2016/07/17 by François Alouges, Giovanni Di Fratta, Alouges, François +1
Computer Science · Engineering · #35B27 #35B40 #74Q05 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #FOS: Physical sciences #Materials Science (cond-mat.mtrl-sci)
paper · doi:10.48550/arxiv.1607.04872
openalex publication_date 2016/07/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The aim of the paper is to introduce an alternative notion of two-scale convergence which gives a more natural modeling approach to the homogenization of partial differential equations with periodically oscillating coefficients: while removing the bother of the admissibility of test functions, it nevertheless simplifies the proof of all the standard compactness results which made classical two-scale convergence very worthy of interest: bounded sequences in L2\sharp[Y,L2(Ω)] and L2\sharp[Y,H1(Ω)] are proven to be relatively compact with respect to this new type of convergence. The strengths of the notion are highlighted on the classical homogenization problem of linear second-order elliptic equations for which first order boundary corrector-type results are also established. Eventually, possible weaknesses of the method are pointed out on a nonlinear problem: the weak two-scale compactness result for \mathbbS2-valued stationary harmonic maps.