A gradient discretisation method to analyse numerical schemes for non-linear variational inequalities, application to the seepage problem

TL;DR

This paper develops a unified gradient discretisation framework for analyzing non-linear variational inequalities, including seepage problems, and demonstrates its effectiveness through convergence proofs and numerical experiments.

Contribution

It introduces a comprehensive GDM-based analysis for non-linear VIs, extending to various models and schemes, with new convergence results and practical numerical validation.

Findings

GDM provides a unified framework for non-linear VIs.

Convergence results are established for multiple schemes.

Numerical experiments confirm accuracy on distorted meshes.

Abstract

Using the gradient discretisation method (GDM), we provide a complete and unified numerical analysis for non-linear variational inequalities (VIs) based on Leray--Lions operators and subject to non-homogeneous Dirichlet and Signorini boundary conditions. This analysis is proved to be easily extended to the obstacle and Bulkley models, which can be formulated as non-linear 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.