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Inf-sup stabilized Scott–Vogelius pairs on general simplicial grids by Raviart–Thomas enrichment

by   Volker John, et al.

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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