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Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem

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Abstract

We introduce and study various discontinuous Galerkin (DG) finite element approximations for a parabolic variational inequality associated with a general obstacle problem in ℝd (d = 2, 3). For the fully-discrete DG scheme, we employ a piecewise linear finite element space for spatial discretization, whereas the time discretization is carried out with the implicit backward Euler method. We present a unified error analysis for all well known symmetric and non-symmetric DG fully discrete schemes, and derive error estimate of optimal order \({\cal O}(h + \Delta t)\) in an energy norm. Moreover, the analysis is performed without any assumptions on the speed of propagation of the free boundary and only the realistic regularity \({u_t} \in {{\cal L}^2}(0,T;{{\cal L}^2}(\Omega ))\) is assumed. Further, we present some numerical experiments to illustrate the performance of the proposed methods.

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Acknowledgements

The author would like to thank anonymous referee and editor for their helpful and constructive comments that lead to the improvement of this article.

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  1. Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi, 110016, India

    Papri Majumder

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  1. Papri Majumder
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Correspondence to Papri Majumder.

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The work was supported by the Council of Scientific and Industrial Research, India [RP03792G].

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Majumder, P. Unified error analysis of discontinuous Galerkin methods for parabolic obstacle problem. Appl Math 66, 673–699 (2021). https://doi.org/10.21136/AM.2021.0030-20

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