Anisotropic Adaptive Finite Elements for a p-Laplacian Problem

The p-Laplacian problem - ∇ ⋅ ( ( μ + | ∇ ⁡ u | p - 2 ) ⁢ ∇ ⁡ u ) = f {-\nabla\cdot((\mu+|\nabla u|^{p-2})\nabla u)=f} is considered, where μ is a given positive number. An anisotropic a posteriori residual-based error estimator is presented. The error estimator is shown to be equivalent, up to higher order terms, to the error in a quasi-norm. The involved constants being independent of μ, the solution, the mesh size and aspect ratio. An adaptive algorithm is proposed and numerical results are presented when p = 3 {p=3} . From this model problem, we propose a simplified error estimator and use it in the framework of an industrial application, namely a nonlinear Navier–Stokes problem arising from aluminium electrolysis.