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A priori error estimates of a discontinuous Galerkin method for the Navier-Stokes equations

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Abstract

This-36pt paper considers a discontinuous Galerkin finite element method for the 2D transient and incompressible Navier-Stokes model. Following the analysis of Heywood and Rannacher (SIAM J. Numer. Anal. 19:275–311, 1982), we derive optimal velocity and pressure error estimates in \(L^{\infty }({\textbf {L}}^{2})\) and \(L^{\infty }(L^{2})\)-norms, respectively, for the discontinuous Galerkin case. We use standard \(L^{2}\)-projection and modified Stokes operator but on appropriate broken Sobolev spaces, and then standard duality arguments to achieve these results. For sufficiently small data, uniform in time estimates are proved. Based on the backward Euler method, time discretization is carried out and fully discrete error estimates are derived. Finally, numerical experiments are conducted to verify our theoretical findings.

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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request

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Funding

The first author received support from Indian Institute of Technology Goa, India, under the start up grant project no. 2019/SG/SB/022. And the third author received financial support from the Council of Scientific & Industrial Research (CSIR) (09/796(0095)/2019-EMR-I).

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Authors and Affiliations

  1. School of Mathematics and Computer Science, Indian Institute of Technology Goa, Ponda, Goa, 403401, India

    Saumya Bajpai

  2. Department of Mathematical Sciences, Tezpur University, Tezpur, Sonitpur, Assam, 784028, India

    Deepjyoti Goswami & Kallol Ray

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  1. Saumya Bajpai
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  2. Deepjyoti Goswami
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  3. Kallol Ray
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All authors contributed equally to this work. And all authors read and approved the final manuscript.

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Correspondence to Deepjyoti Goswami.

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Bajpai, S., Goswami, D. & Ray, K. A priori error estimates of a discontinuous Galerkin method for the Navier-Stokes equations. Numer Algor 94, 937–1002 (2023). https://doi.org/10.1007/s11075-023-01525-w

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  • DOI: https://doi.org/10.1007/s11075-023-01525-w

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