Calculus, algebra, differential equations and functional analysis.

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.

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)

Course learning outcomes
Description
Interpretation of the obtained numerical solution and critical judgment of its quality. Relation with the applied science referred to
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.
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
Additional outcomes
Not established.

• 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

Training Activity	Methodology	Related Competences (only degrees before RD 822/2021)	ECTS	Hours	As	Com	Description
Class Attendance (theory) [ON-SITE]	Lectures		1.5	37.5	Y	N
Computer room practice [ON-SITE]	Problem solving and exercises		0.7	17.5	Y	Y
Individual tutoring sessions [ON-SITE]			0.2	5	Y	N
Writing of reports or projects [OFF-SITE]	Self-study		3.6	90	Y	Y
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).

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%

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

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	https://freefem.org/
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