numérique des équations aux dérivées partielles [Introduction to numerical analysis of partial differential equations]. Paris: Masson.
5. Strikwerda, J. C. (2004) Finite difference schemes and partial differential equations (2nd ed.). Philadelphia, PA: SIAM. (First edition: 1989, Pacific Grove, CA: Wadsworth & Brooks/Cole).
6. Viaño Rey, J. M., and Figueiredo, J. (2000) Implementação do método de elementos finitos [Implementation of the finite element method]. Notes.
COMPLEMENTARY BIBLIOGRAPHY:
1. Ciarlet, P. G. (1991) Basic error estimates for elliptic problems. In: Handbook of numerical analysis (Vol. II, pp. 17–351). (J. L. Lions and P. G. Ciarlet, Eds.). Amsterdam: North-Holland.
2. Godunov, S. K., and Ryabenkii, V. S. (1987) Difference schemes: An introduction to the underlying theory. Amsterdam: North-Holland.
3. LeVeque, R. J. (2007) Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems. Philadelphia, PA: SIAM.
4. Thomas, J. W. (1995) Numerical partial differential equations: Finite difference methods. New York, NY: Springer. Available online.
5. Thomas, J. W. (1999) Numerical partial differential equations: Conservation laws and elliptic equations. New York, NY: Springer. Available online.
Specific Course Competencies:
1. To understand the basic techniques for deriving finite difference schemes for partial differential equations (PDEs).
2. To be familiar with the most common finite difference schemes for PDEs.
3. To grasp the fundamental concepts of numerical scheme analysis for PDEs: consistency, order, stability, and convergence.
4. To understand the theoretical and practical foundations of the finite element method for boundary value problems in PDEs: weak formulations, variational equations, existence analysis, discretization, meshing, implementation, and error estimation.
5. To be able to implement the studied methods using a programming language.
6. To use commercial/academic software to solve problems using the studied methods.
7. To apply, validate, and critically assess the results obtained with the studied methods.
The competencies listed above, as well as those described on page 5 of the degree program’s official document (available at https://www.usc.es/export9/sites/webinstitucional/gl/servizos/sxopra/me…), are developed in class and assessed according to the criteria outlined in the "Assessment System" section.
Lectures.
Interactive lab sessions.
Tutoring.
All grades (CA, PA, AKT, FE, EX, PAnew, and FG) must be understood on the 0-10 scale.
The assessment system combines continuous assessment and a final evaluation.
The continuous assessment (CA) consists of evaluating programming assignments (PA) and, where applicable, up to two additional knowledge tests (AKT) conducted during class time. The CA value is calculated as follows:
If AKT tests are administered, CA = 0.80 * PA + 0.20 * AKT;
Otherwise, CA = PA.
The number of activities contributing to CA does not exceed 3.
The CA grade may be retained for the second assessment opportunity.
The final evaluation (FE) is conducted through a written exam (EX) on the official examination date. The FE value is determined as:
** If PA >= 3, FE = EX.
** If PA is below 3, additional programming assignment-related questions (PAnew) are included in the exam, and FE = max{EX, 0.70 * EX + 0.30 * PAnew}, with the following exception: since this subject must ensure programming competency, FE is capped at 4 if PAnew is below 3.
The final grade (FG) is calculated as FG = max{FE, 0.70 * FE + 0.30 * CA}.
Class attendance will not be factored into the assessment system.
The second assessment opportunity follows these same rules.
+ On-site work (classroom attendance and participation) = 58 hours.
Large-group lectures: 28.
Small-group computer/lab sessions: 28.
Tutorials: 2.
+ Independent work (self-study, exercises, programming, recommended readings) = 92 hours.
- Keep up-to-date with all course content covered in class.
- Complete all assigned exercises and programming tasks.
- Begin practical work from the very first session.
- Clarify any doubts promptly with the instructor.
Programming assignments will be developed using MATLAB®.
In cases of fraudulent completion of exercises or assessments, the University of Santiago de Compostela (USC) regulations outlined in the "Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións" (Regulations for the Assessment of Academic Performance and Grade Review) shall apply.
Óscar López Pouso
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813228
- oscar.lopez [at] usc.es
- Category
- Professor: University Lecturer
José Luis Ferrín González
- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813191
- joseluis.ferrin [at] usc.es
- Category
- Professor: University Lecturer
| Tuesday | |||
|---|---|---|---|
| 09:00-10:00 | Grupo /CLIL_01 | Spanish | Computer room 3 |
| 10:00-11:00 | Grupo /CLIL_01 | Spanish | Computer room 3 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 03 |
| Wednesday | |||
| 09:00-10:00 | Grupo /CLIL_02 | Spanish | Computer room 2 |
| 10:00-11:00 | Grupo /CLIL_02 | Spanish | Computer room 2 |
| Thursday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| 12:00-13:00 | Grupo /CLIL_03 | Spanish | Computer room 2 |
| 13:00-14:00 | Grupo /CLIL_03 | Spanish | Computer room 2 |
| 01.23.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |
| 06.16.2026 10:00-14:00 | Grupo /CLE_01 | Classroom 06 |