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An Overview of the Discontinuous Petrov Galerkin Method

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Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 157))

Abstract

We discuss our current understanding of the discontinuous Petrov Galerkin (DPG) Method with Optimal Test Functions and provide a literature review on the subject.

AMS(MOS) subject classifications. 65N30, 35L15.

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Notes

  1. 1.

    To our credit, the ultraweak formulation was used at that point very formally, without a proper Functional Analysis setting which we established later in [24].

  2. 2.

    See [44], p. 205, and [39], Thm. 5.18.2.

  3. 3.

    For Hilbert space, the supremum is attained and can be replaced with maximum.

  4. 4.

    Functional \(I(\delta u_{h}):= (R_{V }^{-1}(Bu_{h} - l),R_{V }^{-1}B\delta u_{h})_{V }\) is antilinear. Real part of an antilinear functional vanishes if and only if the whole functional vanishes. This follows from the fact that, for any antilinear functional I(v),  Im I(v) = Re  I(iv). 

  5. 5.

    One might say, a generalized least squares method.

  6. 6.

    Note that the local problems are well defined by the assumption that the test norm is localizable.

  7. 7.

    Under the assumption that the traces spaces are equipped with minimum energy extension norms.

  8. 8.

    Neglecting the error due to the approximation of optimal test functions.

  9. 9.

    Actually, BC τ n  = 0 does produce a very weak boundary layer, hard to observe even with very accurate adaptive simulations, see [42].

  10. 10.

    Intuitively speaking, the weights are selected in such a way that they “kill” the effect of the boundary layers.

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Acknowledgements

The work of the author Leszek F. Demkowicz was supported by the Department of Energy under Award Number DE-FC52-08NA28615.

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Authors and Affiliations

  1. Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    Leszek F. Demkowicz

  2. Department of Mathematics, Portland State University, Portland, OR, 97207, USA

    Jay Gopalakrishnan

Authors
  1. Leszek F. Demkowicz
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Correspondence to Leszek F. Demkowicz .

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Editors and Affiliations

  1. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Xiaobing Feng

  2. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Ohannes Karakashian

  3. Department of Mathematics, The University of Tennessee, Knoxville, Tennessee, USA

    Yulong Xing

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Demkowicz, L.F., Gopalakrishnan, J. (2014). An Overview of the Discontinuous Petrov Galerkin Method. In: Feng, X., Karakashian, O., Xing, Y. (eds) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-01818-8_6

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