Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods
Provides a unified theoretical foundation and error analysis for numerical methods tackling variational inequalities with mixed boundary conditions, benefiting computational mechanics and PDE communities.
Extended gradient schemes framework to variational inequalities with Signorini and obstacle problems, enabling unified convergence analysis and error estimates. Achieved theoretical convergence rates confirmed by numerical experiments, with new rates for previously unstudied schemes.
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.