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Two new approaches for solving elliptic obstacle problems using discontinuous Galerkin methods

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Abstract

The main aim of this article is to present two new ways to solve the elliptic obstacle problem by using discontinuous Galerkin finite element methods. In Gaddam and Gudi (Comput Methods Appl Math 18:223–236, 2018. https://doi.org/10.1515/cmam-2017-0018), a bubble enriched conforming quadratic finite element method is introduced and analyzed for the obstacle problem in dimension 3. In this article, without adding bubble functions, we derive optimal order (with respect to regularity) a priori error estimates in dimension 2 and 3 using the localized behavior of DG methods. We consider two different discrete sets, one with integral constraints motivated from Gaddam and Gudi (2018) and the other with nodal constraints at quadrature points. We also discuss the reliability and efficiency of a proposed a posteriori error estimator. The analysis is carried out in a unified setting which holds for several DG methods. Numerical results are presented to illustrate the theoretical findings.

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Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, VIT-AP University, Near Vijayawada, Andhra Pradesh, 522237, India

    Sharat Gaddam

  2. Department of Mathematics, Indian Institute of Science, Bangalore, 560012, India

    Thirupathi Gudi

  3. Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, 110016, India

    Kamana Porwal

Authors
  1. Sharat Gaddam
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  2. Thirupathi Gudi
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  3. Kamana Porwal
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Correspondence to Thirupathi Gudi.

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Communicated by Ralf Hiptmair.

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The first author’s work is supported by University Grants Commission, India. The second author’s work is supported by the DST MATRICS Grant. The third author’s work is supported by DST Inspire Faculty Research Grant.

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Gaddam, S., Gudi, T. & Porwal, K. Two new approaches for solving elliptic obstacle problems using discontinuous Galerkin methods. Bit Numer Math 62, 89–124 (2022). https://doi.org/10.1007/s10543-021-00869-w

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