Convergence analysis of numerical schemes for non-linear variational inequalities, application to the seepage problem

摘要

Using the gradient discretisation method (GDM), we provide a complete andunified numerical analysis for non-linear variational inequalities (VIs) basedon Leray--Lions operators and subject to non-homogeneous Dirichlet andSignorini boundary conditions. This analysis is proved to be easily extended tothe obstacle and Bulkley models, which can be formulated as non-linear VIs. Italso enables us to establish convergence results for many conforming andnonconforming numerical schemes included in the GDM, and not previously studiedfor these models. Our theoretical results are applied to the hybrid mimeticmixed method (HMM), a family of schemes that fit into the GDM. Numericalresults are provided for HMM on the seepage model, and demonstrate that, evenon distorted meshes, this method provides accurate results.