A Hybridized DG / Mixed Method For Nonlinear Convection-Diffusion Problems Aravind Balan, Michael Woopen, Jochen Sch¨ utz and Georg May AICES Graduate School, RWTH Aachen University, Germany WCCM 2012, S ˜ ao Paulo, Brazil July 9, 2012 Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 1 / 24
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A Hybridized DG / Mixed Method For Nonlinear Convection-DiffusionProblems
Aravind Balan, Michael Woopen, Jochen Schutz and Georg May
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 1 / 24
Outline
1 Introduction
2 BDM Mixed Method for Diffusion
3 Hybridized BDM Mixed Method for Diffusion
4 Hybridized DG-BDM (HDG-BDM) for Advection-Diffusion
5 Hybridized DG (HDG) for Advection-Diffusion
6 Numerical Results
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 2 / 24
Background
HDG-BDM method for Advection-Diffusion equations.
∇ · (f(u)− fv(u,∇u)) = 0
Discontinuous Galerkin for Advection; known to work well
∇ · f(u) = 0
BDM Mixed method for Diffusion; known to work well
∇ · fv(u, σ) = 0 σ = ∇u
Hybridization to reduce the global coupled degrees of freedom
λ ≈ u|Γ
Linear case : Proposed by H. Egger and J. Schoberl 1
Non-Linear case : Proposed by J. Schutz and G. May (Promising results for N-Sequations 2 )
1H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 20102J. Schutz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24
Background
HDG-BDM method for Advection-Diffusion equations.
∇ · (f(u)− fv(u,∇u)) = 0
Discontinuous Galerkin for Advection; known to work well
∇ · f(u) = 0
BDM Mixed method for Diffusion; known to work well
∇ · fv(u, σ) = 0 σ = ∇u
Hybridization to reduce the global coupled degrees of freedom
λ ≈ u|Γ
Linear case : Proposed by H. Egger and J. Schoberl 1
Non-Linear case : Proposed by J. Schutz and G. May (Promising results for N-Sequations 2 )
1H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 20102J. Schutz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24
Background
HDG-BDM method for Advection-Diffusion equations.
∇ · (f(u)− fv(u,∇u)) = 0
Discontinuous Galerkin for Advection; known to work well
∇ · f(u) = 0
BDM Mixed method for Diffusion; known to work well
∇ · fv(u, σ) = 0 σ = ∇u
Hybridization to reduce the global coupled degrees of freedom
λ ≈ u|Γ
Linear case : Proposed by H. Egger and J. Schoberl 1
Non-Linear case : Proposed by J. Schutz and G. May (Promising results for N-Sequations 2 )
1H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 20102J. Schutz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24
Background
HDG-BDM method for Advection-Diffusion equations.
∇ · (f(u)− fv(u,∇u)) = 0
Discontinuous Galerkin for Advection; known to work well
∇ · f(u) = 0
BDM Mixed method for Diffusion; known to work well
∇ · fv(u, σ) = 0 σ = ∇u
Hybridization to reduce the global coupled degrees of freedom
λ ≈ u|Γ
Linear case : Proposed by H. Egger and J. Schoberl 1
Non-Linear case : Proposed by J. Schutz and G. May (Promising results for N-Sequations 2 )
1H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 20102J. Schutz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24
Background
HDG-BDM method for Advection-Diffusion equations.
∇ · (f(u)− fv(u,∇u)) = 0
Discontinuous Galerkin for Advection; known to work well
∇ · f(u) = 0
BDM Mixed method for Diffusion; known to work well
∇ · fv(u, σ) = 0 σ = ∇u
Hybridization to reduce the global coupled degrees of freedom
λ ≈ u|Γ
Linear case : Proposed by H. Egger and J. Schoberl 1
Non-Linear case : Proposed by J. Schutz and G. May (Promising results for N-Sequations 2 )
1H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 20102J. Schutz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24
Background
HDG-BDM method for Advection-Diffusion equations.
∇ · (f(u)− fv(u,∇u)) = 0
Discontinuous Galerkin for Advection; known to work well
∇ · f(u) = 0
BDM Mixed method for Diffusion; known to work well
∇ · fv(u, σ) = 0 σ = ∇u
Hybridization to reduce the global coupled degrees of freedom
λ ≈ u|Γ
Linear case : Proposed by H. Egger and J. Schoberl 1
Non-Linear case : Proposed by J. Schutz and G. May (Promising results for N-Sequations 2 )
1H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 20102J. Schutz, M. Woopen and G. May, AIAA Paper 2012-0729, 2012
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 3 / 24
Features of the HDG-BDM scheme
Reduces to DG for pure advection, to BDM mixed for pure diffusion
No additional parameter in the intermediate range
Using local solvers3 to make it a system for λ
Solution can be post-processed to get better convergence
It can be easily modified to the well known Hybridized Discontinuous Galerkin(HDG) scheme 4
It can be even mixed with the HDG scheme due to hybridization.
3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 20044N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24
Features of the HDG-BDM scheme
Reduces to DG for pure advection, to BDM mixed for pure diffusion
No additional parameter in the intermediate range
Using local solvers3 to make it a system for λ
Solution can be post-processed to get better convergence
It can be easily modified to the well known Hybridized Discontinuous Galerkin(HDG) scheme 4
It can be even mixed with the HDG scheme due to hybridization.
3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 20044N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24
Features of the HDG-BDM scheme
Reduces to DG for pure advection, to BDM mixed for pure diffusion
No additional parameter in the intermediate range
Using local solvers3 to make it a system for λ
Solution can be post-processed to get better convergence
It can be easily modified to the well known Hybridized Discontinuous Galerkin(HDG) scheme 4
It can be even mixed with the HDG scheme due to hybridization.
3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 20044N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24
Features of the HDG-BDM scheme
Reduces to DG for pure advection, to BDM mixed for pure diffusion
No additional parameter in the intermediate range
Using local solvers3 to make it a system for λ
Solution can be post-processed to get better convergence
It can be easily modified to the well known Hybridized Discontinuous Galerkin(HDG) scheme 4
It can be even mixed with the HDG scheme due to hybridization.
3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 20044N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24
Features of the HDG-BDM scheme
Reduces to DG for pure advection, to BDM mixed for pure diffusion
No additional parameter in the intermediate range
Using local solvers3 to make it a system for λ
Solution can be post-processed to get better convergence
It can be easily modified to the well known Hybridized Discontinuous Galerkin(HDG) scheme 4
It can be even mixed with the HDG scheme due to hybridization.
3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 20044N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24
Features of the HDG-BDM scheme
Reduces to DG for pure advection, to BDM mixed for pure diffusion
No additional parameter in the intermediate range
Using local solvers3 to make it a system for λ
Solution can be post-processed to get better convergence
It can be easily modified to the well known Hybridized Discontinuous Galerkin(HDG) scheme 4
It can be even mixed with the HDG scheme due to hybridization.
3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 20044N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 4 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇uσ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ HhVh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P
m−1(Ωk)Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ P
m(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇u
σ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ HhVh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P
m−1(Ωk)Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ P
m(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇uσ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ HhVh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P
m−1(Ωk)Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ P
m(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇uσ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ Hh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm−1(Ωk)
Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇uσ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ HhVh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P
m−1(Ωk)
Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇uσ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ HhVh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P
m−1(Ωk)Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ P
m(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
BDM Mixed Method for DiffusionConsider Laplace equation
−∇ · ∇u = S in Ω
u = g in ∂Ω
Introducing new variable, σ = ∇uσ = ∇u in Ω
−∇ · σ = S in Ω
u = g in ∂Ω
The solution spaces : uh ∈ Vh, σh ∈ HhVh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P
m−1(Ωk)Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ P
m(Ωk)× Pm(Ωk)
BDM Mixed method
∫Ω
σh · τ +
∫Ω
(∇ · τ )uh −∫∂Ω
(τ · n)g = 0 ∀τ ∈ Hh
−∫
Ω
∇ · σhϕ =
∫Ω
Sϕ ∀ϕ ∈ Vh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 5 / 24
Hybridizing...
The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm−1(Ωk)
Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
Mh := µ ∈ L2(Γ) : µ|Γk ∈ Pm(Γk)
Hyb. BDM mixed method
∑k
∫Ωk
σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0 ∀τ ∈ Hh
−∑k
∫Ωk
(∇ · σh)ϕ =∑k
∫Ωk
Sϕ ∀ϕ ∈ Vh
∑k
∫∂Ωk
−(σh · n)µ = 0 ∀µ ∈Mh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 6 / 24
Hybridizing...
The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm−1(Ωk)
Hh := τ ∈ L2(Ω)× L2(Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)
Mh := µ ∈ L2(Γ) : µ|Γk ∈ Pm(Γk)
Hyb. BDM mixed method
∑k
∫Ωk
σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0 ∀τ ∈ Hh
−∑k
∫Ωk
(∇ · σh)ϕ =∑k
∫Ωk
Sϕ ∀ϕ ∈ Vh
∑k
∫∂Ωk
−(σh · n)µ = 0 ∀µ ∈Mh
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 6 / 24
Adding DG for Advection
Advection-Diffusion equation
∇ · f(u)− ε∇ · ∇u = S
HDG-BDM method
∑k
∫Ωk
ε−1σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−f(uh) · ∇ϕ+
∫Γk
ϕ (f(λh) · n− α(λh − uh))−∫
Ωk
(∇ · σh)ϕ
=∑k
∫Ωk
Sϕ
∑k
∫∂Ωk
(−σh · n+ f(λh) · n− α(λh − uh))µ = 0
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 7 / 24
Adding DG for Advection
Advection-Diffusion equation
∇ · f(u)− ε∇ · ∇u = S
HDG-BDM method
∑k
∫Ωk
ε−1σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−f(uh) · ∇ϕ+
∫Γk
ϕ (f(λh) · n− α(λh − uh))−∫
Ωk
(∇ · σh)ϕ
=∑k
∫Ωk
Sϕ
∑k
∫∂Ωk
(−σh · n+ f(λh) · n− α(λh − uh))µ = 0
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 7 / 24
Hybridized DG
Proposed by Nguyen et. al 5
The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)
HDG method
∑k
∫Ωk
ε−1σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−f(uh) · ∇ϕ+
∫Γk
ϕ (f(λh) · n− β(λh − uh))−∫
Ωk
(∇ · σh)ϕ
=∑k
∫Ωk
Sϕ
∑k
∫∂Ωk
(−σh · n+ f(λh) · n− β(λh − uh))µ = 0
5N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 8 / 24
Hybridized DG
Proposed by Nguyen et. al 5
The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈Mh
Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)
HDG method
∑k
∫Ωk
ε−1σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−f(uh) · ∇ϕ+
∫Γk
ϕ (f(λh) · n− β(λh − uh))−∫
Ωk
(∇ · σh)ϕ
=∑k
∫Ωk
Sϕ
∑k
∫∂Ωk
(−σh · n+ f(λh) · n− β(λh − uh))µ = 0
5N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 8 / 24
Comparison
HDG-BDM
uh|Ωk ∈ Pm−1
−σh+ fh = −σh+f(λh)−α(λh−uh)n
HDG
uh|Ωk ∈ Pm
−σh+ fh = −σh+f(λh)−β(λh−uh)n
Common method
∑k
∫Ωk
ε−1σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−f(uh) · ∇ϕ+
∫Γk
ϕ (f(λh) · n− (α|β)(λh − uh))−∫
Ωk
(∇ · σh)ϕ
=∑k
∫Ωk
Sϕ
∑k
∫∂Ωk
(−σh · n+ f(λh) · n− (α|β)(λh − uh))µ = 0
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 9 / 24
Comparison
HDG-BDM
uh|Ωk ∈ Pm−1
−σh+ fh = −σh+f(λh)−α(λh−uh)n
HDG
uh|Ωk ∈ Pm
−σh+ fh = −σh+f(λh)−β(λh−uh)n
Common method
∑k
∫Ωk
ε−1σh · τ +
∫Ωk
(∇ · τ )uh −∫∂Ωk
(τ · n)λh = 0
∑k
∫Ωk
−f(uh) · ∇ϕ+
∫Γk
ϕ (f(λh) · n− (α|β)(λh − uh))−∫
Ωk
(∇ · σh)ϕ
=∑k
∫Ωk
Sϕ
∑k
∫∂Ωk
(−σh · n+ f(λh) · n− (α|β)(λh − uh))µ = 0
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 9 / 24
Convergence
Post processing of the solution 6
HDG-RT HDG-BDM HDG 7 Post Proc. Conv.uh Pm Pm−1 Pm m+ 2
σh RTm Pm Pm m+ 1
Same convergence of post-processed solution under optimal conditions.
For HDG-BDM and HDG-RT8 , this optimal condition is when diffusion dominatesand one can put α = 0
6R. Stenberg, Math. Model. Numer. Anal. 25. 151-168, 19917N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 20098H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 10 / 24
Convergence
Post processing of the solution 6
HDG-RT HDG-BDM HDG 7 Post Proc. Conv.uh Pm Pm−1 Pm m+ 2
σh RTm Pm Pm m+ 1
Same convergence of post-processed solution under optimal conditions.
For HDG-BDM and HDG-RT8 , this optimal condition is when diffusion dominatesand one can put α = 0
6R. Stenberg, Math. Model. Numer. Anal. 25. 151-168, 19917N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 20098H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 10 / 24
Convergence
Post processing of the solution 6
HDG-RT HDG-BDM HDG 7 Post Proc. Conv.uh Pm Pm−1 Pm m+ 2
σh RTm Pm Pm m+ 1
Same convergence of post-processed solution under optimal conditions.
For HDG-BDM and HDG-RT8 , this optimal condition is when diffusion dominatesand one can put α = 0
6R. Stenberg, Math. Model. Numer. Anal. 25. 151-168, 19917N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 20098H. Egger and J. Schoberl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 10 / 24
Test case 1 : Boundary Layer
Two dimensional viscous Burgers equation
1
2∇ · (u2, u2)− ε∇ · ∇u = S in Ω
u = 0 in ∂Ω
Solution :
u(x, y) =
(x+
ec1x/ε − 1
1− ec1/ε
)·(y +
ec1y/ε − 1
1− ec1/ε
)
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 11 / 24
Test case 1 : Boundary Layer
Figure: Contours of u, m = 2 (u ∈ P 1), ε = 0.1, HDG-BDM scheme
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 12 / 24
Test case 1 : Boundary Layer
Figure: Contours of u*, m = 2 (u ∈ P 1), ε = 0.1, HDG-BDM scheme
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 13 / 24
Test case 1 : Boundary Layer
Figure: Contours of u*, m = 2 (u ∈ P 1), ε = 0.1, α = 0, HDG-BDM scheme
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 14 / 24
Test case 1 : Boundary Layer
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−8
−7
−6
−5
−4
−3
−2
−1
Log(sqrt(N))
Log(
Erro
r)
Hybridized DG−BDM: Convergence Rate
uu* (α =2)u* (α =0)
4.9
3.0
3.8
Figure: Convergence, m = 3 (u ∈ P 2), ε = 0.1, HDG-BDM scheme
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 15 / 24
Test case 2 : Linear Boundary Layer
Mixing HDG and HDG-BDM methods :
Condition: If Peclect number, Pe = |c|hε< 5, then use HDG-BDM
Contours of u*, m = 2, ε = 0.01 Red : HDG, Blue : HDG-BDM
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 16 / 24
Test case 2 : Linear Boundary Layer
Mixing HDG and HDG-BDM methods :
Condition: If Peclect number, Pe = |c|hε< 5, then use HDG-BDM
Contours of u*, m = 2, ε = 0.01 Red : HDG, Blue : HDG-BDM
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 16 / 24
Test case 2 : Linear Boundary Layer
1 1.2 1.4 1.6 1.8 2 2.2 2.42.5−6.5
−6
−5.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
Log(sqrt(Ne))
Log(Error)
HDGHDG / HDG−BDM
3.9
11 1.2 1.4 1.6 1.8 2 2.2 2.42.50.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Log(sqrt(Ne))R
atio
of n
o. o
f dof
of u
Convergence of u*, m = 2 Reduction of dofs of u
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 17 / 24