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Error analysis for the pseudostress formulation of unsteady Stokes problem

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Abstract

In this paper, we are concerned with error analysis of the semi-discrete and fully discrete approximations to the pseudostress-velocity formulation of the unsteady Stokes problem. The pseudostress-velocity formulation of the Stokes problem allows a Raviart-Thomas mixed finite element. For the semi-discrete approximation, we prove that solution operators of homogeneous Stokes equations have the so-called parabolic smoothing property. For the fully discrete case, backward Euler and Crank-Nicolson schemes in time are considered. We present how to find the initial value of the pseudostress variable which is not given as initial data in Crank-Nicolson algorithm. Matrix equations are derived to show that backward Euler and Crank-Nicolson schemes corresponding to the pseudostress-velocity formulation are unconditionally stable. Finally, numerical examples are presented to test the performance of the algorithm and validity of the theory developed.

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

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Funding

Dongho Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2018R1D1A1B07050583 and NRF-2021R1F1A1062434). Eun-Jae Park was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2015R1A5A1009350). Boyoon Seo was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2020R1I1A1A01070361).

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Authors and Affiliations

  1. University College, Yonsei University, Seoul, 03722, Korea

    Dongho Kim

  2. School of Mathematics and Computing (Computational Science and Engineering), Yonsei University, Seoul, 03722, Korea

    Eun-Jae Park

  3. School of Mathematics and Computing, Yonsei University, Seoul, 03722, Korea

    Boyoon Seo

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  1. Dongho Kim
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  2. Eun-Jae Park
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  3. Boyoon Seo
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Correspondence to Eun-Jae Park.

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Kim, D., Park, EJ. & Seo, B. Error analysis for the pseudostress formulation of unsteady Stokes problem. Numer Algor 91, 959–996 (2022). https://doi.org/10.1007/s11075-022-01288-w

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  • DOI: https://doi.org/10.1007/s11075-022-01288-w

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