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Erschienen in:
A Quick Tutorial on DG Methods for Elliptic Problems Kapitel lesen Erstes Kapitel lesen

2014 | OriginalPaper | Buchkapitel

An Overview of the Discontinuous Petrov Galerkin Method

verfasst von : Leszek F. Demkowicz, Jay Gopalakrishnan

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

Verlag: Springer International Publishing

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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.

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Vorheriges Kapitel A Local Timestepping Runge–Kutta Discontinuous Galerkin Method for Hurricane Storm Surge Modeling
Nächstes Kapitel Discontinuous Galerkin for the Radiative Transport Equation
Fußnoten
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
See [44], p. 205, and [39], Thm. 5.18.2.
 
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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Metadaten
Titel
An Overview of the Discontinuous Petrov Galerkin Method
verfasst von
Leszek F. Demkowicz
Jay Gopalakrishnan
Copyright-Jahr
2014
Buch
Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations

Print ISBN: 978-3-319-01817-1

Electronic ISBN: 978-3-319-01818-8

Copyright-Jahr: 2014

DOI
https://doi.org/10.1007/978-3-319-01818-8_6