Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods

TL;DR

This paper extends the gradient schemes framework to variational inequalities with boundary conditions, enabling error analysis and the development of new hybrid mimetic mixed methods for obstacle and Signorini problems.

Contribution

It introduces an extension of gradient schemes to variational inequalities and proposes a novel hybrid mimetic mixed method for obstacle and Signorini problems.

Findings

Error estimates with known convergence rates

New convergence rates for previously unstudied schemes

Numerical results confirming theoretical accuracy

Abstract

Gradient schemes is a framework which enables the unified convergence analysis of many different methods -- such as finite elements (conforming, non-conforming and mixed) and finite volumes methods -- for $2^{\rm nd}$ order diffusion equations. We show in this work that the gradient schemes framework can be extended to variational inequalities involving mixed Dirichlet, Neumann and Signorini boundary conditions. This extension allows us to provide error estimates for numerical approximations of such models, recovering known convergence rates for some methods, and establishing new convergence rates for schemes not previously studied for variational inequalities. The general framework we develop also enables us to design a new numerical method for the obstacle and Signorini problems, based on hybrid mimetic mixed schemes. We provide numerical results that demonstrate the accuracy of these schemes, and confirm our theoretical rates of convergence.