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A Hybridized DG / Mixed Method For Nonlinear Convection-DiffusionProblems
Aravind Balan, Michael Woopen, Jochen Schutz and Georg May
AICES Graduate School, RWTH Aachen University, Germany
WCCM 2012, Sao Paulo, Brazil
July 9, 2012
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
Test case 3Advection Diffusion equation:
∇ · u−∇ · (ε(x)∇u) = S in Ω
u = g in ΓD
Diffusion Coefficient:
ε =
0.001, x ≤ 0.9
1, x ≥ 1.1
smooth fn., 0.9 < x < 1.1
Solution:u(x, y) = (1− ε(x)) sin(x− y) + ε(x) sin(2πx) sin(2πy)
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 18 / 24
Test case 3Advection Diffusion equation:
∇ · u−∇ · (ε(x)∇u) = S in Ω
u = g in ΓD
Diffusion Coefficient:
ε =
0.001, x ≤ 0.9
1, x ≥ 1.1
smooth fn., 0.9 < x < 1.1
Solution:u(x, y) = (1− ε(x)) sin(x− y) + ε(x) sin(2πx) sin(2πy)
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 18 / 24
Test case 3Advection Diffusion equation:
∇ · u−∇ · (ε(x)∇u) = S in Ω
u = g in ΓD
Diffusion Coefficient:
ε =
0.001, x ≤ 0.9
1, x ≥ 1.1
smooth fn., 0.9 < x < 1.1
Solution:u(x, y) = (1− ε(x)) sin(x− y) + ε(x) sin(2πx) sin(2πy)
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 18 / 24
Test case 3Condition : x > 1.2, use HDG-BDM
Figure: Red : HDG, Blue : HDG-BDM
Figure: Contours of u*, m = 2
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 19 / 24
Test case 3
Condition : x > 1.2, use HDG-BDM
1 1.2 1.4 1.6 1.8 2 2.2 2.42.5−6
−5
−4
−3
−2
−1
0
Log(sqrt(Ne))
Log(Error)
HDGHDG / HDG−BDM
4.5
Figure: Convergence of u*, m = 2
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 20 / 24
Test case 3Condition : Pe < 5, use HDG-BDM
Figure: Red : HDG, Blue : HDG-BDM
Figure: Contours of u*, m = 2Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 21 / 24
Test case 3
Condition : Pe < 5, use HDG-BDM
1 1.2 1.4 1.6 1.8 2 2.2 2.42.5−6
−5
−4
−3
−2
−1
0
Log(sqrt(Ne))
Log(Error)
HDGHDG / HDG−BDM
4.5
3.5
Figure: Convergence of u*, m = 2
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 22 / 24
Conclusions
Present work :
HDG-BDM method and it’s connection with HDG scheme
Mixing of the two methods;
HDG as base scheme and HDG-BDM in diffusion dominated region
Cell peclet number as sensor
Future work :
A robust sensor to determine the region for using HDG-BDM scheme
Shock capturing
Extending to Navier-Stokes equations
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 23 / 24
Conclusions
Present work :
HDG-BDM method and it’s connection with HDG scheme
Mixing of the two methods;
HDG as base scheme and HDG-BDM in diffusion dominated region
Cell peclet number as sensor
Future work :
A robust sensor to determine the region for using HDG-BDM scheme
Shock capturing
Extending to Navier-Stokes equations
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 23 / 24
Conclusions
Present work :
HDG-BDM method and it’s connection with HDG scheme
Mixing of the two methods;
HDG as base scheme and HDG-BDM in diffusion dominated region
Cell peclet number as sensor
Future work :
A robust sensor to determine the region for using HDG-BDM scheme
Shock capturing
Extending to Navier-Stokes equations
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 23 / 24
Conclusions
Present work :
HDG-BDM method and it’s connection with HDG scheme
Mixing of the two methods;
HDG as base scheme and HDG-BDM in diffusion dominated region
Cell peclet number as sensor
Future work :
A robust sensor to determine the region for using HDG-BDM scheme
Shock capturing
Extending to Navier-Stokes equations
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 23 / 24
Conclusions
Present work :
HDG-BDM method and it’s connection with HDG scheme
Mixing of the two methods;
HDG as base scheme and HDG-BDM in diffusion dominated region
Cell peclet number as sensor
Future work :
A robust sensor to determine the region for using HDG-BDM scheme
Shock capturing
Extending to Navier-Stokes equations
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 23 / 24
Acknowledgement
Financial support from the Deutsche Forschungsgemeinschaft (GermanResearch Association) through grant GSC 111 is gratefully acknowledged
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 24 / 24
Post Processing
Cell-wise discretization of the Neumann problem :
ε(∇u∗h,∇φ) = (σh,∇φ) ∀φ ∈ P q0 (Ωk)
(uh, 1) = (u∗h, 1)
whereP q0 (Ωk) := φ ∈ P q(Ωk), (φ, 1) = 0
with q = m+ 1 for HDG and HDG-BDM (α = 0) and q = m for HDG-BDM (α 6= 0).
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 25 / 24
Smooth Function
ε = e(−9+10x)−2
(e(−9+10x)−2
+ e(−11+10x)−2
)−1
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 26 / 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
Log(sqrt(N))
Log(
Erro
r)
Hybridized DG Method : Convergence Rate
uu*
4.88
3.95
Figure: Convergence, m = 3 (u ∈ P 3), ε = 0.1, HDG scheme
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 27 / 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
Log(sqrt(N))
Log(
Erro
r)
Convergence Rate for Hy. DG and Hy. DG−BDM
u* HDGu* HDG−BDM
4.9
Figure: Convergence, m = 3, ε = 0.1, HDG (u ∈ P 3) and HDG-BDM (u ∈ P 2) schemes
Aravind Balan (AICES, RWTH Aachen) Hy. DG-BDM Method July 9, Sao Paulo 28 / 24
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