Adjoint-Based Mesh Adaptation for a Class ofHigh-Order Hybridized Finite-Element

Schemes for Convection-Diffusion Problems

Michael Woopen, Aravind Balan, Georg May, Jochen Schutz

Aachen Institute for Advanced Study in Computational Engineering Science

February 27, 2013

SIAM Conference on Computational Science & Engineering

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Motivation

I Accuracy and efficiency are very important aspects inmodern computational methods

I Higher order finite element methods provide high accuracy

at the cost of very high memory consumption

I Tackle this problem from two sides:

I HybridizationI Output-based adaptation

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Motivation

I Accuracy and efficiency are very important aspects inmodern computational methods

I Higher order finite element methods provide high accuracy

at the cost of very high memory consumption

I Tackle this problem from two sides:

I HybridizationI Output-based adaptation

2 / 30

Motivation

I Accuracy and efficiency are very important aspects inmodern computational methods

I Higher order finite element methods provide high accuracyat the cost of very high memory consumption

I Tackle this problem from two sides:

I HybridizationI Output-based adaptation

2 / 30

Motivation

I Accuracy and efficiency are very important aspects inmodern computational methods

I Higher order finite element methods provide high accuracyat the cost of very high memory consumption

I Tackle this problem from two sides:

I HybridizationI Output-based adaptation

2 / 30

Motivation

I Accuracy and efficiency are very important aspects inmodern computational methods

I Higher order finite element methods provide high accuracyat the cost of very high memory consumption

I Tackle this problem from two sides:I Hybridization

I Output-based adaptation

2 / 30

Motivation

I Accuracy and efficiency are very important aspects inmodern computational methods

I Higher order finite element methods provide high accuracyat the cost of very high memory consumption

I Tackle this problem from two sides:I HybridizationI Output-based adaptation

2 / 30

Idea of Hybridization

I Formulate the global system only on the element interfaces

I Obtain solution within the elements via so-called localsolvers

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(a) Standard DG method

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(b) Hybridized method

Figure: Globally coupled degrees of freedom

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Idea of Hybridization

I Formulate the global system only on the element interfaces

I Obtain solution within the elements via so-called localsolvers

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(a) Standard DG method

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(b) Hybridized method

Figure: Globally coupled degrees of freedom

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Idea of Hybridization

I Formulate the global system only on the element interfaces

I Obtain solution within the elements via so-called localsolvers

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(a) Standard DG method

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(b) Hybridized method

Figure: Globally coupled degrees of freedom

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Idea of Hybridization

I Formulate the global system only on the element interfaces

I Obtain solution within the elements via so-called localsolvers

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(a) Standard DG method

4.3. Finite element methods

Again, by substituting the explicit expression for ‚h, we obtain from (4.17) that

‚uh = uh ≠ 12 –h

[[ruh · n]],

i.e., for an appropriate choice of discrete spaces, the method of Ewing et. al. [67] fromExample 4.6 is equivalent to the IP-HDG method and hence also to a variant of the DGmethod due to Rivière [126]. However, the IP-HDG method yields (after static condensation)a symmetric positive definite global system with fewer unknowns and less coupling than theother discontinuous Galerkin methods. An example is depicted in Figure 4.2.

Figure 4.2.: Comparison of degrees of freedom (fourth order) of an IP-DG (left) and an IP-HDG (right, after static condensation) formulation.

Among the discussed methods, the IP-HDG method, which was considered in Example 4.5combines most of the advantageous features of other methods. The method is locallyconservative, the spaces can be chosen with some flexibility, it involves only primal andhybrid variables and yields symmetric positive definite global systems. Moreover, it sharesthe upwinding capabilities of discontinuous Galerkin methods. In the remaining parts of thiswork, we will therefore concentrate on devising and analyzing such methods for the Stokes,Oseen and Navier-Stokes problems.

33

(b) Hybridized method

Figure: Globally coupled degrees of freedom

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Problem Size

100

1000

10000

100000

0 1 2 3 4 5 6 7 8 9 10

Glo

bally

Couple

d D

OF

s

Polynomial Degree

DG HDG

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Setting

The convection-diffusion equation

∇ · (fc(w)− fv(w,∇w)) = s(w,∇w)

can be written as a first order system

q = ∇w∇ · (fc(w)− fv(w, q)) = s(w, q)

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Setting

The convection-diffusion equation

∇ · (fc(w)− fv(w,∇w)) = s(w,∇w)

can be written as a first order system

q = ∇w∇ · (fc(w)− fv(w, q)) = s(w, q)

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Discretization

Find (qh, wh) ∈ (Vh,Wh) s.t. ∀(τh, ϕh) ∈ (Vh,Wh)

0 = (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, wh〉∂Th

0 = − (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +⟨ϕh, fc − fv

⟩∂Th

where

Vh = v ∈ L2 (Ω)d : v|Ωk∈ P p(Ωk)

d,Ωk ∈ ThWh = w ∈ L2 (Ω) : w|Ωk

∈ P p(Ωk),Ωk ∈ Th

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Discretization

Find (qh, wh) ∈ (Vh,Wh) s.t. ∀(τh, ϕh) ∈ (Vh,Wh)

0 = (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, wh〉∂Th0 = − (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

where

Vh = v ∈ L2 (Ω)d : v|Ωk∈ P p(Ωk)

d,Ωk ∈ ThWh = w ∈ L2 (Ω) : w|Ωk

∈ P p(Ωk),Ωk ∈ Th

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Discretization

Find (qh, wh) ∈ (Vh,Wh) s.t. ∀(τh, ϕh) ∈ (Vh,Wh)

0 = (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, wh〉∂Th0 = − (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

where

Vh = v ∈ L2 (Ω)d : v|Ωk∈ P p(Ωk)

d,Ωk ∈ ThWh = w ∈ L2 (Ω) : w|Ωk

∈ P p(Ωk),Ωk ∈ Th

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Discretization — Introduction of λFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 =Nh (qh, wh, λh; τh, ϕh, µh)

:= (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th− (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

+⟨µh,

rfc − fv

z⟩Γh

where

Vh = v ∈ L2 (Ω)d

: v|Ωk∈ P p(Ωk)d,Ωk ∈ Th

Wh = w ∈ L2 (Ω) : w|Ωk∈ P p(Ωk),Ωk ∈ Th

Mh = µ ∈ L2 (Γh) : µ|e ∈ P p(e), e ∈ Γh

and

fc (λh, wh) = fc (λh)− αc (λh − wh)

fv (λh, wh, qh) = fv (λh, qh)− αv (λh − wh)

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Discretization — Introduction of λFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 =Nh (qh, wh, λh; τh, ϕh, µh)

:= (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th

− (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +⟨ϕh, fc − fv

⟩∂Th

+⟨µh,

rfc − fv

z⟩Γh

where

Vh = v ∈ L2 (Ω)d

: v|Ωk∈ P p(Ωk)d,Ωk ∈ Th

Wh = w ∈ L2 (Ω) : w|Ωk∈ P p(Ωk),Ωk ∈ Th

Mh = µ ∈ L2 (Γh) : µ|e ∈ P p(e), e ∈ Γh

and

fc (λh, wh) = fc (λh)− αc (λh − wh)

fv (λh, wh, qh) = fv (λh, qh)− αv (λh − wh)

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Discretization — Introduction of λFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 =Nh (qh, wh, λh; τh, ϕh, µh)

:= (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th− (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

+⟨µh,

rfc − fv

z⟩Γh

where

Vh = v ∈ L2 (Ω)d

: v|Ωk∈ P p(Ωk)d,Ωk ∈ Th

Wh = w ∈ L2 (Ω) : w|Ωk∈ P p(Ωk),Ωk ∈ Th

Mh = µ ∈ L2 (Γh) : µ|e ∈ P p(e), e ∈ Γh

and

fc (λh, wh) = fc (λh)− αc (λh − wh)

fv (λh, wh, qh) = fv (λh, qh)− αv (λh − wh)

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Discretization — Introduction of λFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 =Nh (qh, wh, λh; τh, ϕh, µh)

:= (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th− (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

+⟨µh,

rfc − fv

z⟩Γh

where

Vh = v ∈ L2 (Ω)d

: v|Ωk∈ P p(Ωk)d,Ωk ∈ Th

Wh = w ∈ L2 (Ω) : w|Ωk∈ P p(Ωk),Ωk ∈ Th

Mh = µ ∈ L2 (Γh) : µ|e ∈ P p(e), e ∈ Γh

and

fc (λh, wh) = fc (λh)− αc (λh − wh)

fv (λh, wh, qh) = fv (λh, qh)− αv (λh − wh)

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Discretization — Introduction of λFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 =Nh (qh, wh, λh; τh, ϕh, µh)

:= (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th− (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

+⟨µh,

rfc − fv

z⟩Γh

where

Vh = v ∈ L2 (Ω)d

: v|Ωk∈ P p(Ωk)d,Ωk ∈ Th

Wh = w ∈ L2 (Ω) : w|Ωk∈ P p(Ωk),Ωk ∈ Th

Mh = µ ∈ L2 (Γh) : µ|e ∈ P p(e), e ∈ Γh

and

fc (λh, wh) = fc (λh)− αc (λh − wh)

fv (λh, wh, qh) = fv (λh, qh)− αv (λh − wh)

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Discretization — Introduction of λFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 =Nh (qh, wh, λh; τh, ϕh, µh)

:= (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th− (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +

⟨ϕh, fc − fv

⟩∂Th

+⟨µh,

rfc − fv

z⟩Γh

where

Vh = v ∈ L2 (Ω)d

: v|Ωk∈ P p(Ωk)d,Ωk ∈ Th

Wh = w ∈ L2 (Ω) : w|Ωk∈ P p(Ωk),Ωk ∈ Th

Mh = µ ∈ L2 (Γh) : µ|e ∈ P p(e), e ∈ Γh

and

fc (λh, wh) = fc (λh)− αc (λh − wh)

fv (λh, wh, qh) = fv (λh, qh)− αv (λh − wh)

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Hybridization

The linearized global system A B RC D SL M N

δQδWδΛ

=

FGH

can be written as[A BC D

] [δQδW

]=

[FG

]−[RS

]δΛ

andLδQ+MδW +NδΛ = H.

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Hybridization

The linearized global system A B RC D SL M N

δQδWδΛ

=

FGH

can be written as[

A BC D

] [δQδW

]=

[FG

]−[RS

]δΛ

andLδQ+MδW +NδΛ = H.

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Hybridization

The linearized global system A B RC D SL M N

δQδWδΛ

=

FGH

can be written as[

A BC D

] [δQδW

]=

[FG

]−[RS

]δΛ

andLδQ+MδW +NδΛ = H.

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Hybridization

Substituting the first into the second equation yields thehybridized system(N − [L,M ]

[A BC D

]−1 [RS

])δΛ = H−[L,M ]

[A BC D

]−1 [FG

]

I The matrix

[A BC D

]is block-diagonal such that the local

problems can be solved element wise

I The global hybridized system is formulated in terms of δΛonly and thus considerably smaller than the usual globalsystem

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Hybridization

Substituting the first into the second equation yields thehybridized system(N − [L,M ]

[A BC D

]−1 [RS

])δΛ = H−[L,M ]

[A BC D

]−1 [FG

]

I The matrix

[A BC D

]is block-diagonal such that the local

problems can be solved element wise

I The global hybridized system is formulated in terms of δΛonly and thus considerably smaller than the usual globalsystem

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Hybridization

Substituting the first into the second equation yields thehybridized system(N − [L,M ]

[A BC D

]−1 [RS

])δΛ = H−[L,M ]

[A BC D

]−1 [FG

]

I The matrix

[A BC D

]is block-diagonal such that the local

problems can be solved element wise

I The global hybridized system is formulated in terms of δΛonly and thus considerably smaller than the usual globalsystem

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Adjoint-based Error Estimation

We are interested in quantifying the error of the computedtarget functional, i. e.

eh := J (x)− J (xh) = J ′ [xh] (x− xh) +O(‖x− xh‖2

)with the primal solution xh = (qh, wh, λh).

The link between variations in the residual and in the targetfunctional is given by the so-called adjoint equation

N ′h [xh] (dxh; zh) = J ′ [xh] (dxh)

with the dual solution zh =(qh, wh, λh

).

Now, one can estimate the error by

eh ≈ Nh (xh; zh)

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Adjoint-based Error Estimation

We are interested in quantifying the error of the computedtarget functional, i. e.

eh := J (x)− J (xh) = J ′ [xh] (x− xh) +O(‖x− xh‖2

)with the primal solution xh = (qh, wh, λh).The link between variations in the residual and in the targetfunctional is given by the so-called adjoint equation

N ′h [xh] (dxh; zh) = J ′ [xh] (dxh)

with the dual solution zh =(qh, wh, λh

).

Now, one can estimate the error by

eh ≈ Nh (xh; zh)

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Adjoint-based Error Estimation

We are interested in quantifying the error of the computedtarget functional, i. e.

eh := J (x)− J (xh) = J ′ [xh] (x− xh) +O(‖x− xh‖2

)with the primal solution xh = (qh, wh, λh).The link between variations in the residual and in the targetfunctional is given by the so-called adjoint equation

N ′h [xh] (dxh; zh) = J ′ [xh] (dxh)

with the dual solution zh =(qh, wh, λh

).

Now, one can estimate the error by

eh ≈ Nh (xh; zh)

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Adjoint-based Error Estimation

In matrix form, the adjoint equation reads as follow A B RC D SL M N

T Q

W

Λ

=

F

G

H

Using again static condensation, one obtains(N − [L,M ]

[A BC D

]−1 [RS

])T

Λ = H−[RT , ST ]

[A BC D

]−T [F

G

]

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Adjoint-based Error Estimation

In matrix form, the adjoint equation reads as follow A B RC D SL M N

T Q

W

Λ

=

F

G

H

Using again static condensation, one obtains(N − [L,M ]

[A BC D

]−1 [RS

])T

Λ = H−[RT , ST ]

[A BC D

]−T [F

G

]

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Pure Convection

∇ · (w,w) = 0 (x, y) ∈ Ω = [0, 1]2

w(x, y) = sin (x− y) (x, y) ∈ Γin = (x, y) : x · y = 0

The target functional computes the outflow over a part of theboundary, i. e.

J(λ) =

0.5∫0

λ(x, 1) dx

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Pure Convection

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Pure Convection (p = 2)

1e-14

1e-12

1e-10

1e-08

1e-06

1e-04

1e-03 1e-02 1e-01

Err

or

h=1/sqrt(ndof)

UniformAdjoint

EstimateCorrectedResidual

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“Confusion”

∇ · (w,w)− ε∆w = s (x, y) ∈ Ω = [0, 1]2

w(x, y) = 0 (x, y) ∈ ∂Ω

We set s such that

w(x, y) =

(x+

ex/ε − 1

1− e1/ε

)·

(y +

ey/ε − 1

1− e1/ε

), ε = 0.01

is a solution to the equation.The target functional of interest is the mean value, i. e.

J(w) =

∫Ωw(x, y) dx

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“Confusion”

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“Confusion” (p = 2)

1e-12

1e-10

1e-08

1e-06

1e-04

1e-02

1e+00

1e-03 1e-02 1e-01

Err

or

h=1/sqrt(ndof)

UniformAdjoint

EstimateCorrectedResidual

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Inviscid Flow over a Smooth Bump

J(w) =

√1

|Ω|

∫Ω

(p/ργ − p∞/ργ∞

p∞/ργ∞

)2

dx

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Inviscid Flow over a Smooth Bump (p = 3)

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03 1e-02 1e-01

||∆

s||

L2

h=1/sqrt(ndof)

UniformAdjoint

EstimateCorrectedResidual

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Subsonic Flow around the NACA0012 Airfoil

Ma = 0.5, α = 2, J(λ) = cD (λ)

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Subsonic Flow around the NACA0012 Airfoil

5.0e-06

1.0e-05

1.5e-05

2.0e-05

2.5e-05

3.0e-05

3.5e-05

4.0e-05

1e-03 1e-02

cD

h=1/sqrt(ndof)

AdjointCorrectedResidual

Reference

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Subsonic Flow around the NACA0012 Airfoil (p = 2)

1e-11

1e-10

1e-09

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03 1e-02

∆c

D

h=1/sqrt(ndof)

AdjointEstimate

CorrectedResidual

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Transonic Flow around the NACA0012 Airfoil

Ma = 0.8, α = 1.25, J(λ) = cD (λ)

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Transonic Flow around the NACA0012 Airfoil (p = 2)

2.25e-02

2.26e-02

2.27e-02

2.28e-02

2.29e-02

2.30e-02

2.31e-02

2.32e-02

1e-03 1e-02

cD

h=1/sqrt(ndof)

AdjointCorrectedResidual

Reference

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Transonic Flow around the NACA0012 Airfoil (p = 2)

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-03 1e-02

∆c

D

h=1/sqrt(ndof)

AdjointEstimate

CorrectedResidual

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Viscous Flow around the NACA0012 Airfoil

Ma = 0.5, α = 1, Re = 5000, J(λ, q) = cD (λ, q)

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Viscous Flow around the NACA0012 Airfoil (p = 2)

5.50e-02

5.52e-02

5.54e-02

5.56e-02

5.58e-02

5.60e-02

1e-03 1e-02

cD

h=1/sqrt(ndof)

AdjointCorrectedResidual

Reference

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Viscous Flow around the NACA0012 Airfoil (p = 2)

1e-08

1e-07

1e-06

1e-05

1e-04

1e-03

1e-03 1e-02

∆c

D

h=1/sqrt(ndof)

AdjointEstimate

CorrectedResidual

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Conclusions

I Reducing degrees of freedom globally via hybridization

I Efficient distribution of degrees of freedoms viaadjoint-based adaptivity

I Enhance computed target functional by error estimate

Future work:

I Full hp-adaptivity

I Turbulent flow

I Parallelism

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Conclusions

I Reducing degrees of freedom globally via hybridization

I Efficient distribution of degrees of freedoms viaadjoint-based adaptivity

I Enhance computed target functional by error estimate

Future work:

I Full hp-adaptivity

I Turbulent flow

I Parallelism

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Conclusions

I Reducing degrees of freedom globally via hybridization

I Efficient distribution of degrees of freedoms viaadjoint-based adaptivity

I Enhance computed target functional by error estimate

Future work:

I Full hp-adaptivity

I Turbulent flow

I Parallelism

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Conclusions

I Reducing degrees of freedom globally via hybridization

I Efficient distribution of degrees of freedoms viaadjoint-based adaptivity

I Enhance computed target functional by error estimate

Future work:

I Full hp-adaptivity

I Turbulent flow

I Parallelism

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Conclusions

I Reducing degrees of freedom globally via hybridization

I Efficient distribution of degrees of freedoms viaadjoint-based adaptivity

I Enhance computed target functional by error estimate

Future work:

I Full hp-adaptivity

I Turbulent flow

I Parallelism

29 / 30

Conclusions

I Reducing degrees of freedom globally via hybridization

I Efficient distribution of degrees of freedoms viaadjoint-based adaptivity

I Enhance computed target functional by error estimate

Future work:

I Full hp-adaptivity

I Turbulent flow

I Parallelism

29 / 30

Acknowledgement

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

30 / 30

Boundary ConditionsFind (qh, wh, λh) ∈ (Vh,Wh,Mh) s.t. ∀(τh, ϕh, µh) ∈ (Vh,Wh,Mh)

0 = (τh, qh)Th + (∇ · τh, wh)Th − 〈τh, λh〉∂Th\∂Ω − 〈τh, w∂Ω (λh)〉∂Th∩∂Ω

0 = − (∇ϕh, fc(wh)− fv(wh, qh))Th − (ϕh, s(wh, qh))Th +⟨ϕh, fc − fv

⟩∂Th\∂Ω

+ 〈ϕh, n · (fc (w∂Ω (λh))− fv (λh, qh))− (λh − w∂Ω (wh))〉∂Th∩∂Ω

0 =⟨µh,

rfc − fv

z⟩Γh\∂Ω

+ 〈µh, n · (fv (λh, qh)− fv,∂Ω (fv (w∂Ω (λh) , qh)))− (λh − w∂Ω (wh))〉Γh∩∂Ω

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