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Generalized local projection stabilized nonconforming finite element methods for Darcy equations

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Abstract

An a priori analysis for a generalized local projection stabilized finite element solution of the Darcy equations is presented in this paper. A first-order nonconforming \(\mathbb {P}^{nc}_{1}\) finite element space is used to approximate the velocity, whereas the pressure is approximated using two different finite elements, namely piecewise constant \(\mathbb {P}_{0}\) and piecewise linear nonconforming \(\mathbb {P}^{nc}_{1}\) elements. The considered finite element pairs, \(\mathbb {P}^{nc}_{1}/\mathbb {P}_{0}\) and \(\mathbb {P}^{nc}_{1}/\mathbb {P}^{nc}_{1} \), are inconsistent and incompatibility, respectively, for the Darcy problem. The stabilized discrete bilinear form satisfies an inf-sup condition with a generalized local projection norm. Moreover, a priori error estimates are established for both finite element pairs. Finally, the validation of the proposed stabilization scheme is demonstrated with appropriate numerical examples.

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Funding

The first author would like to thank the Tata Trusts traveling grants (ODAA/INT/19/189) and the National Mathematics Initiative (NMI), Department of Mathematics, Indian Institute of Science, Bengaluru, India. Furthermore, this work is partially supported by the Science and Engineering Research Board (SERB), Department of Science and Technology, Government of India, with the grant EMR/2016/003412.

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Authors and Affiliations

  1. National Mathematics Initiative, Indian Institute of Science, Bengaluru, 560012, India

    Deepika Garg

  2. Department of Computational and Data Sciences, Indian Institute of Science, Bengaluru, 560012, India

    Sashikumaar Ganesan

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  1. Deepika Garg
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  2. Sashikumaar Ganesan
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Correspondence to Sashikumaar Ganesan.

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Garg, D., Ganesan, S. Generalized local projection stabilized nonconforming finite element methods for Darcy equations. Numer Algor 89, 341–369 (2022). https://doi.org/10.1007/s11075-021-01117-6

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