Guías Docentes Electrónicas
1. General information
ECTS credits:
Academic year:
First semester
Main language:
Second language:
Use of additional languages:
English Friendly:
Web site:
Lecturer: DAMIAN CASTAÑO TORRIJOS - Group(s): 20 
Phone number
Office hours
Edificio Sabatini / 1.53
L y M 16:30-18:00

Lecturer: MARIA CRUZ NAVARRO LERIDA - Group(s): 20 
Phone number
Office hours
Margarita Salas/326
M y J, 18.00h-19.30h

Lecturer: FRANCISCO PLA MARTOS - Group(s): 20 
Phone number
Office hours
Margarita Salas
L y M 16.30-18.00h

2. Pre-Requisites

Calculus, algebra, differential equations and functional analysis.

3. Justification in the curriculum, relation to other subjects and to the profession

Partial differential equations are the main tool for modeling in science and technology. Only a few of these equations have an analytical solution. For this reason, numerical resolution is essential for scientific progress. To acquire knowledge on numerical analysis is relevant in an Applied Mathematics Master of Science.

4. Degree competences achieved in this course
Course competences
Code Description
CB06 Possess and understand knowledge that provides a basis or opportunity to be original in the development and / or application of ideas, often in a research context.
CB07 Apply the achieved knowledge and ability to solve problems in new or unfamiliar environments within broader (or multidisciplinary) contexts related to the area of study
CB08 Be able to integrate knowledge and face the complexity of making judgments based on information that, being incomplete or limited, includes reflections on social and ethical responsibilities linked to the application of knowledge and judgments
CB09 Know how to communicate the conclusions and their supported knowledge and ultimate reasons to specialized and non-specialized audiences in a clear and unambiguous way
CB10 Have the learning skills which allow to continue studying in a self-directed or autonomous way
CE01 Solve physical and mathematical problems, planning their solutions based on the available tools and time and resource constraints
CE02 Develop the ability to decide the appropriate techniques to solve a specific problem with special emphasis on those problems associated with the Modeling in Science and Engineering, Astrophysics, Physics, and Mathematics
CE05 Know how to obtain and interpret physical and/or mathematical data that can be applied in other branches of knowledge
CE07 Ability to understand and apply advanced knowledge of mathematics and numerical or computational methods to problems of biology, physics and astrophysics, as well as to build and develop mathematical models in science, biology and engineering
CE08 Ability to model, interpret and predict from experimental observations and numerical data
CG03 Present publicly the research results or technical reports, to communicate the conclusions to a specialized court, interested persons or organizations, and discuss with their members any aspect related to them
CG04 Know how to communicate with the academic and scientific community as a whole, with the company and with society in general about Physics and/or Mathematics and its academic, productive or social implications
CT03 Develop critical reasoning and the ability to criticize and self-criticize
CT05 Autonomous learning and responsibility (analysis, synthesis, initiative and teamwork)
5. Objectives or Learning Outcomes
Course learning outcomes
Interpretation of the obtained numerical solution and critical judgment of its quality. Relation with the applied science referred to
Understand the theoretic design of finite element, finite difference, finite and spectral volume methods, from known analytic techniques (variational formulations, Taylor developments, integration formulas by parts).
Understand the specific characteristics of the elliptic, parabolic and hyperbolic equations which be solved by numerical methods
Know and understand the basic concepts of consistency, stability and convergence of a numerical scheme in this context, as well as their interrelation
Gain the ability to solve a specific problem as a team: from the choice of an appropriate method to the oral and written presentation of the results obtained after its implementation
Learn to use some tools of Basic Analysis and Functional Analysis to carry out the numerical analysis of a method
Know some software tools that allow to completely solve a problem in the computer, which entails know how to program, generate a computational mesh, apply the appropriate calculus module and visualize the numerical solution. Practical problem solving.
Additional outcomes
Not established.
6. Units / Contents
  • Unit 1: Finite Differences
    • Unit 1.1: Boundary value problems
    • Unit 1.2: Parabolic evolution problems
    • Unit 1.3: Error analysis
    • Unit 1.4: Practice exercises
  • Unit 2: Spectral Methods
    • Unit 2.1: Fourier approximation
    • Unit 2.2: Orthogonal polynomial approximation
    • Unit 2.3: Practice exercises
  • Unit 3: Finite Elements
    • Unit 3.1: Formulation and error analysis
    • Unit 3.2: Effective implementation
    • Unit 3.3: Practice exercises
  • Unit 4: Finte Volumes and Finite Differences
    • Unit 4.1: Finite Differences for hyperbolic evolution problems
    • Unit 4.2: Formulation and error analysis
    • Unit 4.3: Effective implementation
    • Unit 4.4: Practice exercises
  • Unit 5: Courses and seminars
7. Activities, Units/Modules and Methodology
Training Activity Methodology Related Competences (only degrees before RD 822/2021) ECTS Hours As Com Description
Class Attendance (theory) [ON-SITE] Lectures CB06 1.5 37.5 Y N Theoretical development of the course contents and resolution of exercises
Computer room practice [ON-SITE] Problem solving and exercises CB07 CB08 CB09 CB10 CE01 CE02 CE05 0.7 17.5 Y Y Troubleshooting in the computer room
Individual tutoring sessions [ON-SITE] CT03 CT05 0.2 5 Y N Individual tutorials for resolution of doubts
Writing of reports or projects [OFF-SITE] Self-study CE07 CE08 CG03 CG04 3.6 90 Y Y Resolution of problems and preparation of reports and works on the contents of the course
Total: 6 150
Total credits of in-class work: 2.4 Total class time hours: 60
Total credits of out of class work: 3.6 Total hours of out of class work: 90

As: Assessable training activity
Com: Training activity of compulsory overcoming (It will be essential to overcome both continuous and non-continuous assessment).

8. Evaluation criteria and Grading System
Evaluation System Continuous assessment Non-continuous evaluation * Description
Assessment of active participation 20.00% 10.00% Active participation in solving problems
Assessment of activities done in the computer labs 30.00% 20.00% Resolution of practice exercises in the computer lab
Practicum and practical activities reports assessment 50.00% 70.00% Delivery of proposed works
Total: 100.00% 100.00%  
According to art. 4 of the UCLM Student Evaluation Regulations, it must be provided to students who cannot regularly attend face-to-face training activities the passing of the subject, having the right (art. 12.2) to be globally graded, in 2 annual calls per subject , an ordinary and an extraordinary one (evaluating 100% of the competences).

Evaluation criteria for the final exam:
  • Continuous assessment:
    Assessment of active participation, work in the computer lab, and autonomous work
  • Non-continuous evaluation:
    Evaluation criteria not defined

Specifications for the resit/retake exam:
Evaluation criteria not defined
Specifications for the second resit / retake exam:
Evaluation criteria not defined
9. Assignments, course calendar and important dates
Not related to the syllabus/contents
Hours hours

Unit 1 (de 5): Finite Differences
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 2 (de 5): Spectral Methods
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 3 (de 5): Finite Elements
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 4 (de 5): Finte Volumes and Finite Differences
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 8
Computer room practice [PRESENCIAL][Problem solving and exercises] 4
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 20

Unit 5 (de 5): Courses and seminars
Activities Hours
Class Attendance (theory) [PRESENCIAL][Lectures] 5.5
Computer room practice [PRESENCIAL][Problem solving and exercises] 1.5
Individual tutoring sessions [PRESENCIAL][] 1
Writing of reports or projects [AUTÓNOMA][Self-study] 10

Global activity
Activities hours
10. Bibliography and Sources
Author(s) Title Book/Journal Citv Publishing house ISBN Year Description Link Catálogo biblioteca
A.Quarterioni, R.Sacco, F.Saleri Numerical Mathematics Springer-Verlag 2000  
C. Bernardi and Y. Maday Approximations spectrales de problemes aux limites elliptiques Springer 1992  
C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang Spectral Methods for Fluid Dynamics Springer 1988  
C. Johnson Numerical solution of P.D.E. by the Finite Element Method Cambridge University Press 1987  
C.H. Edwards, D. E. Penney Ecuaciones diferenciales y problemas con valores en la frontera. Cómputo y modelado (4ª Ed) Pearson 2009  
D.F.Griffiths, A.R.Mitchell The finite difference method in partial differential equation John Wiley 1980  
E. Godlewski, P.A. Raviart Hyperbolic systems of conservation laws Ellipses 1991  
E. Romera, M.C. Boscá, F. Arias, F.J. Gálvez, J.I. Porras Métodos Matemáticos: Problemas de Espacios de Hilbert, Operadores Lineales y Espectros Paraninfo 2013  
E.Godlewski, P.A. Raviart Numerical Approximation of Hyperbolic Systems of Conservation Laws Springer-Verlag 1996  
Frederic Hecht FreeFEM Documentation. Release 4.6 2021 Pagina oficial de FreeFEM  
G.D. Smith Numerical Solution of Partial Differential Equations: Finite Difference Methods Oxford University Press 1985  
G.F.Forsythe, W.R.Wasow Finite difference methods for partial differential equations John Wiley 1960  
Gwynne A. Evans, G. Evans, G. A. Evans, Jonathan M. Blackledge, J. Blackledge, Peter D. Yardley, P. Yardley Numerical Methods for Partial Differential Equations Springer 2000  
Haïm Brézis Análisis Funcional Alianza Universidad Textos 84-206-8088-5 1983  
J. C. Strikwerda Finite difference Schemes and Partial Differential Pacific Grove, CA: Wadsworth and Brooks 1989  
J.H.Mathews, K.D. Fink Métodos Numéricos con MATLAB Prentice-Hall 2000  
J.M. Sanz-Serna Fourier techniques in numerical methods for evolutionary problems. 3RD Granada Seminar on Computational Physics Springer 1995  
J.N. Reddy An Introduction to the Finite Element Method MCGRAW HILL SERIES IN MECHANICAL ENGINEERING 2005  
L.N. Trefethen Spectral methods in Matlab SIAM 2000  
O.C. Zienkiewicz The Finite Element Method in Engineering Science McGraw-Hill 1971  
P.G.Ciarlet The finite element method for elliptic problems North Holland 1978  
P.M. Gresho, R.L. Sani Incompressible Flow and the Finite Element. Volume Two. Isothermal Laminar Flow John Wiley & Sons 0471492507 2000  
P.M. Gresho, R.L. Sani Incompressible Flow and the Finite Element. Volume One. Advection-Diffusion John Wiley & Sons 0471492493 2000  
R. LeVeque Finite Volume Methods for Hyperbolic Problems Cambridge Univesity Press 2002  
R.B. Richtmyer, K.W. Morton Difference methods for initial-value problems John Wiley & Sons 1967  
R.G. Voigt, D. Gotlieb and M.Y. Hussaini Spectral Methods for Partial Differential Equations SIAM 1984  
Randall LeVeque Finite difference methods for ordinary and partial differential equations SIAM 2007  
Roberto Font, Francisco Periago The Finite Element Method with FreeFem++ for beginners The Electronic Journal of Mathematics and Tecnology, Volume 7 (4) 1933-2823 2013  
S. Nakamura Análisis Numérico y visualización gráfica con MATLAB Pearson Educación/Prentice-Hall Hispanoamerica 1997  
Sandip Mazumder Numerical Methods for Partial Differential Equations: Finite Difference and Finite Volume Methods Elsevier Science 2016  
Stig Larsson, Vidar Thomee Partial Differential Equations with Numerical Methods Springer 2014  

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