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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

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SLIDE 1

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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SLIDE 2

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

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SLIDE 3

Background

HDG-BDM method for Advection-Diffusion equations. ∇ · (f(u) − fv(u, ∇u)) = 0

  • 1H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
  • 2J. Sch¨

utz, 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

slide-4
SLIDE 4

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

  • 1H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
  • 2J. Sch¨

utz, 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

slide-5
SLIDE 5

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

  • 1H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
  • 2J. Sch¨

utz, 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

slide-6
SLIDE 6

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

  • 1H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
  • 2J. Sch¨

utz, 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

slide-7
SLIDE 7

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. Sch¨

  • berl 1
  • 1H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
  • 2J. Sch¨

utz, 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

slide-8
SLIDE 8

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. Sch¨

  • berl 1

Non-Linear case : Proposed by J. Sch¨ utz and G. May (Promising results for N-S equations 2 )

  • 1H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010
  • 2J. Sch¨

utz, 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

slide-9
SLIDE 9

Features of the HDG-BDM scheme

Reduces to DG for pure advection, to BDM mixed for pure diffusion

  • 3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004
  • 4N. 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

slide-10
SLIDE 10

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

  • 3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004
  • 4N. 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

slide-11
SLIDE 11

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

  • 3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004
  • 4N. 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

slide-12
SLIDE 12

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

  • 3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004
  • 4N. 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

slide-13
SLIDE 13

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

  • 3B. Cockburn and J. Gopalakrishnan. SIAM Journal of Num. Analysis. 42, 283-301, 2004
  • 4N. 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

slide-14
SLIDE 14

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, 2004
  • 4N. 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

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SLIDE 15

BDM Mixed Method for Diffusion

Consider Laplace equation −∇ · ∇u = S in Ω u = g in ∂Ω

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

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SLIDE 16

BDM Mixed Method for Diffusion

Consider Laplace equation −∇ · ∇u = S in Ω u = g in ∂Ω Introducing new variable, σ = ∇u

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

slide-17
SLIDE 17

BDM Mixed Method for Diffusion

Consider 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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

slide-18
SLIDE 18

BDM Mixed Method for Diffusion

Consider 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 ∈ ˜ Hh

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

slide-19
SLIDE 19

BDM Mixed Method for Diffusion

Consider 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 ∈ ˜ Hh Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m−1(Ωk)}

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

slide-20
SLIDE 20

BDM Mixed Method for Diffusion

Consider 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 ∈ ˜ Hh Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m−1(Ωk)} ˜ Hh := {τ ∈ H(div, Ω) : τ|Ωk ∈ P m(Ωk) × P m(Ωk)}

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

slide-21
SLIDE 21

BDM Mixed Method for Diffusion

Consider 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 ∈ ˜ Hh Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m−1(Ωk)} ˜ Hh := {τ ∈ H(div, Ω) : τ|Ωk ∈ P m(Ωk) × P m(Ωk)}

BDM Mixed method

  • Ω

σh · τ +

  • Ω

(∇ · τ)uh −

  • ∂Ω

(τ · n)g = ∀τ ∈ ˜ Hh −

  • Ω

∇ · σhϕ =

  • Ω

Sϕ ∀ϕ ∈ Vh

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 5 / 24

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SLIDE 22

Hybridizing...

The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈ Mh Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m−1(Ωk)} Hh := {τ ∈ L2(Ω) × L2(Ω) : τ|Ωk ∈ P m(Ωk) × P m(Ωk)} Mh := {µ ∈ L2(Γ) : µ|Γk ∈ P m(Γk)}

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 6 / 24

slide-23
SLIDE 23

Hybridizing...

The solution spaces : uh ∈ Vh, σh ∈ Hh, λh ∈ Mh Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m−1(Ωk)} Hh := {τ ∈ L2(Ω) × L2(Ω) : τ|Ωk ∈ P m(Ωk) × P m(Ωk)} Mh := {µ ∈ L2(Γ) : µ|Γk ∈ P m(Γk)}

  • Hyb. BDM mixed method
  • k
  • Ωk

σh · τ +

  • Ωk

(∇ · τ)uh −

  • ∂Ωk

(τ · n)λh = ∀τ ∈ Hh −

  • k
  • Ωk

(∇ · σh)ϕ =

  • k
  • Ωk

Sϕ ∀ϕ ∈ Vh

  • k
  • ∂Ωk

−(σh · n)µ = ∀µ ∈ Mh

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 6 / 24

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SLIDE 24

Adding DG for Advection

Advection-Diffusion equation ∇ · f(u) − ǫ∇ · ∇u = S

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 7 / 24

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SLIDE 25

Adding DG for Advection

Advection-Diffusion equation ∇ · f(u) − ǫ∇ · ∇u = S

HDG-BDM method

  • k
  • Ωk

ǫ−1σh · τ +

  • Ωk

(∇ · τ)uh −

  • ∂Ωk

(τ · n)λh =

  • 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 =

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 7 / 24

slide-26
SLIDE 26

Hybridized DG

Proposed by Nguyen et. al 5 The solution spaces : uh ∈ ˜ Vh, σh ∈ Hh, λh ∈ Mh ˜ Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m(Ωk)}

  • 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

slide-27
SLIDE 27

Hybridized DG

Proposed by Nguyen et. al 5 The solution spaces : uh ∈ ˜ Vh, σh ∈ Hh, λh ∈ Mh ˜ Vh := {ϕ ∈ L2(Ω) : ϕ|Ωk ∈ P m(Ωk)}

HDG method

  • k
  • Ωk

ǫ−1σh · τ +

  • Ωk

(∇ · τ)uh −

  • ∂Ωk

(τ · n)λh =

  • 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 =

  • 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

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SLIDE 28

Comparison

HDG-BDM

uh|Ωk ∈ P m−1 −σh + ˆ fh = −σh +f(λh)−α(λh −uh)n

HDG

uh|Ωk ∈ P m − ˆ σh + ˆ fh = −σh +f(λh)−β(λh −uh)n

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 9 / 24

slide-29
SLIDE 29

Comparison

HDG-BDM

uh|Ωk ∈ P m−1 −σh + ˆ fh = −σh +f(λh)−α(λh −uh)n

HDG

uh|Ωk ∈ P m − ˆ σh + ˆ fh = −σh +f(λh)−β(λh −uh)n

Common method

  • k
  • Ωk

ǫ−1σh · τ +

  • Ωk

(∇ · τ)uh −

  • ∂Ωk

(τ · n)λh =

  • 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 =

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 9 / 24

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SLIDE 30

Convergence

Post processing of the solution 6 HDG-RT HDG-BDM HDG 7 Post Proc. Conv. uh P m P m−1 P m m + 2 σh RT m P m P m m + 1

  • 6R. Stenberg, Math. Model. Numer. Anal. 25. 151-168, 1991
  • 7N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
  • 8H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 10 / 24

slide-31
SLIDE 31

Convergence

Post processing of the solution 6 HDG-RT HDG-BDM HDG 7 Post Proc. Conv. uh P m P m−1 P m m + 2 σh RT m P m P m m + 1 Same convergence of post-processed solution under optimal conditions.

  • 6R. Stenberg, Math. Model. Numer. Anal. 25. 151-168, 1991
  • 7N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
  • 8H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 10 / 24

slide-32
SLIDE 32

Convergence

Post processing of the solution 6 HDG-RT HDG-BDM HDG 7 Post Proc. Conv. uh P m P m−1 P m m + 2 σh RT m P m P m m + 1 Same convergence of post-processed solution under optimal conditions. For HDG-BDM and HDG-RT8 , this optimal condition is when diffusion dominates and one can put α = 0

  • 6R. Stenberg, Math. Model. Numer. Anal. 25. 151-168, 1991
  • 7N. C. Nguyen, J. Peraire, and B. Cockburn, J. Comp. Physics, 228, 8841-8855, 2009
  • 8H. Egger and J. Sch¨
  • berl. IMA Journal of Num. Analysis. 30. 1206-1234, 2010

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 10 / 24

slide-33
SLIDE 33

Test case 1 : Boundary Layer

Two dimensional viscous Burgers equation 1 2∇ · (u2, u2) − ǫ∇ · ∇u = S in Ω u = in ∂Ω Solution : u(x, y) =

  • x + ec1x/ǫ − 1

1 − ec1/ǫ

  • ·
  • y + ec1y/ǫ − 1

1 − ec1/ǫ

  • Aravind Balan (AICES, RWTH Aachen)
  • Hy. DG-BDM Method

July 9, Sao Paulo 11 / 24

slide-34
SLIDE 34

Test case 1 : Boundary Layer

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 12 / 24

slide-35
SLIDE 35

Test case 1 : Boundary Layer

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 13 / 24

slide-36
SLIDE 36

Test case 1 : Boundary Layer

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 14 / 24

slide-37
SLIDE 37

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(Error) Hybridized DG−BDM: Convergence Rate u u* (α =2) u* (α =0)

4.9 3.0 3.8

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 15 / 24

slide-38
SLIDE 38

Test case 2 : Linear Boundary Layer

Mixing HDG and HDG-BDM methods : Condition: If Peclect number, Pe = |c|h

ǫ

< 5, then use HDG-BDM

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 16 / 24

slide-39
SLIDE 39

Test case 2 : Linear Boundary Layer

Mixing HDG and HDG-BDM methods : Condition: If Peclect number, Pe = |c|h

ǫ

< 5, then use HDG-BDM Contours of u*, m = 2, ǫ = 0.01 Red : HDG, Blue : HDG-BDM

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 16 / 24

slide-40
SLIDE 40

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) HDG HDG / HDG−BDM

3.9 1 1 1.2 1.4 1.6 1.8 2 2.2 2.42.5 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Log(sqrt(Ne)) Ratio of no. of 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

slide-41
SLIDE 41

Test case 3

Advection Diffusion equation: ∇ · u − ∇ · (ǫ(x)∇u) = S in Ω u = g in ΓD

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 18 / 24

slide-42
SLIDE 42

Test case 3

Advection Diffusion equation: ∇ · u − ∇ · (ǫ(x)∇u) = S in Ω u = g in ΓD Diffusion Coefficient: ǫ =      0.001, x ≤ 0.9 1, x ≥ 1.1 smooth fn., 0.9 < x < 1.1

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 18 / 24

slide-43
SLIDE 43

Test case 3

Advection Diffusion equation: ∇ · u − ∇ · (ǫ(x)∇u) = S in Ω u = g in ΓD Diffusion Coefficient: ǫ =      0.001, x ≤ 0.9 1, x ≥ 1.1 smooth fn., 0.9 < x < 1.1 Solution: u(x, y) = (1 − ǫ(x)) sin(x − y) + ǫ(x) sin(2πx) sin(2πy)

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 18 / 24

slide-44
SLIDE 44

Test case 3

Condition : 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

slide-45
SLIDE 45

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 Log(sqrt(Ne)) Log(Error) HDG HDG / 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

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SLIDE 46

Test case 3

Condition : Pe < 5, 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 21 / 24

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SLIDE 47

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 Log(sqrt(Ne)) Log(Error) HDG HDG / 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

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SLIDE 48

Conclusions

Present work : HDG-BDM method and it’s connection with HDG scheme

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 23 / 24

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SLIDE 49

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 23 / 24

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SLIDE 50

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 23 / 24

slide-51
SLIDE 51

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 23 / 24

slide-52
SLIDE 52

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

slide-53
SLIDE 53

Acknowledgement

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 24 / 24

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SLIDE 54

Post Processing

Cell-wise discretization of the Neumann problem : ǫ(∇u∗

h, ∇φ)

= (σh, ∇φ) ∀φ ∈ P q

0 (Ωk)

(uh, 1) = (u∗

h, 1)

where P q

0 (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 = 0).

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 25 / 24

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SLIDE 55

Smooth Function

ǫ = 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

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SLIDE 56

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(Error) Hybridized DG Method : Convergence Rate u u*

4.88 3.95

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

Aravind Balan (AICES, RWTH Aachen)

  • Hy. DG-BDM Method

July 9, Sao Paulo 27 / 24

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SLIDE 57

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(Error) Convergence Rate for Hy. DG and Hy. DG−BDM u* HDG u* HDG−BDM

4.9

Figure: Convergence, m = 3, ǫ = 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