On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

Abstract

We 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 et al. Nonlinear Anal. 72, 4649–4660, 2010 & Nonlinear Anal. 73, 2–24, 2010] 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.