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Superconvergence error estimates of discontinuous Galerkin time stepping for singularly perturbed parabolic problems

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Abstract

A parabolic convection-diffusion-reaction problem is discretized by the non-symmetric interior penalty Galerkin (NIPG) method in space and discontinuous Galerkin (DG) method in time. To improve the order of convergence of the numerical scheme, we have used piecewise Lagrange interpolation at Gauss points and estimated the error bound in the discrete energy norm. We have shown superconvergence properties of the DG method, i.e., (k + 1)-order convergence in space and (l + 1)-order convergence in time, where k and l are the degrees of piecewise polynomials in the finite element space used in spatial and temporal variables, respectively. Numerical results are given to verify our theoretical findings.

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Acknowledgements

The authors wish to acknowledge the referees for their valuable comments and suggestions, which helped to improve the presentation.

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Authors and Affiliations

  1. Department of Mathematics, Birla Institute of Technology, Mesra, Ranchi, 835215, India

    Gautam Singh

  2. Department of Mathematics, Indian Institute of Technology Guwahati, Assam, 781039, India

    Srinivasan Natesan

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  1. Gautam Singh
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  2. Srinivasan Natesan
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Correspondence to Srinivasan Natesan.

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Singh, G., Natesan, S. Superconvergence error estimates of discontinuous Galerkin time stepping for singularly perturbed parabolic problems. Numer Algor 90, 1073–1090 (2022). https://doi.org/10.1007/s11075-021-01222-6

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