ECTS credits ECTS credits: 3
ECTS Hours Rules/Memories Student's work ECTS: 51 Hours of tutorials: 3 Expository Class: 9 Interactive Classroom: 12 Total: 75
Use languages Spanish, Galician
Type: Ordinary subject Master’s Degree RD 1393/2007 - 822/2021
Departments: Applied Mathematics
Areas: Applied Mathematics
Center Faculty of Mathematics
Call: Second Semester
Teaching: With teaching
Enrolment: Enrollable | 1st year (Yes)
Complete the training of students in the finite element method for partial differential equations, addressing in some depth the following aspects:
i) theoretical and practical foundations of Lagrange finite elements for second order elliptic (scalar and vectorial) boundary value problems in dimension 2 and 3, including the basis for programming it in a high-level language.
ii) Introduction to finite element methods for other problems: evolution, espectral, fourth order, mixed formulations.
1. Approximation of elliptic problems Abstract: Lax-Milgram Lemma, Lemma of Cea.
2. Approximation of elliptic problems of order 2 in dimension 2 and 3 with Lagrange finite elements (triangles, tetrahedra, quadrilaterals and hexahedral): description and construction of finite element spaces, references, basic functions, affine equivalence.
3. A priori error estimates for afin equivalent families of finite element method: quality of meshes, convergence, regular families. Case of curved domains.
4. Computer programming method: elementary matrices and second members, quadrature, assembly, storage profile, boundary conditions. Applications to flexion of membranes, heat conduction, two-and three-dimensional elasticity.
5. Isoparametric finite elements: idea and examples.
6. Finite elements in fourth order problems: bending of elastic beams and plates. Examples of C^1 finite elements.
7. Evolution problems: parabolic and hyperbolic of second order in time: variational formulatión variacional, discretization in space (finite elements) and time (finite differences).
8. Spectral problems: existence of eigenvalues and eigenmodes in elliptic problems, abstract approximation, finite element approximation in elliptic problems, eigenvalues and eigenmodes in the vibration os elastic structures (3D, plates and beams).
9. Mixed finite elements (1): Mixed formulation of Laplace problem. Existence and uniqueness of solution: inf-sup condition. Finite element aproximation: discret inf-sup condition. Examples of finite elements.
10. Mixed finite elements (2): numerical resolution of Stokes equation. A priori error estimates. Discrete inf-sup condition.Examples of finite elements.
BASIC BIBLIOGRAPHY:
Bécache, E., Ciarlet, P. J., Hazard, C., Luneville, E., La méthode des éléments finis: de la théorie a la pratique. Tome II. Compléments., Les Cours, Les Presses de l’ENSTA, Paris, 2010.
Ciarlet, P.G., The finite element method for elliptic problems. North-Holland, 1978.
Ciarlet, P. J., Luneville, E., La méthode des éléments finis: de la théorie a la pratique. Tome I. Concepts généraux., Les Cours, Les Presses de l’ENSTA, Paris, 2009.
Krizek, M., Neittaanmaki, P., Finite element approximation of variational problems and applications. Longman Scientific&Technical, 1984.
Raviart, P.A., Thomas, J.M., Introduction à l’analyse numérique des équations aux derivées partielles. Masson. 1983.
COMPLEMENTARY BIBLIOGRAPHY:
Brenner, S.C., Scott, L.R., The mathematical theory of finite element methods. Springer - Verlag. 1994 (3ª ed., 2008).
Brezzi, F., Fortin, M., Mixed and hybrid finite element methods, vol. 15 of Springer Series in Computational Mathematics, Springer - Verlag, New York, 1991.
Ern, A., Guermond, J.L., Theory and Practice of finite elements. Springer - Verlag. 2004.
Girault, V., Raviart, P.A., Finite element methods for Navier - Stokes equations. Springer - Verlag. 1986.
Glowinski, R, Numerical methos for nonlinear variational problems. Springer. 1984.
Pironneau, O., Finite element methods for fluids. John Wiley - Masson. 1989.
Quarteroni, A., Numerical models for differential problems. Springer - Verlag. 2009 (2ª ed., 2014).
Quarteroni, A., Valli, A., Numerical approximation of Partial Differential Equations. Springer - Verlag. 1997.
Roberts, J.E., Thomas, J.M., Mixed and hybrid methods. Handbook of Numerical Analysis. Vol . II. North Holland. 1991.
Thomee, V., Galerkin finite element methods for parabolic problems. Springer - Verlag. 1997 (2ª ed., 2006).
Verfurth, R., A Review of A Posteriori Error Estimation and Adaptive Mesh - refinement Technique, Wiley & Teubner, 1996.
Basic skills that students should acquire during their studies (established by Royal Decree 861/2010)
General skills
CG3 Being able to integrate knowledge in order to state opinions using information that even incomplete or limited, include reflecting on social and ethical responsibilities linked to the application of their knowledge;
CG5 To have the appropriate learning skills to enable them to continue studying in a way that will be largely self-directed or autonomous, and also to be able to successfully undertake doctoral studies.
Specific skills
CE4: Being able to select a set of numerical techniques, languages and tools, appropriate to solve a mathematical model.
CS2: To adapt, modify and implement software tools for numerical simulation.
The course is developed through theoretical classes taught by videoconference, recorded and reproduced in streaming, backed by written material that is made available to students in the virtual course.
Each student will carry out a supervised practical task on the solution using the finite element method of one or more problems proposed by the teacher that includes everything from theoretical formulation to resolution using the existing software in the systems of the universities to which they will have access, complemented by its own programs. A report with results and comments on the work must be delivered and presented to the teacher in an oral presentation of a maximum of 20 minutes that will be attended by all students of the course.
Face-to-face tutoring, through a virtual course, by email or by any audiovisual platform.
The CG3, CG5, CE4 and CS2 competences will be evaluated with the procedures indicated below.
In cases of fraudulent performance of exercises or tests, the provisions of the Regulations for the evaluation of the academic performance of students and the review of grades will apply.
The evaluation of the practical task carried out throughout the course will have a value of 80% of the final grade (8/10). The evaluation will be made based on the document delivered and the oral presentation given. Any student who does not submit the work within the deadlines established for this purpose is considered "Not presented."
The remaining 20% of the grade (2/10) will be obtained through an individual written or oral test on the theoretical content of the course. This test is mandatory and it will be carried out in person or remotely through the same videoconference as the classes.
There are two opportunities in each exam. Work and exam grades can be retained from first to second chance.
If the practical task has not been passed on the first opportunity, the same task, suitably revised, must be presented on the second opportunity.
Hours of teacher activity: 21 hours
-Teaching Expository: 15 hours
- Interactive Teaching: 6 hours
Student activities individually or in groups: 54 hours
- Exam Preparation: 9 hours
- Works: 40 hours
- Reviews: 5 hours
Total hours of student work: 75 hours
To have studied a basic course in finite elements and a course of partial differential equations and variational theory
Juan Manuel Viaño Rey
Coordinador/a- Department
- Applied Mathematics
- Area
- Applied Mathematics
- Phone
- 881813188
- juan.viano [at] usc.es
- Category
- Professor: University Professor
| Thursday | |||
|---|---|---|---|
| 09:00-11:00 | Grupo /CLE_01 | Spanish | Computer room 5 |
| Friday | |||
| 10:00-11:00 | Grupo /CLE_01 | Spanish | Computer room 5 |