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Analysis of Hybrid Discontinuous Galerkin Methodsfor Incompressible Flow Problems
Christian Waluga1
advised by Prof. Herbert Egger2 Prof. Wolfgang Dahmen3
1Aachen Institute for Advanced Study in Computational Engineering Science, RWTH Aachen University2M2 – Center of Mathematics, Technische Universitat Munchen
3 Institut fur Geometrie und Praktische Mathematik, RWTH Aachen University
February 3, 2012
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 1 / 34
Outline
Outline
1 IntroductionMotivationGoverning equationsDiscretizationOverview of the thesis
2 A Hybrid Discontinuous Galerkin MethodPreliminariesPoisson problemStokes problemNavier-Stokes problem
3 A posteriori error estimators and adaptivityError estimationAn adaptive algorithmNumerical results
4 Conclusions
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 2 / 34
Introduction
Outline
1 IntroductionMotivationGoverning equationsDiscretizationOverview of the thesis
2 A Hybrid Discontinuous Galerkin Method
3 A posteriori error estimators and adaptivity
4 Conclusions
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 3 / 34
Introduction Motivation
Simulation aims to predict physical phenomena which aredifficult, expensive, or even impossibleto observe in conventional experiments.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 4 / 34
Introduction Motivation
Source: Wikimedia Commons
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 5 / 34
Introduction Motivation
Real-world problems
⇓
Mathematical language
⇓Computable problems
⇓Approximate solutions
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 6 / 34
Introduction Governing equations
Motion of fluids
Incompressible Navier-Stokes equations:
−ν∆u + u ·∇u + ∇p = fdivu = 0
on Ω ⊂ Rd
velocity: u = [u1, . . . , ud] pressure: p
Boundary conditions:
u = gD on ∂ΩD (Dirichlet)
ν∂nu− pn = gN on ∂ΩN (Neumann)
For simplicity, let us suppose that u = 0 on ∂Ω.
∂ΩN
∂ΩD
Ω
R2
Difficulties? incompressibility, convective terms, nonlinearity
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 7 / 34
Introduction Governing equations
Motion of fluids
Incompressible Navier-Stokes equations:
−ν∆u + u ·∇u + ∇p = fdivu = 0
on Ω ⊂ Rd
velocity: u = [u1, . . . , ud] pressure: p
Boundary conditions:
u = gD on ∂ΩD (Dirichlet)
ν∂nu− pn = gN on ∂ΩN (Neumann)
For simplicity, let us suppose that u = 0 on ∂Ω.
∂ΩN
∂ΩD
Ω
R2
Difficulties? incompressibility, convective terms, nonlinearity
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 7 / 34
Introduction Governing equations
Motion of fluids
Incompressible Navier-Stokes equations:
−ν∆u + u ·∇u + ∇p = fdivu = 0
on Ω ⊂ Rd
velocity: u = [u1, . . . , ud] pressure: p
Boundary conditions:
u = gD on ∂ΩD (Dirichlet)
ν∂nu− pn = gN on ∂ΩN (Neumann)
For simplicity, let us suppose that u = 0 on ∂Ω.
∂ΩN
∂ΩD
Ω
R2
Difficulties? incompressibility, convective terms, nonlinearity
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 7 / 34
Introduction Discretization
Popular discretization methods
Finite Volume (FV)
• locally conservative
• suitable for convection dominated flow
• extension to higher orders is complicated
Finite Element (FE)
• straightforward extension to higher orders
• not locally conservative
• unstable for dominant convection on coarsemeshes
Discontinuous Galerkin (DG)
• combines advantages of FV and FE methods
• very suitable for adaptivity
• increased number of degrees of freedom
• reduced sparsity in the discrete system
FE
DG
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 8 / 34
Introduction Discretization
Circumventing the drawbacks of DG...
add additional unknowns at the interfaces (hybridization*).
relax the coupling across interfaces.
eliminate element unknowns (static condensation).
DG HDG* HDG
Literature: Cockburn, Gopalakrishnan and Lazarov. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methods forsecond order elliptic problems. 2009.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 9 / 34
Introduction Discretization
Hybrid Discontinuous Galerkin (HDG) Methods
DG methods with some (algorithmic) advantages
better sparsity structure (for higher orders)static condensationelement-based assembly
but: Implementation more involved!
Overview: Cockburn, Gopalakrishnan, Lazarov. Unified hybridization of discontinuousGalerkin, mixed and continuous Galerkin methods for second order elliptic problems.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 10 / 34
Introduction Overview of the thesis
Overview of the thesis
A priori analysis for different flow model-problems.
Technical results for the hp analysis.
Suitable error estimators to drive adaptive algorithms.
Hybrid mortar methods for domain decomposition.
Numerical experiments.
Analysis of Hybrid Discontinuous Galerkin Methods forIncompressible Flow Problems
Von der Fakultät für Mathematik, Informatik undNaturwissenschaften der RWTH Aachen University zurErlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte
D i s s e r t a t i o n
vorgelegt von
Diplom-Ingenieur
Christian Waluga
aus Würselen
Berichter: Univ.-Prof. Dr. Herbert EggerUniv.-Prof. Dr. Wolfgang Dahmen
Tag der mündlichen Prüfung: 3. Februar 2012
Diese Dissertation ist auf den Internetseitender Hochschulbibliothek online verfügbar.
Updated and corrected version: May 23, 2013
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 11 / 34
Introduction Overview of the thesis
Overview of the thesis
A priori analysis for different flow model-problems.
Technical results for the hp analysis.
Suitable error estimators to drive adaptive algorithms.
Hybrid mortar methods for domain decomposition.
Numerical experiments.
Analysis of Hybrid Discontinuous Galerkin Methods forIncompressible Flow Problems
Von der Fakultät für Mathematik, Informatik undNaturwissenschaften der RWTH Aachen University zurErlangung des akademischen Grades eines Doktors der
Naturwissenschaften genehmigte
D i s s e r t a t i o n
vorgelegt von
Diplom-Ingenieur
Christian Waluga
aus Würselen
Berichter: Univ.-Prof. Dr. Herbert EggerUniv.-Prof. Dr. Wolfgang Dahmen
Tag der mündlichen Prüfung: 3. Februar 2012
Diese Dissertation ist auf den Internetseitender Hochschulbibliothek online verfügbar.
Updated and corrected version: May 23, 2013
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 11 / 34
HDG for incompressible flow
Outline
1 Introduction
2 A Hybrid Discontinuous Galerkin MethodPreliminariesPoisson problemStokes problemNavier-Stokes problem
3 A posteriori error estimators and adaptivity
4 Conclusions
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 12 / 34
HDG for incompressible flow Preliminaries
Triangulations of the domain Ω
Hybrid meshes with a bounded level of nonconformity (shape regular, quasi-uniform)
T1 T2 T3
T4 T5
T6
T7
T8T9
T10
T11 T12
E0,8E0,9E0,10
E′0,10
E0,1
E′0,1 E0,2
E0,3
E0,5
E0,6
E′0,6
E1,2 E2,3
E3,4
E4,7
E4,5
E5,6
E6,7E7,8
E7,12
E2,12E1,10 E2,11
E11,12
E8,9
E8,12E9,11
E10,11
E9,10
Collection of elements: Th = T1, T2, . . .
Boundary + interior facets: Eh := E0,1, E0,2, . . . , E1,2, E2,3, . . .
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 13 / 34
HDG for incompressible flow Preliminaries
Hybrid Discontinuous Galerkin?
Galerkin?
seek a weak solution in a finite dimensional spacesolution can be computed by solving a linear system of equations
AU = F A : stiffness matrix, U : unknowns, F : right hand side
Discontinuous?
find a discrete solution uh ∈ Vh on Thspace Vh consists of piecewise discontinuous polynomial functions.
Vh =vh ∈ L2(Ω) : vh|T ∈ Pk(T ), T ∈ Th
Hybrid?
also approximate the trace uh ∈ Vh on Eh
Vh =vh ∈ L2(Eh) : vh|E ∈ Pk(E), E ∈ Eh, vh = 0 on ∂Ω
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 14 / 34
HDG for incompressible flow Poisson problem
HDG for Poisson
−∆u = f in Ω and u = 0 on ∂Ω.
Discrete problem
Find (uh, uh) ∈ Vh × Vh, such that
ah(uh, uh; vh, vh) = fh(vh, vh) for all (vh, vh) ∈ Vh × Vh.
where we define
ah(uh, uh; vh, vh) :=∑
T∈Th
(∫T∇uh ·∇vh dx−
∫∂T
∂nuh · (vh − vh) ds
−∫∂T
(uh − uh) · ∂nvh ds+
∫∂T
γTk2T
hT(uh − uh) · (vh − vh) ds
)fh(vh, vh) :=
∑T∈Th
∫Tf · vh dx
Consistency For u ∈ H10 (Ω) ∩H2(Th), there holds
ah(u, u|Eh; vh, vh) = fh(vh, vh) for all (vh, vh) ∈ Vh × Vh.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 15 / 34
HDG for incompressible flow Poisson problem
A priori analysis for Poisson
Theorem (Coercivity):
For γT sufficiently large, there holds
ah(vh, vh; vh, vh) ≥ 12‖(vh, vh)‖21,h, ∀ (vh, vh) ∈ Vh × Vh.
Energy norm:
‖(vh, vh)‖21,h :=∑
T∈Th
(‖∇vh‖2T + γT
k2T
hT|vh − vh|2∂T
)1/2.
Remarks:
Optimal γT is explicitly given by sharp trace inverse estimates.
Existence, uniqueness and stability bounds of discrete solution (Lax-Milgram).
Standard arguments and approximation results yield order-optimal error estimates.
Literature: Arnold et. al. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. 2002.Warburton, Hesthaven. On the constants in hp-finite element trace inverse inequalities, 2003.Burman, Ern. Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations, 2007.Egger. A class of hybrid mortar finite element methods for interface problems with non-matching meshes, 2009.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 16 / 34
HDG for incompressible flow Stokes problem
HDG for Stokes
System of equations for u = [u1, . . . , ud] with incompressibility constraint.
−∆u + ∇p = f and divu = 0 in Ω and u = 0 on ∂Ω.
Finite dimensional spaces
discrete velocity (uh, uh): Vh := Vdh (on Th), Vh := Vd
h (on Eh)
discrete pressure ph: Qh :=qh ∈ L2
0(Ω) : qh|T ∈ Pk−1(T )
Discrete (saddle-point) problem
Find (uh, uh) ∈ Vh × Vh and ph ∈ Qh, such that
ah(uh, uh;vh, vh) + bh(vh, vh; ph) = fh(vh, vh) for all (vh, vh) ∈ Vh × Vh,
bh(uh, uh; qh) = 0 for all qh ∈ Qh.
The bilinear form associated with the incompressibility constraint is defined as
bh(uh, uh; qh) :=∑
T∈Th
(−∫T
divuh · qh dx−∫∂T
(uh − uh) · n · qh ds)
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 17 / 34
HDG for incompressible flow Stokes problem
A priori analysis for Stokes
The crucial part of the analysis is the following stability condition.
Theorem (discrete inf-sup stability):
There exists a constant β independent of the mesh and the polynomial degree k, such that
sup(vh,vh)∈Vh×Vh
bh(vh, vh; qh)
‖(vh, vh)‖1,h≥ βk−1/2‖qh‖0,h, ∀ qh ∈ Qh.
Remarks:
The proof is based on an argument due to Fortin.
We employ new hp-estimates for the L2-orthogonal projections:
ΠT : H1(T )→ Pk(T ) ΠE : H1(T )→ Pk(E).
The analysis applies to hybrid meshes (e.g. tri/quad or tet/hex) with hanging nodes.
Error estimates of such mixed methods depend on the approximation properties of the finiteelement spaces and the discrete inf-sup estimate.
Hence, the k-dependence is important for high order discretizations.
Literature: Fortin. Analysis of the Convergence of Mixed Finite Element Methods, 1977.Brezzi, Fortin. Mixed and Hybrid Finite Element Methods. Springer, 1991.Boffi et. al. Mixed finite elements, compatibility conditions, and applications, Springer, 2008.Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 18 / 34
HDG for incompressible flow Stokes problem
Comparison with related work
The discrete inf-sup constant is usually of the order k−s (optimal method: s = 0).
supvh∈Vh
bh(vh, qh)
‖vh‖Vh
≥ βk−s ‖qh‖Qh∀qh ∈ Qh,
Type Reference Suboptimality s Element types Balanced approx.Spectral [BM:99] 0 quad, hex yes
CG [AC:02] 0 quad yesCG [SS:96] ε quad, hex yesCG [S:98] 1/2 quad no
HDG [EW:11] 1/2 tri, quad, tet, hex yes
DG [T:02] d−12
quad, hex noDG [SST:03] 1 quad, hex yesCG [S:98] ( 3 ) tri no
Literature: [SS:96] Stenberg, Suri. Mixed finite element methods for problems in elasticity and Stokes flow, 1996.[S:98] Schwab. p- and hp- finite element methods: theory and applications in solid and fluid mechanics, 1998.[BM:99] Bernardi, Maday. Uniform inf-sup conditions for the spectral discretization of the Stokes problem, 1999.[AC:02] Ainsworth, Coggins. A uniformly stable family of mixed hp-finite elements with continuous pressures for incompressible flow.[T:02] Toselli. hp discontinuous Galerkin approximations for the Stokes problem, 2002.[SST:03] Schotzau, Schwab, Toselli. Mixed hp-DGFEM for incompressible flows, 2003.[EW:11] Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 19 / 34
HDG for incompressible flow Stokes problem
Convergence rates for Stokes
We can prove order-optimal convergence rates; i.e, for (u, p) ∈Hm+1(Th)×Hm(Th):
‖(u− uh,u|Eh− uh)‖1,h + 1√
k‖p− ph‖0,h
hm
km−1‖u‖m+1,Ω +
hm
km−1/2‖p‖m,Ω
k level velocity error rate pressure error rate
1
0 2.2052 · 101 − 9.9237 · 100 −1 1.1569 · 101 0.93 4.9250 · 100 1.012 5.8722 · 100 0.98 2.4036 · 100 1.033 2.9487 · 100 0.99 1.1837 · 100 1.02
2
0 3.3991 · 100 − 1.8315 · 100 −1 8.6146 · 10−1 1.98 4.6010 · 10−1 1.99
2 2.1494 · 10−1 2.00 1.1302 · 10−1 2.03
3 5.3539 · 10−2 2.01 2.7859 · 10−2 2.02
3
0 2.5261 · 10−1 − 1.7153 · 10−1 −1 3.1045 · 10−2 3.02 2.0734 · 10−2 3.05
2 3.8526 · 10−3 3.01 2.5606 · 10−3 3.02
3 4.8003 · 10−4 3.00 3.1885 · 10−4 3.01
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 20 / 34
HDG for incompressible flow Navier-Stokes problem
HDG for Navier-Stokes
Add nonlinear convective terms...
−ν∆u + u ·∇u + ∇p = f and divu = 0 in Ω and u = 0 on ∂Ω.
Discrete (nonlinear) problem
Find (uh, uh) ∈ Vh × Vh and ph ∈ Qh, such that
νah(uh, uh;vh, vh) + ch(uh, uh;uh, uh;vh, vh) + bh(vh, vh; ph) = fh(vh, vh),
bh(uh, uh; qh) = 0.
for all (vh, vh) ∈ Vh × Vh and all qh ∈ Qh.
The form associated with the convective terms is defined as
ch(wh, wh;uh, uh;vh, vh)
:=∑
T∈Th
(−∫Tuh · (wh ·∇vh) dx+
∫∂T
(wh · n) uh/uh · (vh − vh) ds
+ 12bh(wh, wh;uh · vh).
where uh/uh = f(wh, wh;uh, uh) denotes an upwind value.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 21 / 34
HDG for incompressible flow Navier-Stokes problem
What is upwinding?
Numerical scheme adapts to the direction of propagation of information in the flow field.
uh/uh :=
uh if wh · n ≤ 0,
uh if wh · n > 0.
wh
uh/uh = uh
uh/uh = uh
Literature: Reed and Hill. Triangular mesh methods for the neutron transport equation. 1973.Egger and Schoberl. A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusion problems. 2010.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 22 / 34
HDG for incompressible flow Navier-Stokes problem
Dealing with the nonlinearity...
The discrete Navier-Stokes problem is equivalent to a fixed point problem
(uh, uh) = Φh(uh, uh)
The operator Φh : (wh, wh)→ (uh, uh) is defined by the following discrete Oseen problem
Find (uh, uh) ∈ Vh × Vh and ph ∈ Qh, such that
νah(uh, uh;vh, vh) + ch(wh, wh;uh, uh;vh, vh) + bh(ph;vh, vh) = fh(vh, vh),
bh(qh;uh, uh) = 0.
for all (vh, vh) ∈ Vh × Vh and all qh ∈ Qh.
Remarks:
The fixed-point operator is well-defined.
Existence of fixed points (thus discrete solutions) by Leray-Schauder principle.
For small ‖f‖Ω, we also obtain
Uniqueness of a discrete solution (Banach)Order-optimal convergence rates.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 23 / 34
HDG for incompressible flow Navier-Stokes problem
Lid-driven cavity flow
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 24 / 34
HDG for incompressible flow Navier-Stokes problem
Lid-driven cavity flow (ν = 1/100)
Comparison of a third order solution with reference data by Ghia et. al.
0 0.5 10
0.2
0.4
0.6
0.8
1
u1(x = 0.5)
y
Literature: Ghia et. al. High-Re Solutions for Incompressible Flow using the Navier-Stokes Equations and a Multigrid Method, 1982.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 25 / 34
HDG for incompressible flow Navier-Stokes problem
Lid-driven cavity flow (ν = 1/1000)
Comparison of a third order solution with reference data by Ghia et. al.
−0.5 0 0.5 10
0.2
0.4
0.6
0.8
1
u1(x = 0.5)
y
Literature: Ghia et. al. High-Re Solutions for Incompressible Flow using the Navier-Stokes Equations and a Multigrid Method, 1982.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 25 / 34
A posteriori error estimators and adaptivity
Outline
1 Introduction
2 A Hybrid Discontinuous Galerkin Method
3 A posteriori error estimators and adaptivityError estimationAn adaptive algorithmNumerical results
4 Conclusions
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 26 / 34
A posteriori error estimators and adaptivity Error estimation
Error estimation
Exact solution of relevant problems is usually unknown.
The jump error estimator ηJ is given by a sum of local contributions
ηJ :=(∑
T∈Th η2T
)1/2where η2
T := γTk2T
hT
∣∣uh − uh
∣∣2L2(∂T )
Estimator bounds the error from below (efficiency) and above (reliability)
ηJ ≤ ‖(u− uh,u− uh)‖1,h + ‖p− ph‖0,h k ηJ + osc
Reliability proved for HDG Methods for Poisson and Stokes.
ηT can be used as error indicator to drive adaptive refinement strategies.
Literature: Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 27 / 34
A posteriori error estimators and adaptivity An adaptive algorithm
The adaptive algorithm
Given an initial mesh T 0h of Ω, we invoke the algorithm: For i = 0, 1, . . .
SOLVE→ ESTIMATE→ MARK→ REFINE,
(SOLVE) The discrete problem on T ih is solved by the HDG Method.
(ESTIMATE) For each T ∈ T ih , we compute the local error indicator ηT .
(MARK) Dorfler: obtain minimal M(Th) ⊆ Th, such that∑T∈M(Th)
η2T ≥ θ2
∑T∈Th
η2T , here: θ = 0.5
(REFINE) Refine T ih by subdividing all marked triangles in M(Th) into four
similar ones. Ensure that the maximal difference of the refinementlevels between two neighboring elements is one (1-irregular).
Literature: Dorfler. A convergent adaptive algorithm for Poissons equation, 1996.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 28 / 34
A posteriori error estimators and adaptivity Numerical results
Stokes in L-shape domain
Exact solution due to Verfurth exhibits corner singularity.
Initial mesh and adaptively refined meshes after 20 refinement steps.
initial k = 1
k = 2 k = 3
Literature: Verfurth. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Teubner, 1996.Egger, W. hp-Analysis of a Hybrid DG Method for Stokes Flow, 2011.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 29 / 34
A posteriori error estimators and adaptivity Numerical results
Convergence rates and effectivity index
102 103 104
10−1
100
number of elements (k=1)
energy error
ηJ estimate
102 103
10−2
10−1
100
number of elements (k=2)
energy error
ηJ estimate
102 103
10−2
100
number of elements (k=3)
energy error
ηJ estimate
102 103 1040
2
4
6
8
10
number of elements (k=1)
effectivity index
102 1030
2
4
6
8
10
number of elements (k=2)
effectivity index
102 1030
2
4
6
8
10
number of elements (k=3)
effectivity index
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 30 / 34
Conclusions
Outline
1 Introduction
2 A Hybrid Discontinuous Galerkin Method
3 A posteriori error estimators and adaptivity
4 Conclusions
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 31 / 34
Conclusions
Conclusions
We derived analyzed HDG methods for a class of incompressible flow problems.
Some technical results may also be useful for related work.
HDG methods are promising for high order simulations.
Reliable, efficient and simple error estimators.
Outlook: HDG methods for other interesting physical models.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 32 / 34
Financial support from theDeutsche Forschungsgemeinschaft
through grant GSC 111is gratefully acknowledged
Conclusions
Selected references
Discontinuous Galerkin (DG)
Arnold et. al. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. 2002.
Di Pietro, Ern. Mathematical Aspects of Discontinuous Galerkin Methods. 2011.
Cockburn et. al. A locally conservative LDG method for the incompressible Navier-Stokes equations. 2005.
Girault et. al. A discontinuous Galerkin method with non-overlapping domain decomposition for theStokes and Navier-Stokes problems. 2005.
Hybrid Discontinuous Galerkin (HDG)
Cockburn et. al. Unified hybridization of discontinuous Galerkin, mixed and continuous Galerkin methodsfor second order elliptic problems. 2009.
Egger and Schoberl. A mixed-hybrid-discontinuous Galerkin finite element method for convection-diffusionproblems. 2010.
Nguyen et. al. An implicit high-order hybridizable discontinuous Galerkin method for the incompressibleNavier-Stokes equations. 2010.
Cockburn et. al. Analysis of HDG methods for Stokes flow. 2011.
Egger and W. hp-analysis of a hybrid DG method for Stokes flow. 2011.
Christian Waluga (AICES) HDG Methods for Incompressible Flow February 3, 2012 34 / 34