Workshop on Numerical Methods and Analysis in CFD - Abstract

Merdon, Christian

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

This talk presents a novel way to stabilize the Scott--Vogelius finite element method for the Stokes equations on arbitrary regular simplicial grids.
The key idea, inspired by [1] for k = 1, consists in enriching the continuous polynomials of order k of the Scott--Vogelius velocity space with appropriately chosen and explicitly given Raviart--Thomas functions. The proposed method is inf-sup stable and pressure-robust. The optimally converging H ¹ -conforming velocity comes with a small H(div)-conforming correction that renders the full velocity divergence-free. For k ≥ d in d dimensions, the method is parameter-free.
The additional degrees of freedom for the Raviart--Thomas enrichment and all non-constant pressure degrees of freedom can be condensated, which effectively results in a pressure-robust, inf-sup stable, optimally convergent P_k × P₀ scheme. Some aspects are discussed and numerical studies confirm the analytic results.

References:
[1] Xu Li, Hongxing Rui, A low-order divergence-free H(div)-conforming finite element method for Stokes flows, IMA Journal of Numerical Analysis, 2021