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

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.

The gradient discretisation method for the chemical reactions of biochemical systems

“…We refer the reader to Refs. [7][8][9][10][11][12][13][14] and the monograph [15] for a complete presentation. The main purpose of this paper is to introduce the GDM to a system of reaction-diffusion equations subject to non-homogeneous Dirichlet boundary conditions.…”

Convergence analysis for a nonlinear system of parabolic variational inequalities

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

“…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 nonconforming numerical schemes can be included in the GDM, see [2,3,6,[9][10][11][12][13] for more details.…”

On Numerical Approximation of Diffusion Problems Governed by Variable-Exponent Nonlinear Elliptic Operators

“…Numerical analysis for the p-laplacian and more general Leray-Lions problems is well developed (see in particular [18,19,27] for the finite element analysis, [4,[9][10][11]34] and references therein for different finite volume schemes, [12] for mimetic schemes, [3,36] for gradient schemes (encompassing many of the previous ones), [31] for a recent hybrid high-order strategy. The analysis highlights the importance of strongly consistent gradient approximation and exploits in the essential way the L p − L p duality for proving convergence of such gradient approximations via the Minty-Browder argument [24,46,47].…”