A counterexample for the inf‐sup stability of the RT0 − P1 ⊂ L2(Ω) × H10(Ω) finite element combination for the mixed Poisson equation

Fleurianne Bertrand,Daniele Boffi

doi:10.1002/pamm.201900426

Abstract

AbstractThis paper investigates the inf‐sup stability of a dual mixed discretization of the Poisson problem using the lowest‐order (Hdiv‐conforming) Raviart–Thomas space combined with the space of standard linear (continuous) Lagrange elements.

Highlights

In this paper we deal with the following dual mixed formulation of the Poisson problem: find (σ, u) ∈ (L2(Ω))2 × H01(Ω) such that (σ, τ ) − (τ, ∇u) = 0 ∀τ ∈ (L2(Ω))2 and (σ, ∇v) = −(f, v) ∀v ∈ H01(Ω).It is well known that finite element spaces in the approximation of mixed formulations have to satisfy the inf-sup condition, see e.g. [1]
Finland This paper investigates the inf-sup stability of a dual mixed discretization of the Poisson problem using the lowest-order (Hdiv-conforming) Raviart–Thomas space combined with the space of standard linear Lagrange elements
This paper presents numerical evidence that the inf sup condition in the case of the Poisson problem is not satisfied

Summary

Introduction

In this paper we deal with the following dual mixed formulation of the Poisson problem: find (σ, u) ∈ (L2(Ω))2 × H01(Ω) such that (σ, τ ) − (τ , ∇u) = 0 ∀τ ∈ (L2(Ω)) and (σ, ∇v) = −(f, v) ∀v ∈ H01(Ω). It is well known that finite element spaces in the approximation of mixed formulations have to satisfy the inf-sup condition, see e.g. It is more natural to consider the space of discontinuous Raviart– Thomas spaces (for which the inf-sup condition is almost immediate to prove), the reason for using Hdiv(Ω)-conforming Raviart–Thomas elements comes from the success of the analogous discretization of linear elasticity presented in [2]. This paper presents numerical evidence that the inf sup condition in the case of the Poisson problem is not satisfied

Structured simple meshes

Section 18: Numerical methods of differential equations

Counterexample