We study quasilinear elliptic double obstacle problems with a variable exponent growth when the right-hand side is measure. A global Calder\'{o}n-Zygmund estimate for gradient of an approximable solution obtained in terms associated obstacles and given measure, identifying minimal requirements regularity estimate.

Nonlinear multigrid methods such as the Full Approximation Scheme (FAS) and Newton-multigrid (Newton-MG) are well established as fast solvers for nonlinear PDEs of elliptic and parabolic type. In this paper we consider Newton-MG and FAS iterations applied to second order differential operators with nonlinear diffusion coefficient. Under mild assumptions arising in practical applications, an app...