An introduction to scientific computing using free software FreeFem++
[ Abstract ]
This course provides a hands-on introduction to the foundations and implementation of numerical algorithms to solve various PDE problems in 1D/2D/3D domains with the finite element method. Each session contains a short theoretical presentation of the numerical method, directly followed by its implementation in the form of computer programs. The free software FreeFem++ (www.freefem.org) offers an ideal framework to start a scientific computing activity: all the technicalities of the finite-element implementation are hidden from the user and the syntax is very close to mathematical formulations. For advanced users, FreeFem++ allows to focus on numerical algorithms to solve complex problems. A graduate-level presentation of theoretical and technical aspects will enable a fast progress in mastering numerical methods and the software: from linear to non-linear PDEs, from 2D to 3D problems, from sequential to parallel computing. A large variety of PDEs will be addressed: heat and wave equation, problems arising in linear elasticity, Stokes and Navier-Stokes-Boussinesq systems, etc.
The sessions of the second week will be followed by short technical presentations of complex problems solved with FreeFem++: Navier-Stokes-Boussinesq equations for phase-change materials, Schrodinger equation for Bose-Einstein condensates, fluid-structure interaction.
Instructors: Prof I. Danaila (University of Rouen, France), Prof F. Hecht (University Pierre et Marie Curie, France)
Features: Theory and Examples, Computer Implementation.
Prerequisites: basic programming skills, elementary calculus. Participants are advised to bring their laptops.
Week 1: Introduction and basic notions: solving 2D linear and non-linear PDEs
Week 2: Advanced notions: non-linear PDEs, 3D domains, parallel computing
[ Course 8 ]
Problems solved with FreeFem++. General presentation of the software.
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Abstract:
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general presentation of FreeFem++, |
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install FreeFem++ on each computer,
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examples of problems solved with FreeFem++,
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test of examples from the documentation of the software.
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[ Course 7 ]
Building a finite-element triangular mesh
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Abstract:
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building a 2D finite element mesh,
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identify boundaries,
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identify subdomains,
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define finite-element spaces. |
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[ Course 6 ]
Solving the 2D Poisson/Laplace problem
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Abstract:
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weak formulation of the Poisson problem,
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solving the problem, visualize the solution,
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convergence of the method,
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boundary conditions.
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[ Course 5 ]
Solving stationary or time-dependent linear PDEs in 2D domains
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Abstract:
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solving the steady heat equation with various BC, |
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solving the steady heat equation in complex domains,
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solving the time-dependent heat equation,
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solving the wave equation.
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[ Course 4 ]
From linear to non-linear PDEs
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Abstract:
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solving the linear elasticity problem (vectorial FE spaces), |
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solving the non-linear problem of the minimal surface (Newton method).
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[ Course 3 ]
Advanced notions and tricks
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Abstract:
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matrix formulation and optimization of programs,
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mesh adaptivity.
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[ Course 2 ]
Incompressible Fluid Dynamics
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Abstract:
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solving the convection equation,
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solving the Stokes and Navier-Stokes equations.
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[ Course 1 ]
Soving 3D problems, eigenvalue problems, parallel computing
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Abstract:
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solving the eigenvalue problem (2D and 3D),
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moving boundaries, parallel computing.
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