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.
Keywords
AMS(MOS) subject classifications. 65N30, 35L15.
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- 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.
- 3.
For Hilbert space, the supremum is attained and can be replaced with maximum.
- 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.
One might say, a generalized least squares method.
- 6.
Note that the local problems are well defined by the assumption that the test norm is localizable.
- 7.
Under the assumption that the traces spaces are equipped with minimum energy extension norms.
- 8.
Neglecting the error due to the approximation of optimal test functions.
- 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.
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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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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