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A numerical scheme for computing approximations to the inf-sup constant of the divergence operator in bounded Lipschitz polytopes in R is proposed. The method is based on a conforming approximation of the pressure space based on piecewise polynomials of some fixed degree k ≥ 0. The scheme can be viewed as a Rayleigh–Ritz method and it gives monotonically decreasing approximations of the inf-sup constant under mesh refinement. The new approximation replaces the H norm of a gradient by a discrete H norm which behaves monotonically under mesh refinement. By discretizing the pressure space with piecewise polynomials, upper bounds to the inf-sup constant are obtained. Error estimates are presented that prove convergence rates for the approximation of the inf-sup constant provided it is an isolated eigenvalue of the corresponding non-compact eigenvalue problem; otherwise, plain convergence is achieved. Numerical computations on uniform and adaptive meshes are provided. 1. The inf-sup constant of the divergence Let Ω ⊆ R for n ≥ 2 be an open, bounded, connected Lipschitz polytope, let V := H 0 (Ω;R ) denote the L vector fields over Ω with generalized first derivatives in L(Ω) and vanishing trace on the boundary, and let and Q := L0(Ω) denote the L functions with vanishing average over Ω. In fluid models, V usually refers to the space of velocity fields and Q is the pressure space. The inf-sup constant β = β(Ω) for the divergence is defined as (1) β := inf p∈Q\{0} |∇p|−1 ‖p‖ = inf p∈Q\{0} sup v∈V \{0} (p, div v)L2(Ω) ‖p‖‖Dv‖ where |∇p|−1 denotes the H norm of ∇p and ‖ · ‖ denotes the L norm over the domain Ω. Throughout this article, the space V is equipped with the norm ‖D · ‖. The inf-sup constant, sometimes also called Ladyzhenskaya–Babuška–Brezzi (LBB) constant, and its variants are in close connection to stability considerations in several applications such as fluid mechanics [21, 29], Korn’s inequalities in the theory of elasticity [12], or the splitting of polyharmonic functions into second-order systems [20]. It is known [1, 8] that for a broad class of domains (including Lipschitz domains) the constant β is positive, β > 0, which implies that the divergence operator div : V → Q possesses a continuous right-inverse. The numerical approximation of β is challenging because the Rayleigh quotient from (1) belongs to a non-compact Date: 23rd June 2017. 1991 Mathematics Subject Classification. 65N12, 65N15, 65N30, 76D07.

Rayleigh–Ritz approximation of the inf-sup constant for the divergence