
Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods
Hartmann, Ralf
(2008)
Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods.
In:
VKI Lecture Series, 20080.
pp. 1107.
ISBN 13 9782930389885.
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AbstractAfter the introduction in Section 1 this lecture starts off with
recalling wellknown results from the numerical analysis of the
continuous finite element methods. In particular, we recall <em>a
priori</em> error estimates in the energy norm and the
L<sup>2</sup>norm including their proofs for higher order standard
finite element methods of Poisson's equation in Section 2 and for the
standard and the streamline diffusion finite element method of the
linear advection equation in Section 3.
<br>
</br>
We then introduce the discontinuous Galerkin discretization of the
linear advection equation in Section 4. Following
[BrezziMariniSüli2004] we consider two numerical flux
functions, the meanvalue flux and the upwind flux, and derive the
corresponding <em>a priori</em> error estimates. Whereas the standard
Galerkin discretization of the linear advection equation is unstable
and requires e.g. streamline diffusion for stabilization, we will see
in Section 4 that the discontinuous Galerkin discretization of the
linear advection based on upwind is stable without addition of
streamline diffusion.
<br>
</br>
Then in Section 5, we follow [ArnoldBrezziCockburnMarini2002] and
derive and analyze a variety of discontinuous Galerkin discretizations
of Poisson's equations. In particular, we derive the symmetric and
nonsymmetric interior penalty Galerkin method (SIPG and NIPG), the
method of BaumannOden (BO) and the first and second method of Bassi
and Rebay (BR1 and BR2). The analysis of the methods includes the
consistency and adjoint consistency of the schemes, continuity and
coercivity of the respective bilinear forms and <em>a priori</em>
error estimates for the interior penalty methods. In particular, we
will see that the adjoint consistent SIPG scheme is of optimal order
in the L<sup>2</sup>norm whereas the adjoint inconsistent NIPG scheme
is not.
<br>
</br>
Motivated by the connection of adjoint consistency of DG
discretizations to the availability of optimal order error estimates
in the L<sup>2</sup>norm we concentrate on the adjoint consistency
property in Section 6. In particular, here we follow [Hartmann2007]
and give a general framework for analyzing the consistency and adjoint
consistency of DG discretizations for linear problems with
inhomogeneous boundary conditions. This includes the derivation of
continuous adjoint problems associated to specific target quantities,
the derivation of primal and adjoint residual forms of the
discretizations and the discussion whether the discretizations in
combination with specific target quantities J(.) are adjoint
consistent or not. This analysis is performed in Sections 6.3 and 6.4
for the interior penalty DG discretization of the DirichletNeumann
boundary value problem of Poisson's equations and for the upwind DG
discretization of the linear advection equation, respectively.
<br>
</br>
Then in Section 7 the previously shown properties and estimates for
the interior penalty and the upwind DG discretization are used to
derive <em>a priori</em> estimates for the error measured in terms of
target quantities J(.). Here again, we will see that a discretization
must be consistent and adjoint consistent in order to provide optimal
error estimates in J(.).
<br>
</br>
This lecture is finalized with the Sections 8 and 9 which introduce
the DG discretizations of the compressible Euler and NavierStokes
equations. Additionally, the consistency and adjoint consistency
analysis which has been introduced in Section 6 for linear problems is
now generalized to nonlinear problems in Section 8.5. This analysis is
performed for the compressible Euler and NavierStokes equations in
Sections 8.6 and 9.3, respectively. This includes the derivation of an
adjoint consistent discretization of boundary conditions and of target
functionals. Here particular emphasis is placed on the aerodynamic
force coefficients like the drag, lift and moment coefficients.
<br>
</br>
Various examples in Sections 5.6, 7.3, 8.7 and 9.4 illustrate the
numerical methods described.
<br>
</br>
In particular, the contents of this lecture is given as follows
<br>
</br>
<br>
</br>
1) Introduction
<br>
</br>
1.1) Higher order discretization methods
<br>
</br>
1.2) Discontinuous Galerkin discretizations
<br>
</br>
1.3) Numerical analysis of finite element methods
<br>
</br>
1.4) Outline
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</br>
<br>
</br>
2) Higher order continuous FE methods for Poisson's equation
<br>
</br>
2.1) Poisson's equation
<br>
</br>
2.1.1) The homogeneous Dirichlet problem
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</br>
2.1.2) The inhomogeneous Dirichlet problem
<br>
</br>
2.1.3) The Neumann problem
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</br>
2.2) The standard finite element method for Poisson's equation
<br>
</br>
2.2.1) Consistency
<br>
</br>
2.2.2) Existence and uniqueness of discrete solutions
<br>
</br>
2.2.3) Best approximation property
<br>
</br>
2.2.4) Interpolation estimates
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</br>
2.2.5) <em>A priori</em> error estimates in the H<sup>1</sup> and L<sup>2</sup>norm
<br>
</br>
<br>
</br>
3) Higher order continuous FE methods for the linear advection equation
<br>
</br>
3.1) The linear advection equation
<br>
</br>
3.1.1) Variational formulation with strong boundary conditions
<br>
</br>
3.1.2) Variational formulation with weak boundary conditions
<br>
</br>
3.2) The standard Galerkin method with weak boundary conditions
<br>
</br>
3.3) The streamline diffusion method with weak boundary conditions
<br>
</br>
<br>
</br>
4) Higher order DG discretizations of the linear advection equation
<br>
</br>
4.1) Mesh related function spaces
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</br>
4.2) A variational formulation of the linear advection equation
<br>
</br>
4.3) Consistency, conservation property, coercivity and stability
<br>
</br>
4.4) The discontinuous Galerkin discretization
<br>
</br>
4.5) The local L<sup>2</sup>projection and approximation estimates
<br>
</br>
4.6) <em>A priori</em> error estimates
<br>
</br>
4.7) The discontinuous Galerkin discretization based on upwind
<br>
</br>
4.7.1) The importance of the interelement jump terms
<br>
</br>
4.7.2) The global and local conservation property
<br>
</br>
4.7.3) Consistency
<br>
</br>
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</br>
5) Higher order DG discretizations of Poisson's equation
<br>
</br>
5.1) The system and primal flux formulation
<br>
</br>
5.2) The DG discretization: Consistency and adjoint consistency
<br>
</br>
5.3) Derivation of various DG discretization methods
<br>
</br>
5.3.1) The SIPG and NIPG methods and the method of BaumannOden
<br>
</br>
5.3.2) The original DG discretization of Bassi and Rebay (BR1)
<br>
</br>
5.3.3) The modified DG discretization of Bassi and Rebay (BR2)
<br>
</br>
5.4) Consistency, adjoint consistency, continuity and coercivity
<br>
</br>
5.5) <em>A priori</em> error estimates
<br>
</br>
5.6) Numerical results
<br>
</br>
<br>
</br>
6) Consistency and adjoint consistency for linear problems
<br>
</br>
6.1) Definition of consistency and adjoint consistency
<br>
</br>
6.2) The consistency and adjoint consistency analysis
<br>
</br>
6.3) Adjoint consistency analysis of the IP discretization
<br>
</br>
6.3.1) The continuous adjoint problem to Poisson's equation
<br>
</br>
6.3.2) Primal residual form of the interior penalty DG discretization
<br>
</br>
6.3.3) Adjoint residual form of the interior penalty DG discretization
<br>
</br>
6.4) Adjoint consistency analysis of the upwind DG discretization
<br>
</br>
6.4.1) The continuous adjoint problem to the linear advection equation
<br>
</br>
6.4.2) Primal residual form of the DG discretization based on upwind
<br>
</br>
6.4.3) Adjoint residual form of the DG discretization based on upwind
<br>
</br>
<br>
</br>
7) <em>A priori</em> error estimates for target functionals J(.)
<br>
</br>
7.1) Upwind DG of the linear advection equation: Estimates in J(.)
<br>
</br>
7.2) IP DG discretization for Poisson's equation: Estimates in J(.)
<br>
</br>
7.3) Numerical results
<br>
</br>
<br>
</br>
8) Discontinuous Galerkin discretizations of the compressible Euler equations
<br>
</br>
8.1) Hyperbolic conservation equations
<br>
</br>
8.2) The compressible Euler equations
<br>
</br>
8.3) The DG discretization of the compressible Euler equations
<br>
</br>
8.4) Boundary conditions
<br>
</br>
8.5) Consistency and adjoint consistency for nonlinear problems
<br>
</br>
8.5.1) The consistency and adjoint consistency analysis
<br>
</br>
8.6) Adjoint consistency analysis of DG for the compressible Euler equations
<br>
</br>
8.6.1) The continuous adjoint problem to the compressible Euler equations
<br>
</br>
8.6.2) Primal residual form of DG for the compressible Euler equations
<br>
</br>
8.6.3) Adjoint residual form of DG for the compressible Euler equations
<br>
</br>
8.7) Numerical results
<br>
</br>
<br>
</br>
9) DG discretizations of the compressible NavierStokes equations
<br>
</br>
9.1) The compressible NavierStokes equations
<br>
</br>
9.2) DG discretizations of the compressible NavierStokes equations
<br>
</br>
9.3) Adjoint consistency analysis of DG for the compressible NavierStokes equations
<br>
</br>
9.3.1) The continuous adjoint problem to the compressible NS equations
<br>
</br>
9.3.2) Primal residual form of DG for the compressible NS equations
<br>
</br>
9.3.3) Adjoint residual form of DG for the compressible NS equations
<br>
</br>
9.4) Numerical results
<br>
</br>
<br>
</br>
Acknowledgements
<br>
</br>
Bibliography Item URL in elib:  https://elib.dlr.de/57074/ 

Document Type:  Contribution to a Collection 

Title:  Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods 

Authors:  Authors  Institution or Email of Authors  Authors ORCID iD 

Hartmann, Ralf  UNSPECIFIED  UNSPECIFIED 


Date:  October 2008 

Open Access:  Yes 

Gold Open Access:  No 

In SCOPUS:  No 

In ISI Web of Science:  No 

Volume:  20080 

Page Range:  pp. 1107 

Editors:  Editors  Email 

Deconinck, Herman  UNSPECIFIED 


Series Name:  VKI Lecture Series 

ISBN:  13 9782930389885 

Status:  Published 

Keywords:  Numerical Analysis, Higher Order Discretization Methods, Discontinuous Galerkin Methods, Error estimation, adjoint consistency, target functionals 

HGF  Research field:  Aeronautics, Space and Transport 

HGF  Program:  Aeronautics 

HGF  Program Themes:  Aircraft Research (old) 

DLR  Research area:  Aeronautics 

DLR  Program:  L AR  Aircraft Research 

DLR  Research theme (Project):  L  Concepts & Integration (old) 

Location: 
Braunschweig


Institutes and Institutions:  Institute of Aerodynamics and Flow Technology > CASE 

Deposited By: 
Hartmann, Dr.rer.nat. Ralf


Deposited On:  06 Jan 2009 

Last Modified:  31 Jul 2019 19:23 

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