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Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number

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Abstract

A preasymptotic error analysis of the interior penalty discontinuous Galerkin (IPDG) method of high order for Helmholtz equation with the first order absorbing boundary condition in two and three dimensions is proposed. We derive the \(H^1\)- and \(L^2\)- error estimates with explicit dependence on the wave number k. In particular, it is shown that if \(k(kh)^{2p}\) is sufficiently small, then the pollution errors of IPDG method in \(H^1\)-norm are bounded by \(O(k(kh)^{2p})\), which coincides with the phase error of the finite element method obtained by existent dispersion analyses on Cartesian grids, where h is the mesh size, p is the order of the approximation space and is fixed. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the symmetric IPDG method in reducing the pollution effect.

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Acknowledgments

The authors thank Professor Haijun Wu for his valuable comments leading to an improvement of the original results.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, Jiangsu Province, People’s Republic of China

    Yu Du

  2. Department of Mathematics, Jinling Institute of Technology, Nanjing, Jiangsu Province, People’s Republic of China

    Lingxue Zhu

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  1. Yu Du
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  2. Lingxue Zhu
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Correspondence to Lingxue Zhu.

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The work was partially supported by the National Natural Science Foundation of China Grant 11401272 and by the Natural Science Foundation of Jiangsu Province of China Grant BK20140105 and by the doctoral scientific research foundation of Jinling Institute of Technology Grant jit-b-201413.

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Du, Y., Zhu, L. Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number. J Sci Comput 67, 130–152 (2016). https://doi.org/10.1007/s10915-015-0074-8

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