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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

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

Background

∇ · (f(u)− fv(u,∇u)) = 0

∇ · f(u) = 0

∇ · fv(u, σ) = 0 σ = ∇u

λ ≈ u|Γ

Background

∇ · (f(u)− fv(u,∇u)) = 0

∇ · f(u) = 0

∇ · fv(u, σ) = 0 σ = ∇u

λ ≈ u|Γ

Background

∇ · (f(u)− fv(u,∇u)) = 0

∇ · f(u) = 0

∇ · fv(u, σ) = 0 σ = ∇u

λ ≈ u|Γ

Background

∇ · (f(u)− fv(u,∇u)) = 0

∇ · f(u) = 0

∇ · fv(u, σ) = 0 σ = ∇u

λ ≈ u|Γ

Background

∇ · (f(u)− fv(u,∇u)) = 0

∇ · f(u) = 0

∇ · fv(u, σ) = 0 σ = ∇u

λ ≈ u|Γ

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

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

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

−∇ · ∇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

m(Ωk)× Pm(Ωk)

BDM Mixed method

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

−∇ · ∇u = S in Ω

u = g in ∂Ω

−∇ · σ = S in Ω

u = g in ∂Ω

m(Ωk)× Pm(Ωk)

BDM Mixed method

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

−∇ · ∇u = S in Ω

u = g 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

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

−∇ · ∇u = S in Ω

u = g in ∂Ω

−∇ · σ = S in Ω

u = g in ∂Ω

m−1(Ωk)

Hh := τ ∈ H(div,Ω) : τ |Ωk ∈ Pm(Ωk)× Pm(Ωk)

BDM Mixed method

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

−∇ · ∇u = S in Ω

u = g in ∂Ω

−∇ · σ = S in Ω

u = g in ∂Ω

m(Ωk)× Pm(Ωk)

BDM Mixed method

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

−∇ · ∇u = S in Ω

u = g in ∂Ω

−∇ · σ = S in Ω

u = g in ∂Ω

m(Ωk)× Pm(Ωk)

BDM Mixed method

σh · τ +

(∇ · τ )uh −∫∂Ω

(τ · n)g = 0 ∀τ ∈ Hh

−∫

∇ · σhϕ =

Sϕ ∀ϕ ∈ Vh

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

∫Ωk

σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0 ∀τ ∈ Hh

−∑k

∫Ωk

(∇ · σh)ϕ =∑k

∫Ωk

Sϕ ∀ϕ ∈ Vh

∫∂Ωk

−(σh · n)µ = 0 ∀µ ∈Mh

Hybridizing...

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

∫Ωk

σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0 ∀τ ∈ Hh

−∑k

∫Ωk

(∇ · σh)ϕ =∑k

∫Ωk

Sϕ ∀ϕ ∈ Vh

∫∂Ωk

−(σh · n)µ = 0 ∀µ ∈Mh

Adding DG for Advection

Advection-Diffusion equation

∇ · f(u)− ε∇ · ∇u = S

HDG-BDM method

∫Ωk

ε−1σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∫Ωk

−f(uh) · ∇ϕ+

∫Γk

ϕ (f(λh) · n− α(λh − uh))−∫

(∇ · σh)ϕ

∫Ωk

∫∂Ωk

(−σh · n+ f(λh) · n− α(λh − uh))µ = 0

Adding DG for Advection

Advection-Diffusion equation

∇ · f(u)− ε∇ · ∇u = S

HDG-BDM method

∫Ωk

ε−1σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∫Ωk

−f(uh) · ∇ϕ+

∫Γk

ϕ (f(λh) · n− α(λh − uh))−∫

(∇ · σh)ϕ

∫Ωk

∫∂Ωk

(−σh · n+ f(λh) · n− α(λh − uh))µ = 0

Hybridized DG

Proposed by Nguyen et. al 5

Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)

HDG method

∫Ωk

ε−1σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∫Ωk

−f(uh) · ∇ϕ+

∫Γk

ϕ (f(λh) · n− β(λh − uh))−∫

(∇ · σh)ϕ

∫Ωk

∫∂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

Hybridized DG

Proposed by Nguyen et. al 5

Vh := ϕ ∈ L2(Ω) : ϕ|Ωk ∈ Pm(Ωk)

HDG method

∫Ωk

ε−1σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∫Ωk

−f(uh) · ∇ϕ+

∫Γk

ϕ (f(λh) · n− β(λh − uh))−∫

(∇ · σh)ϕ

∫Ωk

∫∂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

Comparison

HDG-BDM

uh|Ωk ∈ Pm−1

−σh+ fh = −σh+f(λh)−α(λh−uh)n

uh|Ωk ∈ Pm

−σh+ fh = −σh+f(λh)−β(λh−uh)n

Common method

∫Ωk

ε−1σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∫Ωk

−f(uh) · ∇ϕ+

∫Γk

ϕ (f(λh) · n− (α|β)(λh − uh))−∫

(∇ · σh)ϕ

∫Ωk

∫∂Ωk

(−σh · n+ f(λh) · n− (α|β)(λh − uh))µ = 0

Comparison

HDG-BDM

uh|Ωk ∈ Pm−1

−σh+ fh = −σh+f(λh)−α(λh−uh)n

uh|Ωk ∈ Pm

−σh+ fh = −σh+f(λh)−β(λh−uh)n

Common method

∫Ωk

ε−1σh · τ +

∫Ωk

(∇ · τ )uh −∫∂Ωk

(τ · n)λh = 0

∫Ωk

−f(uh) · ∇ϕ+

∫Γk

ϕ (f(λh) · n− (α|β)(λh − uh))−∫

(∇ · σh)ϕ

∫Ωk

∫∂Ωk

(−σh · n+ f(λh) · n− (α|β)(λh − uh))µ = 0

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

Convergence

σh RTm Pm Pm m+ 1

Convergence

σh RTm Pm Pm m+ 1

Test case 1 : Boundary Layer

Two dimensional viscous Burgers equation

2∇ · (u2, u2)− ε∇ · ∇u = S in Ω

u = 0 in ∂Ω

Solution :

u(x, y) =

ec1x/ε − 1

1− ec1/ε

)·(y +

ec1y/ε − 1

1− ec1/ε

Figure: Contours of u, m = 2 (u ∈ P 1), ε = 0.1, HDG-BDM scheme

Figure: Contours of u*, m = 2 (u ∈ P 1), ε = 0.1, HDG-BDM scheme

Figure: Contours of u*, m = 2 (u ∈ P 1), ε = 0.1, α = 0, HDG-BDM scheme

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−8

Log(sqrt(N))

Hybridized DG−BDM: Convergence Rate

uu* (α =2)u* (α =0)

Figure: Convergence, m = 3 (u ∈ P 2), ε = 0.1, HDG-BDM scheme

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

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

1 1.2 1.4 1.6 1.8 2 2.2 2.42.5−6.5

−5.5

−4.5

−3.5

−2.5

Log(sqrt(Ne))

Log(Error)

HDGHDG / HDG−BDM

11 1.2 1.4 1.6 1.8 2 2.2 2.42.50.3

Log(sqrt(Ne))R

Convergence of u*, m = 2 Reduction of dofs of u

Test case 3Advection Diffusion equation:

∇ · u−∇ · (ε(x)∇u) = S in Ω

u = g in ΓD

Diffusion Coefficient:

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)

∇ · u−∇ · (ε(x)∇u) = S in Ω

u = g in ΓD

0.001, x ≤ 0.9

1, x ≥ 1.1

smooth fn., 0.9 < x < 1.1

∇ · u−∇ · (ε(x)∇u) = S in Ω

u = g in ΓD

0.001, x ≤ 0.9

1, x ≥ 1.1

smooth fn., 0.9 < x < 1.1

Test case 3Condition : x > 1.2, use HDG-BDM

Figure: Red : HDG, Blue : HDG-BDM

Figure: Contours of u*, m = 2

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

Log(sqrt(Ne))

Log(Error)

HDGHDG / HDG−BDM

Figure: Convergence of u*, m = 2

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

Log(sqrt(Ne))

Log(Error)

HDGHDG / HDG−BDM

Figure: Convergence of u*, m = 2

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

Conclusions

Present work :

Future work :

Shock capturing

Conclusions

Present work :

Future work :

Shock capturing

Conclusions

Present work :

Future work :

Shock capturing

Conclusions

Present work :

Future work :

Shock capturing

Acknowledgement

Financial support from the Deutsche Forschungsgemeinschaft (GermanResearch Association) through grant GSC 111 is gratefully acknowledged

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).

Smooth Function

ε = e(−9+10x)−2

(e(−9+10x)−2

+ e(−11+10x)−2

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−8

Log(sqrt(N))

Hybridized DG Method : Convergence Rate

Figure: Convergence, m = 3 (u ∈ P 3), ε = 0.1, HDG scheme

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−8

Log(sqrt(N))

Convergence Rate for Hy. DG and Hy. DG−BDM

u* HDGu* HDG−BDM

Figure: Convergence, m = 3, ε = 0.1, HDG (u ∈ P 3) and HDG-BDM (u ∈ P 2) schemes

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