IPSO - 2012 - Annual activity report

IPSO

IPSO - 2012

Project-Team Ipso

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Overall Objectives

Scientific Foundations

Application Domains

New Results

PIROCK: a swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise
Mean-square A-stable diagonally drift-implicit integrators of weak second order for stiff Itô stochastic differential equations
Weak second order explicit stabilized methods for stiff stochastic differential equations
High weak order methods for stochastic differential equations based on modified equations
Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems
Coupling heterogeneous multiscale FEM with Runge-Kutta methods for parabolic homogenization problems: a fully discrete space-time analysis
A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems
An Isogeometric Analysis Approach for the study of the gyrokinetic quasi-neutrality equation
Guiding-center simulations on curvilinear meshes using semi-Lagrangian conservative methods
Quasi-periodic solutions of the 2D Euler equation
Kinetic/fluid micro-macro numerical schemes for Vlasov-Poisson-BGK equation using particles
Two-Scale Macro-Micro decomposition of the Vlasov equation with a strong magnetic field
A dynamic multi-scale model for transient radiative transfer calculations
Accuracy of unperturbed motion of particles in a gyrokinetic semi-Lagrangian code
High order Runge-Kutta-Nyström splitting methods for the Vlasov-Poisson equation
A Discontinuous Galerkin semi-Lagrangian solver for the guiding-center problem
Asymptotic preserving schemes for highly oscillatory kinetic equation
Asymptotic preserving schemes for the Wigner-Poisson-BGK equations in the diffusion limit
Orbital stability of spherical galactic models
Stable ground states and self-similar blow-up solutions for the gravitational Vlasov-Manev system
Micro-macro schemes for kinetic equations including boundary layers
Stroboscopic averaging for the nonlinear Schrödinger equation
An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit
Analysis of a large number of Markov chains competing for transitions
High frequency behavior of the Maxwell-Bloch mdel with relaxations: convergence to the Schrödinger-rate system
Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion
Coexistence phenomena and global bifurcation structure in a chemostat-like model with species-dependent diffusion rates
Markov Chains Competing for Transitions: Application to Large-Scale Distributed Systems
Optimized high-order splitting methods for some classes of parabolic equations
A formal series approach to averaging: exponentially small error estimates
Higher-order averaging, formal series and numerical integration II: the quasi-periodic case
Existence of densities for the 3D Navier-Stokes equations driven by Gaussian noise
Diffusion limit for a stochastic kinetic problem
Global Existence and Regularity for the 3D Stochastic Primitive Equations of the Ocean and Atmosphere with Multiplicative White Noise
Weak backward error analysis for SDEs
Convergence of stochastic gene networks to hybrid piecewise deterministic processes
Exponential mixing of the 3D stochastic Navier-Stokes equations driven by mildly degenerate noises
Existence and stability of solitons for fully discrete approximations of the nonlinear Schrödinger equation
Fast Weak-Kam Integrators
Sparse spectral approximations for computing polynomial functionals

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Section: New Results

Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems

In [13] , an analysis of the finite element heterogeneous multiscale method for a class of quasilinear elliptic homogenization problems of nonmonotone type is proposed. We obtain optimal convergence results for dimension $d \leq 3$ . Our results, which also take into account the microscale discretization, are valid for both simplicial and quadrilateral finite elements. Optimal a-priori error estimates are obtained for the $H^{1}$ and $L^{2}$ norms, error bounds similar as for linear elliptic problems are derived for the resonance error. Uniqueness of a numerical solution is proved. Moreover, the Newton method used to compute the solution is shown to converge. Numerical experiments confirm the theoretical convergence rates and illustrate the behavior of the numerical method for various nonlinear problems.

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