梯度方案用于Signorini问题和障碍问题，及其在混合仿照混合方法中的应用
Gradient schemes for the Signorini and the obstacle problems, and application to hybrid mimetic mixed methods
AI总结：
本文提出了一种梯度方案框架，用于分析涉及混合Dirichlet、Neumann和Signorini边界条件的变分不等式问题，并设计了一种新的数值方法来解决障碍和Signorini问题，同时验证了该方法的准确性和收敛性。
摘要：
梯度方案是一种框架，能够统一分析多种不同方法（如有限元法（一致性和非一致性和混合型）和有限体积法）对于二阶扩散方程的收敛性。本文展示了该框架可以扩展到涉及混合Dirichlet、Neumann和Signorini边界条件的变分不等式问题。这种扩展使我们能够为这类模型的数值近似提供误差估计，恢复一些方法已知的收敛速率，并为之前未研究过的变分不等式方案建立新的收敛速率。我们开发的通用框架还使我们能够设计一种新的数值方法来解决障碍和Signorini问题，基于混合仿照混合方案。我们提供了数值结果，证明了这些方案的准确性和理论收敛速率。
英文摘要：
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.
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