Inf-sup stabilized Scott–Vogelius pairs on general simplicial grids by Raviart–Thomas enrichment
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This paper considers the discretization of the Stokes equations with Scott–Vogelius pairs of finite element spaces on arbitrary shape-regular simplicial grids. A novel way of stabilizing these pairs with respect to the discrete inf-sup condition is proposed and analyzed. The key idea consists in enriching the continuous polynomials of order k of the Scott–Vogelius velocity space with appropriately chosen and explicitly given Raviart–Thomas bubbles. This approach is inspired by [Li/Rui, IMA J. Numer. Anal, 2021], where the case k=1 was studied. The proposed method is pressure-robust, with optimally converging H^1-conforming velocity and a small H(div)-conforming correction rendering the full velocity divergence-free. For k≥ d, with d being the dimension, the method is parameter-free. Furthermore, it is shown that the additional degrees of freedom for the Raviart–Thomas enrichment and also all non-constant pressure degrees of freedom can be condensated, effectively leading to a pressure-robust, inf-sup stable, optimally convergent P_k × P_0 scheme. Aspects of the implementation are discussed and numerical studies confirm the analytic results.
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