There are three research groups working in analysis: Numerical Analysis, Partial Differential Equations and Mechanics.

## Numerical Analysis

The mathematicians in this group (fr) [1] work on the following topics:

*Development and analysis of numerical methods.*Finite element methods, integral equations, nonlinear hyperbolic problems, kinetic equations, Lattice-Boltzmann methods, optimisation problems with equilibrium constraints, Krylov methods, approximation.*Scientific computing, code designs, numerical simulations.*In particular design and maintenance of the finite element library MELINA++ [2] (fr) [1], then XLiFE++ [3]; of the advection-reaction-diffusion equations integrator PIROCK, and more generally the modelling and numerical simulations interplaying with other fields.*High-frequency problems, confinement and quantum models.*Schrödinger operators with magnetic fields, quantum waveguides, nonlinear Schrödinger equation and quantum confinement, high-frequency Helmholtz equation or Maxwell system, superconductivity.*Hamiltonian PDEs.*Stability, large time behavior, geometric integrators, highly oscillatory problems and averaging, gravitational Vlasov-Poisson system (INRIA team MINGuS [4]).*Elliptic Problems.*Homogenization, boundary conditions, singular perturbations.*Shape optimization and control.**Transient phenomena and front propagation.*Reaction-diffusion equations, front propagation, Hamilton-Jacobi equations, nonlinear parabolic problems, gradient fields identification, modelling ecological networks.*Operators and applied functional analysis.*Singular integral operators, numerical range of operators, Taylorian fields.

## Partial Differential Equations

The mathematicians in this group (fr) [5] work on the following topics:

*Spectral theory. S*cattering theory, quantum diffusion, quantum fields theory, non self-adjoint operators.*Phase-space analysis.*Microlocal analysis, semiclassical methods, symplectic geometry and quantification, integrable systems, mean fields evolution in quantum fields theory.*Multiscale analysis.*WKB method, nonlinear geometrical optics, semi-quantum models, homogenization, Dirichlet forms, fractal structures, turbulence theory.*Analysis of nonlinear PDEs.*Hyperbolic systems, stability of dispersive waves, nonlinear quantum mechanics, fluid mechanics and oceanography.

## Mechanics

The mathematicians in this group (fr) [6] work on the following topics:

*Generalized continuum mechanics.**Biomechanics.*Analysis of brain-CSF-skull and cardiovascular systems during shocks. Diagnosis and orthopedic treatment of idiopathic scoliosis. Biological tissue and artificial transplant thermodynamics.*Vibration and wave propagation.*Structure vibration through Timoshenko beam theory. Vibro-acoustic analysis. Waves and fluid-structure coupling.*Homogenization.*Multi-diffusion, mathematical homogenization.*Modelling and analysis of turbulence models of incompressible flows*(INRIA team Fluminance [7]).