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A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem

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Abstract

We develop a formal construction of a pointwise divergence-free basis in the nonconforming virtual element method of arbitrary order for the Stokes problem introduced in Zhao et al. (SIAM J. Numer. Anal. 57(6):2730–2759, 2019). The proposed construction can be seen as a generalization of the divergence-free basis in Crouzeix-Raviart finite element space (Brenner, Math. Comp. 55(192):411–437, 1990; Thomasset, 1981) to the virtual element space. Using the divergence-free basis obtained from our construction, we can eliminate the pressure variable from the mixed system and obtain a symmetric positive definite system. Several numerical tests are presented to confirm the efficiency and the accuracy of our construction.

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Funding

This work is partially supported by NRF, contract no. 2021R1A2C1003340.

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  1. Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea

    Do Y. Kwak & Hyeokjoo Park

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  1. Do Y. Kwak
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  2. Hyeokjoo Park
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Correspondence to Do Y. Kwak.

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Kwak, D.Y., Park, H. A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem. Numer Algor 91, 449–471 (2022). https://doi.org/10.1007/s11075-022-01269-z

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