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

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 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.

A Gradient Discretization Method to Analyze Numerical Schemes for Nonlinear Variational Inequalities, Application to the Seepage Problem

“…The schemes (2.10) or (3.4) are then modified by replacing the set K D by K D,a D in the Signorini case, or by K D,ψ D in the obstacle case. The convergence results for this case of approximate barriers are given in the following theorems, whose proofs are identical to that of Theorem 2.9 (see [2,Section 6] for the case of approximate barriers in gradient schemes for linear VIs). Under the assumptions of Theorem 2.9, let (D m ) m∈N be a sequence of gradient discretisations in the sense of Definition 2.3, such that (D m ) m∈N is coercive, limitconforming, compact, and GD-consistent (with S D defined using K D,a D instead of K D ).…”

“…We presented in [2] three properties called coercivity, GD-consistency and limitconformity to assess the accuracy of gradient schemes for VIs. These properties were sufficient to establish error estimates and prove the convergence of the GDM for VIs based on linear differential operator.…”

Convergence Analysis For Non Linear System Of Parabolic Variational Inequalities

“…We begin with defining the discrete space and operators. These discrete elements are slightly different from the ones defined in [2,3], in particular, χ D , I D , and J D are introduced to deal with the non constant barrier χ and the initial solutions A ini and B ini . (1) The discrete set X D,0 is a finite-dimensional vector space on R, taking into account the homogenous Dirichlet boundary condition (1.1e).…”

“…( Ā, B) = (0, 0) on (∂Ω × (0, T )) 2 , (1.1e) ( Ā(x x x, 0), B(x x x, 0)) = (A ini , B ini ) in (Ω × {0}) 2 .…”

“…The GDM is a generic framework to unify the numerical analysis for diffusion partial differential equations and their corresponding problems. Due to the variety of choice of the discrete elements in the GDM, a series of conforming and non conforming numerical schemes can be included in the GDM, se [2,3,6,[9][10][11][12][13] for more details.…”