Gradient Schemes for Stokes problem

Abstract

The gradient scheme framework encompasses several conforming and non-conforming numerical schemes for diffusion equations. We develop here this framework for the approximation of the steady state and transient incompressible Stokes equations with homogeneous Dirichlet boundary conditions. Using this framework, we establish generic convergence results-by error estimates in the case of the steady problem, and by compactness arguments in the case of the transient problem-that are applicable to both old and new schemes for Stokes' equations. Three classical methods (MAC, Taylor-Hood and Crouzeix-Raviart schemes) are shown to fit into the gradient schemes framework; some of the convergence results obtained for those through the framework are new. We also show that a Hybrid Mixed Mimetic scheme, extension of the Crouzeix-Raviart scheme to any polyhedral mesh, can be designed within the gradient scheme framework; this scheme is new for Stokes' equations, and our abstract analysis establishes its convergence along with error estimates.

Gradient schemes: Generic tools for the numerical analysis of diffusion equations

“…Ongoing works concern the adaptation of the gradient scheme framework to some more general operators. In the case of the incompressible Stokes equations [26], it has been possible to obtain convergence results which simultaneously hold for the Taylor-Hood scheme, the Crouzeix-Raviart scheme and the MAC scheme. Results in this direction have also been obtained on the elasticity problems [30].…”

“…-but it also enables complete convergence analyses for a wide variety of models of 2nd order diffusion PDEs (Task 2) -linear, non-linear, non-local, degenerate, etc. [29,39,31,26,25,2,27,13,40,15] -through the verification of a very small number of properties (3 for linear models, 4 or 5 for non-linear models). The purpose of this article is to bring gradient schemes one step further towards a unification theory.…”

Introduction to discrete functional analysis techniques for the numerical study of diffusion equations with irregular data

“…Generic discrete functional analysis tools exist to ensure that several well-known schemes -including meshless methods -satisfy these properties [23], and therefore that the aforementioned convergence results apply to these schemes. The gradient scheme framework covers several boundary conditions, and also guided the design of new schemes [31,18].…”

The Gradient Discretization Method for Optimal Control Problems, with Superconvergence for Nonconforming Finite Elements and Mixed-Hybrid Mimetic Finite Differences

“…The variety of possible choices of GDs results in as many different GSs, allowing the GDM to cover a wide range of numerical methods (finite elements, mixed finite elements, finite volume, mimetic methods, etc.). Reference [19] presents the methods known to be GDMs, and [14,15,18,20,[22][23][24] provide a few models on which the convergence analysis can be carried out within this framework; see also [16] for a complete presentation of the GDM for various boundary conditions and models.…”