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. Introduce available software for the numerical solution of partial differential equations by the methods of finite differences and finite elements.
1. Finite differences (FROM THE BEGINNING OF THE COURSE UNTIL THE PRINCIPLES OF NOVEMBER, APPROXIMATELY 13 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 (3.7 HOURS: explicit method, implicit method, theta-methods, Crank-Nicolson), transport equation (3.7 HOURS: explicit schemes: FTFS, FTBS, Lax-Wendroff; implicit schemes: BTFS, BTBS, BTCS), wave equation (3.7 HOURS: standard explicit, schemes of order O(k^2) + O(h^4), theta-methods, Crank-Nicolson).
- ELLIPTIC PDEs IN TWO SPATIAL DIMENSIONS (1.9 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 13 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. (9.5 HOURS.)
- Variational formulation of a two-dimensional elliptic problem. (1.5 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.]
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.
5. THOMAS, J. W. (1999) Numerical partial differential equations: conservation laws and elliptic equations. Springer, New York, NY.
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 marks (CA, FE, PA, OT and FG) must be understood in the 0-10 scale.
The evaluation system includes, on one hand, a continuous assessment (CA) and, secondly, a final evaluation (FE).
Continuous assessment is the control of the programming assignments (PA) and, if appropriate, other tests of knowledge (OT) which would be effected, without prior notice, during the hours reserved for the subject. The score for the continuous assessment will be calculated using the following formula:
CA = 0.80*PA + 0.20*OT if tests of knowledge other than programming assignments have been done;
CA = PA otherwise.
The CA mark may be preserved for the second call of course.
The final evaluation is done through a written exam, which is done at the end of the teaching activities, on the officially scheduled dates. Below, FE will be the grade obtained in this written exam.
The final grade (FG) is calculated taking into account that this subject has to provide the student with computer programming skills, and it will therefore be required:
- To hand and to defend programming practices within the prescribed period. Otherwise, CA = 0.
- Not exceed 10% (6 h.) unexcused absences in controls that the teacher will randomly do during the blackboard classes and the small group activities. Otherwise, CA = 0.
The final grade will be calculated using the following formula:
FG = MAX{FE, 0.75*FE + 0.25*CA} if CA >= 3;
FG = MIN{4, MAX{FE, 0.75*FE + 0.25*CA}} otherwise.
+ On-site work at the classroom (attendance to classes and participation on them) = 54 hours.
Classes on blackboard to big groups: 26.
Classes in computer / laboratory in small group: 26.
Tutorials: 2.
+ Personal work (autonomous study, doing exercises, programming, recommended readings) = 90 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®.
CONTINGENCY PLAN for the adaptation of this guide to the document "Bases para o desenvolvemento dunha docencia presencial segura no curso 2020-2021", approved by the "Consello de Goberno" of the USC in ordinary session held on June 19, 2020:
Hereinafter, scenarios 1, 2 and 3 are those described in the document "DIRECTRICES PARA O DESENVOLVEMENTO DUNHA DOCENCIA PRESENCIAL SEGURA" of the USC.
** Bibliography: if necessary due to a health emergency, the teacher will be in charge of providing the students with notes or other suitable bibliography so that they can prepare the subject and carry out the tasks assigned to them.
** Teaching methodology: In any of the scenarios, the virtual course of the USC platform will be active.
Scenario 2: Students will receive study material, and will be asked to solve a certain number of exercises (EXER) and also do some programming practices (PP). These tasks can be done in a group, but only if the teacher so determines. The teacher will make use of the class schedule so that, in rotating shifts, he meets with the students in face-to-face sessions in order to resolve doubts and explain the most difficult parts of the course. Non-classroom training will be asynchronous except for virtual tutorials. EXER will be both the set of exercises and their grade, and an analogous comment applies to PP.
Scenario 3: The same methodology will be followed as that described above for scenario 2, except that appointments with students to resolve doubts and explain the most difficult parts of the course will be virtual rather than face-to-face.
** Assessment system: In any of the scenarios, a student who does not attend the test that we call “final evaluation” (FE) will be considered as “no presentado”.
In either scenario, the “continuous assessment” (CA) grade may be retained for the second assessment opportunity.
In cases of fraudulent performance of exercises or tests, the provisions of the “Normativa de avaliación do rendemento académico dos estudantes e de revisión de cualificacións” will apply.
Each student may be asked for a statement of responsibility for each of the asynchronous tasks that he or she submits.
All the grades (CA, EXER, FE, FG, and PP) must be understood on the 0-10 scale.
Scenario 2 (the continuous assessment will have a weight of 30%): The continuous assessment (CA) will consist of the control of the programming practices (PP) and of the exercises (EXER) that were described in the previous section (“Teaching methodology”) for scenario 2, according to the formula
CA = 0.30*PP + 0.70*EXER.
All these tasks (PP and EXER) must be sent to the teacher within the deadlines that will be established at the time, and their qualification may be canceled if the delivery deadlines are not met. The final evaluation grade (FE) will come from the students' answers to questions asked about the mentioned works in a telematic session (synchronous oral test).
The final grade (FG) will be calculated using the following formula, which takes into account that this subject must give programming skills:
FG = 0.70* FE + 0.30*CA if PP >= 3;
FG = MIN {4, 0.70*FE + 0.30*CA} otherwise.
Scenario 3 (the continuous assessment will have a weight of 30%): The same system as the one described above for scenario 2.
The second evaluation opportunity will be governed by the same system as the first, with the corresponding adaptation in case the scenario is different.
** Study time and individual work: The total work load will be, in any of the scenarios, that corresponding to a 6 ECTS subject.
Óscar López Pouso
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813228
- oscar.lopez [at] usc.es
- Category
- Professor: University Lecturer
| Tuesday | |||
|---|---|---|---|
| 10:00-11:00 | Grupo /CLIL_01 | Spanish | Computer room 3 |
| 12:00-13:00 | Grupo /CLE_01 | Spanish | Classroom 02 |
| Wednesday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Classroom 07 |
| 17:00-18:00 | Grupo /CLIL_01 | Spanish | Computer room 4 |
| 01.12.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 02 |
| 01.12.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |
| 07.01.2021 10:00-14:00 | Grupo /CLE_01 | Classroom 03 |