Organisation of Scientific Computing 2
Table of Contents
1 Organisation of Scientific Computing 2
1.1 Introduction
1.1.1 Scientific Computing 2 (Galerkin h/p methods)
 The course is an extension of Scientific Computing 1 (CS1) given the first semester of the 2nd year master. It uses some of the ingredients of CS1 and presents new ones in the context of the finite element software Life written in C++^{1}
 We have 6 weeks from January 4th to February 9th, 2 days per week, 2 courses and 2 practical sessions.
1.1.2 Courses
 Courses given by C. Prud'homme^{2}

When/Where
 Mondays: (C) 11:3013:00 Room F211
 Tuesdays: (C) 13:3015:00 Room F211
 Course notes in english will be provided.
1.1.3 Practical Sessions
 Practical session given by M. Ismail^{3}

When/Where
 Mondays: (P) 17:0018:30 Room F212
 Tuesdays: (P) 11:3013:00 Room F211
 Will be in computer rooms with bi processors and 4Go of RAM
 Must use the Linux OS (no windows), Debian or Ubuntu distros are preferred
 Requirements: knowledge in C++ programming environment (compilation of large software, …).
 A course on software tools is being developed by C. Prud'homme and F. Faure that may help you with this course. Check out this page http://coursescicomp.www.ljkforge.imag.fr Some of the slides/courses notes are already available
 Visualisation will be done with Paraview^{4}
1.1.4 Time table
Sessions  Course(h)  Practical(h)  Total(h eq TD) 

Introduction  1  1.5  
Intro Life  2  1.5  4.5 
Mesh  3  1.5  6. 
Large Sparse Systems  3  1.5  6. 
Galerkin Methods  6  3  12. 
Models  3  1.5  6. 
Project  0  9  9. 
Total  18  18.  45. 
1.2 Weekly organisation
1.2.1 Week 1:
 Course
 Session 1 : introduction (1h) + introduction life
 Session 2 : introduction life
 Practical session
 Session 1 : introduction to life
 Session 2 : introduction to life + presentation of the project
1.2.2 Week 2:
 Course

Session 1 : Mesh
 mesh generation
 localisation, localize points in a mesh, see kdtrees and ann^{5}

Session 2 : Mesh
 localisation,
 adaptation

Session 1 : Mesh
 Practical session
 Session 1 : Use the Chain OpenCascade^{6}, Gmsh^{7} and MadLib^{8}
Use OpenCascade to define the geometry, import the geometry in Gmsh and adapt the mesh using Madlib according to a given metric (e.g. given by life, or something else)
 Session 2 : Project
1.2.3 Week 3:
 Course
 Session 1 : Large sparse linear systems
 Session 2 : Large sparse linear and nonlinear systems
 Practical session

Session 1 : Manipulation and comparison of various methods
 provide one linear example and one nonlinear example
 exercise the solvers/preconditioners from the course
 look for information on Trilinos^{9}/Petsc^{10} websites to get help on the solver/preconditioner availability
 test the code/solvers with respect to solver type, preconditioner type, dimension and polynomials order.
 measure number of iterations and computing time. Report them in tables
 Session 2 : Project

Session 1 : Manipulation and comparison of various methods
1.2.4 Week 4:
 Course

Session 1 : Galerkin methods
 high order approximation in nD, n=1,2,3

Session 2 : Galerkin methods
 high order approximation in nD, n=1,2,3
 other elements: Raviart Thomas, Nedelec, CrouzeixRaviart, …

Session 1 : Galerkin methods
 Practical session

Session 1 : Numerical tests
 convergence order plots with respect to and ,
 choose quadratures properly,
 use only one domain and very high order approximation. Compare it with low order and many elements,
 test with respect to geometry order and observe convergence with respect to and
 Session 2 : Project

Session 1 : Numerical tests
1.2.5 Week 5:
 Course

Session 1 : Galerkin Methods
 Mixed problems: Saddle point formulations, Stokes and Darcy

Session 2 : Galerkin Methods
 Mixed problems
 Galerkin Least Square Methods

Session 1 : Galerkin Methods
 Practical session

Session 1 : Numerical Tests
 Invalid mixed elements wrt to infsup and show the chessboard effect
 observe convergence wrt to and for some families of space pairs (e.g. , )
 For darcy (i) compare various strategies, RT+Lag, RT+CrouzeizRaviart, VLag+CrouzeizRaviart, (ii) observe the local conservation (or not) of mass or
the fact that the Darcy laws verified locally
 Session 2 : Project

Session 1 : Numerical Tests
1.2.6 Week 6:
 Course

Session 1 : Advanced Models
 Flows : fluid mechanics (NavierStokes) and flows in porous media (Darcy)

Session 2 : Advanced Models
 Flows
 Elasticity
Note that if necessary this session may actually be used for Galerkin Methods and make sure we cover mixed problems and Stabilisation methods properly. It could be also that we extend Galerkin methods with a section on a posteriori error estimate and mesh adaption.

Session 1 : Advanced Models
 Practical session
 Session 1 : Project
 Session 2 : Project
1.3 Project
1.3.1 Project
 The project is presented the first week at the second session. Every week, one of the practical session is dedicated to the project.
 The project is the simulation of a cooling system for electronic components. This is a coupling between heattransfer (heat equation) and fluid flow (poiseuille or NavierStokes flows). Ideally this problem will be treated fully nonlinearly using a Newton solver in both steady and unsteady cases. Alternatives approximation spaces may be used for comparisons. Performance indices will be computed to check the behavior of the system.
1.4 Validation of the Course
1.4.1 Validation
The course is validated as follows:
 Report in LaTeX on the project with at least two of the practical sessions in an appendix
 20 minutes of oral presentation where the students present the project and the associated results and one of the practical sessions they have written in the appendices

Grade is given as follows:
 quality of report
 quality of oral presentation
 advancement in the project
 answers to questions on the practical sessions and the project
 failing to write one of the two practical sessions will automatically reduce the grade
 Students can be in groups of at most 2 persons
Footnotes:
^{1} The Life project is available on the LJKForge plateform http://ljkforge.imag.fr/projects/life
^{2} christophe.prudhomme@ujfgrenoble.fr
^{3} mourad.ismail@ujfgrenoble.fr
^{5} http://www.cs.umd.edu/~mount/ANN/
^{6} http://www.opencascade.org/
^{8} http://www.madlib.be
Date: <20080825 Mo>
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