A posteriori computation of parameters in stabilized methods for - - PowerPoint PPT Presentation

A posteriori computation of parameters in stabilized methods for convection–diffusion problems

Petr Knobloch Charles University in Prague, Czech Republic joint work with Volker John WIAS, Berlin, Germany Workshop Numerical Analysis for Singularly Perturbed Problems Dedicated to the 60th Birthday of Martin Stynes Dresden, November 16–18, 2011

Contents – numerical methods for convection–diffusion–reaction equations – computation of stabilization parameters by minimization

• f a functional

– Fr´ echet derivative of the functional – choice of suitable functionals – numerical results

Contents – numerical methods for convection–diffusion–reaction equations – computation of stabilization parameters by minimization

• f a functional

– Fr´ echet derivative of the functional – choice of suitable functionals – numerical results Basic ideas published in John, K., Savescu, CMAME 200 (2011)

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Ω ⊂ Rd, d = 2,3 . . . bounded domain with a polyhedral

Lipschitz–continuous boundary ∂Ω

ε > 0 constant b ∈ W 1,∞(Ω)d, c ∈ L∞(Ω), f ∈ L2(Ω), ub ∈ H1/2(∂Ω)

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Ω ⊂ Rd, d = 2,3 . . . bounded domain with a polyhedral

Lipschitz–continuous boundary ∂Ω

ε > 0 constant b ∈ W 1,∞(Ω)d, c ∈ L∞(Ω), f ∈ L2(Ω), ub ∈ H1/2(∂Ω)

simple model problem for many more complicated applications

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

⇒

narrow layers in u

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

⇒

narrow layers in u

⇒

standard discretizations lead to global spurious oscillations unless the layers are resolved by the mesh

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

⇒

narrow layers in u

⇒

standard discretizations lead to global spurious oscillations unless the layers are resolved by the mesh Two options: 1) layer–adapted mesh – piecewise uniform mesh

• r mesh obtained by anisotropic adaptive refinement

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

⇒

narrow layers in u

⇒

standard discretizations lead to global spurious oscillations unless the layers are resolved by the mesh Two options: 1) layer–adapted mesh – piecewise uniform mesh

• r mesh obtained by anisotropic adaptive refinement
• ften not feasible

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

⇒

narrow layers in u

⇒

standard discretizations lead to global spurious oscillations unless the layers are resolved by the mesh Two options: 1) layer–adapted mesh – piecewise uniform mesh

• r mesh obtained by anisotropic adaptive refinement
• ften not feasible

2) coarse mesh + modifications of a standard discretization

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

• ften ε ≪ |b|

⇒

narrow layers in u

⇒

standard discretizations lead to global spurious oscillations unless the layers are resolved by the mesh Two options: 1) layer–adapted mesh – piecewise uniform mesh

• r mesh obtained by anisotropic adaptive refinement
• ften not feasible

2) coarse mesh + modifications of a standard discretization – special discretization of the convective term (upwinding) – introduction of additional terms (stabilization) – manipulations at algebraic level (FEMTVD schemes)

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Galerkin FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh) = (f,vh) ∀ vh ∈ Vh ,

where

a(u,v) = ε (∇u,∇v)+(b·∇u,v)+(cu,v) .

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Galerkin FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh) = (f,vh) ∀ vh ∈ Vh ,

where

a(u,v) = ε (∇u,∇v)+(b·∇u,v)+(cu,v) .

Stabilized FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh)+ ∑

K∈Th

τK sK(uh,vh) = (f,vh) ∀ vh ∈ Vh

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Galerkin FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh) = (f,vh) ∀ vh ∈ Vh ,

where

a(u,v) = ε (∇u,∇v)+(b·∇u,v)+(cu,v) .

Stabilized FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh)+ ∑

K∈Th

τK sK(uh,vh) = (f,vh) ∀ vh ∈ Vh τK determines the added artificial diffusion which should be:

Steady convection–diffusion–reaction equation

−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Galerkin FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh) = (f,vh) ∀ vh ∈ Vh ,

where

a(u,v) = ε (∇u,∇v)+(b·∇u,v)+(cu,v) .

Stabilized FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh)+ ∑

K∈Th

τK sK(uh,vh) = (f,vh) ∀ vh ∈ Vh τK determines the added artificial diffusion which should be:

• not ‘too small’ to remove oscillations
• not ‘too large’ to avoid excessive smearing

Steady convection–diffusion–reaction equation

L u :=−ε ∆u+b·∇u+cu = f in Ω, u = ub on ∂Ω

Galerkin FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh) = (f,vh) ∀ vh ∈ Vh ,

where

a(u,v) = ε (∇u,∇v)+(b·∇u,v)+(cu,v) .

Stabilized FEM: Find uh ∈ Wh such that uh = ubh on ∂Ω and

a(uh,vh)+ ∑

K∈Th

τK sK(uh,vh) = (f,vh) ∀ vh ∈ Vh τK determines the added artificial diffusion which should be:

• not ‘too small’ to remove oscillations
• not ‘too large’ to avoid excessive smearing

Examples of sK(u,v) SUPG method:

sK(u,v) = (L u− f,b·∇v)K

Brooks, Hughes (1982)

GLS method:

sK(u,v) = (L u− f,L v)K

Hughes, Franca, Hulbert (1989)

USFEM:

sK(u,v) = (L u− f,−L ∗ v)K

Franca, Frey, Hughes (1992), Franca, Farhat (1995)

GGLS method:

sK(u,v) = (∇(L u− f),∇L v)K

Franca, do Carmo (1989)

Local projection method:

sK(u,v) = (κK(b·∇u),κK(b·∇v))K

Becker, Braack (2004)

κK = id −πK

Edge stabilization:

sK(u,v) = ([∇u],[∇v])∂K

Burman, Hansbo (2004)

Examples of sK(u,v) SUPG method:

sK(u,v) = (L u− f,b·∇v)K

Brooks, Hughes (1982)

• ne of the most popular finite element approaches for

convection–dominated problems

Examples of sK(u,v) SUPG method:

sK(u,v) = (L u− f,b·∇v)K

Brooks, Hughes (1982)

• ne of the most popular finite element approaches for

convection–dominated problems

τK = hK 2|b|

• cothPeK − 1

PeK

• with

PeK = |b|hK 2ε

Examples of sK(u,v) SUPG method:

sK(u,v) = (L u− f,b·∇v)K

Brooks, Hughes (1982)

• ne of the most popular finite element approaches for

convection–dominated problems

τK = hK 2|b|

• cothPeK − 1

PeK

• with

PeK = |b|hK 2ε

typically still spurious oscillations localized in narrow regions along sharp layers

SOLD methods

(spurious oscillations at layers diminishing methods)

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method – isotropic artificial diffusion:

( ε ∇uh,∇vh)

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method – isotropic artificial diffusion:

( ε ∇uh,∇vh)

– crosswind artificial diffusion: (

ε b⊥ ·∇uh,b⊥ ·∇vh)

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method – isotropic artificial diffusion:

( ε ∇uh,∇vh)

– crosswind artificial diffusion: (

ε b⊥ ·∇uh,b⊥ ·∇vh)

– edge stabilization:

∑

K∈Th

• ∂K
• εK sign

∂uh ∂t∂K ∂vh ∂t∂K dσ

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method – isotropic artificial diffusion:

( ε ∇uh,∇vh)

– crosswind artificial diffusion: (

ε b⊥ ·∇uh,b⊥ ·∇vh)

– edge stabilization:

∑

K∈Th

• ∂K
• εK sign

∂uh ∂t∂K ∂vh ∂t∂K dσ

typically

ε = ε(uh) ⇒

resulting method is nonlinear

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method – isotropic artificial diffusion:

( ε ∇uh,∇vh)

– crosswind artificial diffusion: (

ε b⊥ ·∇uh,b⊥ ·∇vh)

– edge stabilization:

∑

K∈Th

• ∂K
• εK sign

∂uh ∂t∂K ∂vh ∂t∂K dσ

typically

ε = ε(uh) ⇒

resulting method is nonlinear many proposals for

ε in the literature

SOLD methods

(spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method – isotropic artificial diffusion:

( ε ∇uh,∇vh)

– crosswind artificial diffusion: (

ε b⊥ ·∇uh,b⊥ ·∇vh)

– edge stabilization:

∑

K∈Th

• ∂K
• εK sign

∂uh ∂t∂K ∂vh ∂t∂K dσ

typically

ε = ε(uh) ⇒

resulting method is nonlinear many proposals for

ε in the literature

review and computational comparison: John, K., CMAME (2007), CMAME (2008)

Examples of

ε

do Carmo, Gale˜ ao, CMAME (1991)

• ε|K = max
• 0, hK |Rh(uh)|

2|∇uh| − hK 2|b| |Rh(uh)|2 |∇uh|2

• Codina, CMAME (1993) (modified)
• ε|K = max
• 0,η diam(K)|Rh(uh)|

2|∇uh| −ε

• η = 0.7

recommended Burman, Ern, CMAME (2002) (modified)

• ε|K = hK |Rh(uh)|

2|∇uh| 1 1+ |Rh(uh)|

|b||∇uh|

Burman, Ern (2005)

∑

K∈Th

|K|

• ∂K C
• Rh(uh)|K
• ∂uh

∂t∂K

• ∂uh

∂t∂K ∂vh ∂t∂K dσ

Examples of

ε

do Carmo, Gale˜ ao, CMAME (1991)

• ε|K = max
• 0, hK |Rh(uh)|

2|∇uh| − hK 2|b| |Rh(uh)|2 |∇uh|2

• Codina, CMAME (1993) (modified)
• ε|K = max
• 0,η diam(K)|Rh(uh)|

2|∇uh| −ε

• η = 0.7

recommended Burman, Ern, CMAME (2002) (modified)

• ε|K = hK |Rh(uh)|

2|∇uh| 1 1+ |Rh(uh)|

|b||∇uh|

Burman, Ern (2005)

∑

K∈Th

|K|

• ∂K C
• Rh(uh)|K
• ∂uh

∂t∂K

• ∂uh

∂t∂K ∂vh ∂t∂K dσ

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations – SOLD methods involving a parameter (modified method of Codina, edge stabilization): – in model cases, oscillations removed for appropriate parameters

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations – SOLD methods involving a parameter (modified method of Codina, edge stabilization): – in model cases, oscillations removed for appropriate parameters – different values of parameters in different regions of Ω

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations – SOLD methods involving a parameter (modified method of Codina, edge stabilization): – in model cases, oscillations removed for appropriate parameters – different values of parameters in different regions of Ω – optimal parameters depend on the data and on the grid

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations – SOLD methods involving a parameter (modified method of Codina, edge stabilization): – in model cases, oscillations removed for appropriate parameters – different values of parameters in different regions of Ω – optimal parameters depend on the data and on the grid – not clear how to choose parameters for more complicated problems

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations – SOLD methods involving a parameter (modified method of Codina, edge stabilization): – in model cases, oscillations removed for appropriate parameters – different values of parameters in different regions of Ω – optimal parameters depend on the data and on the grid – not clear how to choose parameters for more complicated problems – sometimes very difficult to compute the solution of the nonlinear discretization

Properties of the SOLD methods

– SOLD methods without user–chosen parameters: – generally not able to remove oscillations – SOLD methods involving a parameter (modified method of Codina, edge stabilization): – in model cases, oscillations removed for appropriate parameters – different values of parameters in different regions of Ω – optimal parameters depend on the data and on the grid – not clear how to choose parameters for more complicated problems – sometimes very difficult to compute the solution of the nonlinear discretization – oscillation–free solutions cannot be guaranteed

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0.

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

Example – SUPG method:

Rh(uh,yh),vh = a(uh,vh)+(L uh − f,yh b·∇vh)−(f,vh), Yh ... piecewise constant functions on Ω

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

Example – SUPG method:

Rh(uh,yh),vh = a(uh,vh)+(L uh − f,yh b·∇vh)−(f,vh), Yh ... piecewise constant functions on Ω

We shall write uh(yh) instead of uh.

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

Example – SUPG method:

Rh(uh,yh),vh = a(uh,vh)+(L uh − f,yh b·∇vh)−(f,vh), Yh ... piecewise constant functions on Ω

We shall write uh(yh) instead of uh.

Ih(uh(yh))

. . . measure for the error or the quality of uh(yh)

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

Example – SUPG method:

Rh(uh,yh),vh = a(uh,vh)+(L uh − f,yh b·∇vh)−(f,vh), Yh ... piecewise constant functions on Ω

We shall write uh(yh) instead of uh.

Ih(uh(yh))

. . . measure for the error or the quality of uh(yh) Aim: find yh such that Ih(uh(yh)) is small

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

Example – SUPG method:

Rh(uh,yh),vh = a(uh,vh)+(L uh − f,yh b·∇vh)−(f,vh), Yh ... piecewise constant functions on Ω

We shall write uh(yh) instead of uh.

Ih(uh(yh))

. . . measure for the error or the quality of uh(yh) Aim: find yh such that Ih(uh(yh)) is small

⇒

constrained nonlinear optimization problem

A posteriori optimization of stabilization parameters For simplicity:

ub = 0

Stabilized FEM: Given a stabilization parameter yh ∈ Yh, find uh ∈ Vh such that Rh(uh,yh) = 0. Here:

Rh : Vh ×Yh → V ′

h

Example – SUPG method:

Rh(uh,yh),vh = a(uh,vh)+(L uh − f,yh b·∇vh)−(f,vh), Yh ... piecewise constant functions on Ω

We shall write uh(yh) instead of uh.

Ih(uh(yh))

. . . measure for the error or the quality of uh(yh) Aim: find yh such that Ih(uh(yh)) is small Example – SUPG method:

0 ≤ yh|K ≤ 10τK ∀ K ∈ Th

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

not efficient!

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

not efficient! Adjoint problem: Find ψh(yh) ∈ Vh such that

(∂wRh)′(uh(yh),yh)ψh(yh) = DIh(uh(yh)).

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

not efficient! Adjoint problem: Find ψh(yh) ∈ Vh such that

(∂wRh)′(uh(yh),yh)ψh(yh) = DIh(uh(yh)).

Since Rh(uh(yh),yh) = 0, we have

∂wRh(uh(yh),yh)Duh(yh)+∂yRh(uh(yh),yh) = 0.

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

not efficient! Adjoint problem: Find ψh(yh) ∈ Vh such that

(∂wRh)′(uh(yh),yh)ψh(yh) = DIh(uh(yh)).

Since Rh(uh(yh),yh) = 0, we have

∂wRh(uh(yh),yh)Duh(yh)+∂yRh(uh(yh),yh) = 0.

Thus,

DΦh(yh) = −(∂yRh)′(uh(yh),yh)ψh(yh).

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

not efficient! Adjoint problem: Find ψh(yh) ∈ Vh such that

(∂wRh)′(uh(yh),yh)ψh(yh) = DIh(uh(yh)).

Since Rh(uh(yh),yh) = 0, we have

∂wRh(uh(yh),yh)Duh(yh)+∂yRh(uh(yh),yh) = 0.

Thus,

DΦh(yh) = −(∂yRh)′(uh(yh),yh)ψh(yh).

Example – SUPG method:

ψh(yh) solves a(vh,ψh(yh))+(L vh,yh b·∇ψh(yh)) = DIh(uh(yh)),vh ∀ vh ∈ Vh

Fr´ echet derivative of Φh(yh) := Ih(uh(yh))

DΦh(yh) = DIh(uh(yh))Duh(yh)

not efficient! Adjoint problem: Find ψh(yh) ∈ Vh such that

(∂wRh)′(uh(yh),yh)ψh(yh) = DIh(uh(yh)).

Since Rh(uh(yh),yh) = 0, we have

∂wRh(uh(yh),yh)Duh(yh)+∂yRh(uh(yh),yh) = 0.

Thus,

DΦh(yh) = −(∂yRh)′(uh(yh),yh)ψh(yh).

Example – SUPG method:

DΦh(yh) is given by DΦh(yh), ˜ yh = −(L uh(yh)− f, ˜ yh b·∇ψh(yh)).

Choice of Ih

Γ+ = {x ∈ ∂Ω; (b·n)(x) > 0} Γ0 = {x ∈ ∂Ω; (b·n)(x) = 0}

Choice of Ih

Γ+ = {x ∈ ∂Ω; (b·n)(x) > 0} Γ0 = {x ∈ ∂Ω; (b·n)(x) = 0} Gh = {K ∈ Th ; K ∩Γ+ = /

• r

K ∩Γ0 = / 0}

Choice of Ih

Γ+ = {x ∈ ∂Ω; (b·n)(x) > 0} Γ0 = {x ∈ ∂Ω; (b·n)(x) = 0} Gh = {K ∈ Th ; K ∩Γ+ = /

• r

K ∩Γ0 = / 0} Gh =

• K∈Gh

K

Choice of Ih

Γ+ = {x ∈ ∂Ω; (b·n)(x) > 0} Γ0 = {x ∈ ∂Ω; (b·n)(x) = 0} Gh = {K ∈ Th ; K ∩Γ+ = /

• r

K ∩Γ0 = / 0} Gh =

• K∈Gh

K Ires

h (uh) = L uh − f2 0,Ω\Gh

Numerical results

Numerical results – BFGS method for minimization of Φh(yh)

Numerical results – BFGS method for minimization of Φh(yh) – initialization of the SUPG parameter by τK

Numerical results – BFGS method for minimization of Φh(yh) – initialization of the SUPG parameter by τK – Ω = (0,1)2

Numerical results – BFGS method for minimization of Φh(yh) – initialization of the SUPG parameter by τK – Ω = (0,1)2 – conforming P

1 finite element space on the triangulation

(2048 triangles)

Example 1 (convection skew to the mesh)

u = 1 u = 0 ε = 10−8 |b| = 1 c = 0 f = 0

Example 1, SUPG method

Example 1, SUPG method optimized using Ires

h

Choice of Ih A B C We need a functional that prefers C.

Choice of Ih A B C We need a functional that prefers C. A candidate:

1

0 |u′|pdx

Choice of Ih A B C We need a functional that prefers C. A candidate:

1

0 |u′|pdx= d

• 1

d

• p with d the width of the layer

Choice of Ih A B C We need a functional that prefers C. A candidate:

1

0 |u′|pdx= d

• 1

d

• p with d the width of the layer

⇒ use p < 1

Choice of Ih A B C We need a functional that prefers C. A candidate:

1

0 |u′|pdx= d

• 1

d

• p with d the width of the layer

⇒ use p < 1 ⇒ Icross

h

(uh) =

• Ω\Gh
• |b⊥ ·∇uh|dx

√

regularized near 0

Example 1, SUPG method optimized using Ires

h

+α Icross

h

SOLD method

SUPG method + additional term

( ε(uh, ˜ yh)b⊥ ·∇uh,b⊥ ·∇vh)

SOLD method

SUPG method + additional term

( ε(uh, ˜ yh)b⊥ ·∇uh,b⊥ ·∇vh)

• ε(uh, ˜

yh)|K = ˜ yh|K diam(K)|L uh − f| 2|∇uh|

SOLD method

SUPG method + additional term

( ε(uh, ˜ yh)b⊥ ·∇uh,b⊥ ·∇vh)

• ε(uh, ˜

yh)|K = ˜ yh|K diam(K)|L uh − f| 2|∇uh| 0 ≤ ˜ yh|K ≤ 1 ∀ K ∈ Th

SOLD method

SUPG method + additional term

( ε(uh, ˜ yh)b⊥ ·∇uh,b⊥ ·∇vh)

• ε(uh, ˜

yh)|K = ˜ yh|K diam(K)|L uh − f| 2|∇uh| 0 ≤ ˜ yh|K ≤ 1 ∀ K ∈ Th

initialization:

˜ yh := 0

Example 1, SOLD method optimized using Ires

h

+α Icross

h

Example 1, SUPG method

Example 1, SUPG method optimized using Ires

h

+α Icross

h

Example 1, SOLD method optimized using Ires

h

+α Icross

h

Example 1, SUPG method optimized using Ires

h

Example 2 (convection with a constant nonzero source term)

u = 0 ε = 10−8 |b| = 1 c = 0 f = 1

Example 2, SUPG method

Example 2, SOLD method optimized using Ires

h

Example 3 (problem with two interior layers)

u = 1 ∂u ∂n = 0 u = 0 ε = 10−8 b = (−y,x) c = 0 f = 0

Example 3, SUPG method

Example 3, SOLD method

Example 3, SOLD method optimized using Icross

h

Example 3, outflow profiles

Example 4 (problem with two interior layers)

u = 1+x ∂u ∂n = 0 u = 0 ε = 10−8 b = (−y,x) c = 0 f = 0

Choice of Ih

Ilim

h (uh) =

• Ω\Gh

ϕ(|L uh − f|2)dx ϕ(x) is increasing and concave for x ∈ (0,x0), ϕ(x) = 1 for x > x0

Example 4, SUPG method

Example 4, SOLD method

Example 4, SOLD method optimized using Ilim

h

Example 4, outflow profiles

Conclusions

– a posteriori optimization of parameters can significantly improve solutions of stabilized methods – further development of appropriate target functionals needed – more efficient minimization techniques have to be applied