A gradient discretization method to analyze numerical schemes for nonlinear variational inequalities, application to the seepage problem

Abstract

Using the gradient discretization method (GDM), we provide a complete and unified numerical analysis for nonlinear variational inequalities (VIs) based on Leray–Lions operators and subject to nonhomogeneous Dirichlet and Signorini boundary conditions. This analysis is proved to be easily extended to the obstacle and Bulkley models, which can be formulated as nonlinear VIs. It also enables us to establish convergence results for many conforming and nonconforming numerical schemes included in the GDM, and not previously studied for these models. Our theoretical results are applied to the hybrid mimetic mixed method (HMM), a family of schemes that fit into the GDM. Numerical results are provided for HMM on the seepage model and demonstrate that, even on distorted meshes, this method provides accurate results.