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A divergence-free finite element method for the Stokes problem with boundary correction

  • Haoran Liu , Michael Neilan EMAIL logo and M. Baris Otus


This paper constructs and analyzes a boundary correction finite element method for the Stokes problem based on the Scott–Vogelius pair on Clough–Tocher splits. The velocity space consists of continuous piecewise polynomials of degree k, and the pressure space consists of piecewise polynomials of degree (k – 1) without continuity constraints. A Lagrange multiplier space that consists of continuous piecewise polynomials with respect to the boundary partition is introduced to enforce boundary conditions and to mitigate the lack of pressure-robustness. We prove several inf-sup conditions, leading to the well-posedness of the method. In addition, we show that the method converges with optimal order and the velocity approximation is divergence-free.

Classification: 65N30; 65N12; 76M10
  1. Funding: This work was supported in part by the National Science Foundation through grant number DMS-2011733.


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A Details of estimate (4.8)

Here we provide the calculations in the estimate (4.8) that ensures the bilinear form ah is coercive. As a first step, we provide an explicit estimate of the constant C > 0 in the first estimate of (4.1). To this end, we use the discrete trace inequality in [22, Thm. 3] to obtain


Combining this estimate with the inverse estimate in [33, Thm. 2] yields


where C > 0 is the maximum eigenvalue of of a matrix defined in [33, Sect. 3], which numerically scales as O(4):


Combining this result with the same discrete trace inequality, we have


where C is given by (4.8). Applying this estimate in (4.7) of the proof of Lemma 4.3 shows that ah is coercive provided cδ < C1 .

B Proof of Lemma 5.1

For a boundary edge e EhB with endpoints a1, a2, let x(t)=a1+the1(a2a1)(0the) be its parameterization, and introduce the 2D parameterization φ(t, s) = x(t) + s d(x(t)) for 0 ⩽ the and 0 ⩽ sδ(x(t)). The Taylor remainder estimation with Shu + Rh u = 0 yields


Applying the Cauchy–Schwarz inequality, we obtain


and therefore


where we used assumption A in the last inequality. The estimate in Lemma 5.1 now follows from a change of variables (cf. [4, 31]) and summing over e EhB .

Received: 2021-11-03
Revised: 2022-03-17
Accepted: 2022-03-18
Published Online: 2022-04-14
Published in Print: 2023-06-27

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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