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 of 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 of 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 = u b on ∂ Ω Ω ⊂ R d , d = 2 , 3 . . . bounded domain with a polyhedral Lipschitz–continuous boundary ∂ Ω ε > 0 constant b ∈ W 1 , ∞ ( Ω ) d , c ∈ L ∞ ( Ω ) , f ∈ L 2 ( Ω ) , u b ∈ H 1 / 2 ( ∂ Ω )

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω Ω ⊂ R d , d = 2 , 3 . . . bounded domain with a polyhedral Lipschitz–continuous boundary ∂ Ω ε > 0 constant b ∈ W 1 , ∞ ( Ω ) d , c ∈ L ∞ ( Ω ) , f ∈ L 2 ( Ω ) , u b ∈ H 1 / 2 ( ∂ Ω ) simple model problem for many more complicated applications

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω often ε ≪ | b |

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω often ε ≪ | b | ⇒ narrow layers in u

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω often 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 = u b on ∂ Ω often 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 or mesh obtained by anisotropic adaptive refinement

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω often 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 or mesh obtained by anisotropic adaptive refinement often not feasible

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω often 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 or mesh obtained by anisotropic adaptive refinement often not feasible 2) coarse mesh + modifications of a standard discretization

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω often 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 or mesh obtained by anisotropic adaptive refinement often 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 = u b on ∂ Ω Find u h ∈ W h such that u h = u bh on ∂ Ω and Galerkin FEM: a ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h , a ( u , v ) = ε ( ∇ u , ∇ v )+( b · ∇ u , v )+( cu , v ) . where

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω Find u h ∈ W h such that u h = u bh on ∂ Ω and Galerkin FEM: a ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h , a ( u , v ) = ε ( ∇ u , ∇ v )+( b · ∇ u , v )+( cu , v ) . where Find u h ∈ W h such that u h = u bh on ∂ Ω and Stabilized FEM: a ( u h , v h )+ ∑ τ K s K ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h K ∈ T h

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω Find u h ∈ W h such that u h = u bh on ∂ Ω and Galerkin FEM: a ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h , a ( u , v ) = ε ( ∇ u , ∇ v )+( b · ∇ u , v )+( cu , v ) . where Find u h ∈ W h such that u h = u bh on ∂ Ω and Stabilized FEM: a ( u h , v h )+ ∑ τ K s K ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h K ∈ T h τ K determines the added artificial diffusion which should be:

Steady convection–diffusion–reaction equation − ε ∆ u + b · ∇ u + cu = f in Ω , u = u b on ∂ Ω Find u h ∈ W h such that u h = u bh on ∂ Ω and Galerkin FEM: a ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h , a ( u , v ) = ε ( ∇ u , ∇ v )+( b · ∇ u , v )+( cu , v ) . where Find u h ∈ W h such that u h = u bh on ∂ Ω and Stabilized FEM: a ( u h , v h )+ ∑ τ K s K ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h K ∈ T h τ K 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 = u b on ∂ Ω Find u h ∈ W h such that u h = u bh on ∂ Ω and Galerkin FEM: a ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h , a ( u , v ) = ε ( ∇ u , ∇ v )+( b · ∇ u , v )+( cu , v ) . where Find u h ∈ W h such that u h = u bh on ∂ Ω and Stabilized FEM: a ( u h , v h )+ ∑ τ K s K ( u h , v h ) = ( f , v h ) ∀ v h ∈ V h K ∈ T h τ K determines the added artificial diffusion which should be: - not ‘too small’ to remove oscillations - not ‘too large’ to avoid excessive smearing

Examples of s K ( u , v ) SUPG method: s K ( u , v ) = ( L u − f , b · ∇ v ) K Brooks, Hughes (1982) s K ( u , v ) = ( L u − f , L v ) K GLS method: Hughes, Franca, Hulbert (1989) s K ( u , v ) = ( L u − f , − L ∗ v ) K USFEM: Franca, Frey, Hughes (1992), Franca, Farhat (1995) s K ( u , v ) = ( ∇ ( L u − f ) , ∇ L v ) K GGLS method: Franca, do Carmo (1989) Local projection method: s K ( u , v ) = ( κ K ( b · ∇ u ) , κ K ( b · ∇ v )) K κ K = id − π K Becker, Braack (2004) s K ( u , v ) = ([ ∇ u ] , [ ∇ v ]) ∂ K Edge stabilization: Burman, Hansbo (2004)

Examples of s K ( u , v ) SUPG method: s K ( u , v ) = ( L u − f , b · ∇ v ) K Brooks, Hughes (1982) one of the most popular finite element approaches for convection–dominated problems

Examples of s K ( u , v ) SUPG method: s K ( u , v ) = ( L u − f , b · ∇ v ) K Brooks, Hughes (1982) one of the most popular finite element approaches for convection–dominated problems � � Pe K = | b | h K τ K = h K cothPe K − 1 with 2 | b | 2 ε Pe K

Examples of s K ( u , v ) SUPG method: s K ( u , v ) = ( L u − f , b · ∇ v ) K Brooks, Hughes (1982) one of the most popular finite element approaches for convection–dominated problems � � Pe K = | b | h K τ K = h K cothPe K − 1 with 2 | b | 2 ε Pe K 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 ( � ε ∇ u h , ∇ v h ) – isotropic artificial diffusion:

SOLD methods (spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method ( � ε ∇ u h , ∇ v h ) – isotropic artificial diffusion: ε b ⊥ · ∇ u h , b ⊥ · ∇ v h ) – crosswind artificial diffusion: ( �

SOLD methods (spurious oscillations at layers diminishing methods) add a suitable artificial diffusion term to the SUPG method ( � ε ∇ u h , ∇ v h ) – isotropic artificial diffusion: ε b ⊥ · ∇ u h , b ⊥ · ∇ v h ) – crosswind artificial diffusion: ( � � ∂ u h � ∂ v h � ∑ � ε K sign d σ – edge stabilization: ∂ t ∂ K ∂ t ∂ K ∂ K K ∈ T h