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

Haoran Liu; Michael Neilan; M. Baris Otus

doi:10.1515/jnma-2021-0125

Published by De Gruyter April 14, 2022

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

Haoran Liu , Michael Neilan and M. Baris Otus

From the journal Journal of Numerical Mathematics

https://doi.org/10.1515/jnma-2021-0125

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Abstract

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.

Keywords: finite elements; Stokes problem; boundary correction; divergence-free

Classification: 65N30; 65N12; 76M10

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 a_h 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

he−1∫e|δ|2j∂jv∂dj2ds⩽he−1δe2j∥Djv∥L2(e)2⩽he−2δe2j(k+1)(k+2)2∥Djv∥L2(Te)2,j=1,…,k.

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

he−1∫e|δ|2j∂jv∂dj2ds⩽∏ℓ=k−j+1k−1Cℓ(k+1)(k+2)2he−2jδ2j∥∇v∥L2(Te)2⩽∏ℓ=k−j+1k−1Cℓ(k+1)(k+2)2cδ2j∥∇v∥L2(Te)2

where C_ℓ > 0 is the maximum eigenvalue of of a matrix defined in [33, Sect. 3], which numerically scales as O(ℓ⁴):

∑e∈EhBhe−1∥Shv−v∥L2(e)2⩽∑j=1k∏ℓ=k−j+1k−1Cℓ(k+1)(k+2)2cδ2∥∇v∥L2(Ωh)2.

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

∑e∈EhB∫e∂v∂nh⋅(Shv−v)dsC†−1cδ∥∇v∥L2(Ωh)2

where C_† is given by (4.8). Applying this estimate in (4.7) of the proof of Lemma 4.3 shows that a_h is coercive provided c_δ < C†−1 .

B Proof of Lemma 5.1

For a boundary edge e ∈ EhB with endpoints a₁, a₂, let x(t)=a1+the−1(a2−a1)(0⩽t⩽he) be its parameterization, and introduce the 2D parameterization φ(t, s) = x(t) + s d(x(t)) for 0 ⩽ t ⩽ h_e and 0 ⩽ s ⩽ δ(x(t)). The Taylor remainder estimation with S_hu + R_h u = 0 yields

|Shu(x(t))|=|Rhu(x(t))|=1k!∫0δ(x(t))∂k+1u∂dk+1(φ(t,s))(δ(x(t))−s)kds.

Applying the Cauchy–Schwarz inequality, we obtain

|Shu(x(t))|⩽Cδ(x(t))k+1/2∫0δ(x(t))|∂k+1u∂dk+1(φ(t,s))|2ds1/2

and therefore

he−1∥Shu∥L2(e)2⩽Che−1δe2k+1∫0he∫0δ(x(t))∂k+1u∂dk+1(φ(t,s))2dsdt⩽Che2k∫0he∫0δ(x(t))∂k+1u∂dk+1(φ(t,s))2dsdt

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

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

Abstract

References

A Details of estimate (4.8)

B Proof of Lemma 5.1

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