CIMPA: Summer School on Multiscale Computational Methods and Error Control

July 10, 2017  to  July 21, 2017  




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Course Material




(10-July-2017  to  21-July-017)

Topic 1: Numerical Homogenization

Random homogenization, theoretical and numerical aspects.
Frederic Legoll, France

The aim of this course is to introduce the students to random homogenization and the associated numerical approaches that have been recently proposed in that context. Consider a heterogeneous material modelled at the continuum scale with PDEs, e.g. by continuum mechanics. The properties of the materials are supposed to be varying on a small characteristic lengthscale. Think e.g. of composite materials (used e.g. in the aeronautics industry), of porous media in subsurface modelling (modeled by the Darcy law), etc. The aim of homogenization theory is to replace this heterogeneous material by a homogeneous, effective material, for which the computation of the material response is inexpensive. We will first briefly review the case when the microstructure of the heterogeneous material is periodic. In that case, computing the homogenized coefficient is inexpensive.
The periodicity assumption may be restrictive. In a second stage, we will consider other cases, such as the random case. In that setting, computing the homogenized coefficient is very challenging. We will describe several practical approaches to address this question. Students are expected to be familiar with simple linear elliptic PDEs (such as the Poisson problem). No background on multiscale problems is expected.

Numerical techniques for PDEs with rapidly varying coefficients.
Alexei Lozinski, France

Partial differential equations (PDEs) with with rapidly varying coefficients arising, for example, in modelling of heterogeneous materials as mentioned above, present a challenge for numerical simulations. Indeed, classical finite element (or other discretization) methods would require a computational mesh with a step sufficiently small with respect to the length scale of coefficient variation, which can make the simulation prohibitivly expensive. We shall present so called multi-scale numerical methods (MsFEM, MsFEM with oversampling, Crouzeix-Raviart-MsFEM, etc) which aim to overcome this issue by incorporating the typical behaviour of multi-scale solutions in the basis functions used in the approximation (unlike piecewise polynomial basis function used in standard finite elements). Although these methods can be implemented and used on a relatively general microstructure, their theoretical analysis can typically be performed only for the periodic microstructure and is based on the results from periodic homogenization. We shall finally discuss some methods (such as local orthogonal decomposition) with theoretically guaranteed convergence on a general microstructure.


Topic 2: Optimal Control and Navier Stokes Equations

Mythily Ramaswamy, India

First of all, divergence free function spaces and their properties, required to develop the existence theory for Navier-Stokes equations, will be introduced. Using this, a basic existence theory for time dependent NavierStokes equations will be covered. The aim of these lectures is to pose an optimal control problem and derive the optimality conditions in simple cases.


Topic 3: A posteriori error estimation and adaptivity

Goal oriented error estimation for multi-physics problems
Thomas Richter, Germany

1. Introduction to goal oriented error estimators for linear problems, output functionals, adjoint solutions and first examples.
2. The general concept of the Dual Weighted Residual method for nonlinear problems with application to flow models.
3. Application to coupled multi-physics problem.
4. Advanced topics: nonstationary problems and space-time adaptivity, relation to optimization problems, anisotropic meshes.

A posteriori error estimation based on equilibrated fluxes
Martin Vohralik, France
- A unified framework for a posteriori error estimation based on equilibrated fluxes (covers conforming, nonconforming, and mixed finite elements, linear/nonlinear and steady/unsteady partial differential equations)
- Robustness in a posteriori error estimation (with respect to the approximation polynomial degree and to problem data)
- Adaptive solvers (adaptive stopping criteria, adaptive regularization, model choice)
- Multiscale approximations (multiscale and mortar discretizations, multigrid solvers
- Applications to flows in porous media


Topic 4: Finite volume methods for dissipative problems

Claire Chainais, France

We shall study finite volume schemes for some problems stemming from physics or engineering (semiconductors, corrosion, porous medium, etc). We shall emphasize the importance of preserving certain physical proprieties (positivity, Maximum principle, energy or entropy inequalities) to prove the convergence of the numerical schemes. We shall demonstrate how to apply the entropy dissipation methods at the discrete level to study the long time behavior of the schemes or to ensure the correct asymptotics in the limit of certain parameters going to 0.  


Topic 5: Simulations of thin structures and fluid-structure interaction

Modeling and simulations of thin structures: application in biology
Olivier Pantz, France

- Non linear elasticity: hyperelastic solids.

- Asymptotic limits: formal asymptotic expansion. Nonlinear membrane models, isometric plates, VonKarman model.

- Gamma convergence.

- Modeling of vesicles and red blood cells: strongly  solids.

- Numerical Simulations: Finite elements, FreeFem++, Gradient methods, Newton method.

Monolithic Methods for Fluid-Structure Interactions.
Olivier Pironneau, France and Hiroshi Suito, Japan

- Fundamental equations of continuum mechanics for fluids and structures. Eulerian, Lagrangian and Arbitrary Lagrangian-Eulerian formulations (OP).
- Fluid-structure interaction with immersed boundary method and its applications for 3D problems (HS).
- Formulation with Lagrangian multipliers and error estimates (OP).
- Small displacements and/or thin structures: transpiration approximation (OP).
- Large displacement : Fully Eulerian formulation and discretizations. Incompressible materials. Generalisation to compressible materials (OP).
- Pre- and post-processing techniques for FEM applications (HS).
- Test cases for FSI (OP).
- Application of FEM and fluid structure interaction to medical problems (HS).


Topic 6: From a microscopic description of matter to a macroscopic one on a computer: computational statistical physics

Gabriel Stoltz, France

In this series of lectures, I will first introduce the fundamental principles of statistical physics from a mathematical viewpoint, by emphasizing how the computation of macroscopic properties of matter (transport coefficients such as the shear viscosity or the thermal conductivity, equation of states linking the pressure, density and temperature of a fluid, etc) reduces to the evaluation of high dimensional integrals with respect to appropriate probability measures. I will then present several strategies to approximate these integrals, relying on ergodic averages of deterministic or stochastic differential equations (with a pedagogical survey of stochastic differential equations, seen from a numerical perspective). I will also discuss the numerical analysis underpinning the error estimations of the computed quantities, by distinguishing the two main sources of errors: statistical errors, related to central limit theorems, and systematic errors (biases).




(June 26, 2017 – July 08, 2017)


The aim of this pre-school will be to provide the basic background required to follow the main courses of the summer school. This pre-school will also train the participants in Freefem++ and Scilab as this would be necessary for the main school. In this way we hope that the students from less developed areas in India and neighbouring countries will fully benefit the main workshop. Moreover, we would definitely give a preference to candidates who are working in the areas (related areas) which will be focussed in the CIMPA School.


The topics which will be covered in the pre-school are: 


A quick introduction to Sobolev spaces; Review of Elliptic PDE: existence of unique weak solution, regularity. Quick review of FEM: Cea's Lemma, Aubin-Nitsche duality technique, convergence rates. A posteriori error analysis, reliability, efficiency, Adaptive finite element methods for elliptic PDEs: basics, Discontinuous Galerkin FEM for elliptic PDE and FEM for Navier Stokes Equations and Introduction to Optimal Control Problems governed by PDE. Mathematical introduction to Continuum Mechanics with an emphasis on Fluids. An introduction to homogenization method for PDEs. Basics on Stochastic Differential Equations.


There will also be lab sessions during the pre-school to enable to Indian participants to learn the lab basics and get prepared for the main school.


The tentative resource persons for the pre-school are:


Prof. Alexei Lozinski, Prof. B.V.Rathish Kumar (organisers),

Prof. Pravir Dutt (IIT Kanpur),

Prof. S. Ghorai (IIT Kanpur),

Prof. A.K. Pani (IIT Bombay),

Prof. Neela Nataraj, (IIT Bombay)

Prof. Imran Biswas (TIFR CAM),

Prof. T.Muthukumar (IIT Kanpur),

Prof. Rajen Sinha (IIT Guwahati),

Prof. D. Palla(BITS, Goa),

Prof. Sashi Kumar Ganeshan (IISC, Bangalore)

Prof. Nandakumaran (IISc, Bangalore)
Prof. Mythily Ramaswamy (TIFR, Bangalore)