https://hal.science/hal-03644203v2Andreianov, BorisBorisAndreianovIDP - Institut Denis Poisson - UO - Université d'Orléans - UT - Université de Tours - CNRS - Centre National de la Recherche ScientifiqueRUDN - Peoples Friendship University of Russia [RUDN University]Houssaine Quenjel, ElElHoussaine QuenjelLGPM - Laboratoire de Génie des Procédés et Matériaux - CentraleSupélec - Université Paris-SaclayCEBB - Centre Européen de Biotechnologies et BioéconomieOn numerical approximation of diffusion problems governed by variable-exponent nonlinear elliptic operatorsHAL CCSD2022[MATH] Mathematics [math]Andreianov, Boris2022-08-02 17:38:272023-03-24 14:53:272022-08-03 10:50:49enPreprints, Working Papers, ...https://hal.science/hal-03644203v1application/pdf2We highlight the interest and the limitations of the L1-based Young measure technique for studying convergence of numerical approximations for diffusion problems of the variable-exponent p(x)- and p(u)- laplacian kind. CVFE (Control Volume Finite Element) and DDFV (Discrete Duality Finite Volume) schemes are analyzed and tested. In the situation where the variable exponent is log-Hölder continuous, convergence is proved along the guidelines elaborated in [Andreianov, Bendahmane, Ouaro, Nonlinear Analysis, 2010, Vol. 72&73] while investigating the structural stability of weak solutions for this class of PDEs. In general, the lack of density of the smooth functions in the energy space, related to the Lavrentiev phenomenon for the associated variational problems, makes it necessary to distinguish two notions of solutions, the narrow ones (the H-solutions) and the broad ones (the W-solutions). Some situations where approximation methods "select" the one or the other of these two solution notions are described and illustrated by numerical tests.