Motivation

Theoreticalresults

Numericalresults

Summary

Albert-Ludwigs-Universitat Freiburg

Compact and Stable Discontinuous GalerkinMethods for Convection-Diffusion Problems

S. BRDAR

JOINT WORK WITH R. KLOFKORN AND A. DEDNER

Int. MetStrom Conference,June 7, 2011

Berlin

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 1 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Outline

Motivation

Theoretical results

Numerical results

Summary

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 2 / 27

Motivation

Theoretical results

Numerical results

Summary

Motivation

Theoreticalresults

Numericalresults

Summary

DG methods for convection-diffusion problems

Brief history

I Douglas and Dupont 1976: elliptic and parabolic PDEsI Cockburn, Shu et al 1998: nonlinear parabolic PDE for conservation laws

Advantages of DG methodsI easy to achieve higher order without enlarging stencilI easy construction of discrete function spacesI easy extension on non-conforming meshesI efficient parallelization

Disadvantages of DG methodsI high number of unknowns (Navier-Stokes in 3d, 3rd order→ 50 unknowns

per mesh element)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 4 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG methods for convection-diffusion problems

Brief history

I Douglas and Dupont 1976: elliptic and parabolic PDEsI Cockburn, Shu et al 1998: nonlinear parabolic PDE for conservation laws

Advantages of DG methodsI easy to achieve higher order without enlarging stencilI easy construction of discrete function spacesI easy extension on non-conforming meshesI efficient parallelization

Disadvantages of DG methodsI high number of unknowns (Navier-Stokes in 3d, 3rd order→ 50 unknowns

per mesh element)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 4 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG methods for convection-diffusion problems

Brief history

I Douglas and Dupont 1976: elliptic and parabolic PDEsI Cockburn, Shu et al 1998: nonlinear parabolic PDE for conservation laws

Advantages of DG methodsI easy to achieve higher order without enlarging stencilI easy construction of discrete function spacesI easy extension on non-conforming meshesI efficient parallelization

Disadvantages of DG methodsI high number of unknowns (Navier-Stokes in 3d, 3rd order→ 50 unknowns

per mesh element)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 4 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Stencils of DG methods

I 1st order PDEs: compact stencil even for higher order

I 2nd order PDEs: compact stencil for higher order?

I Interior Penalty (IP)

J. Douglas; T. Dupont Interior penalty procedures for elliptic and parabolic Galerkin methods(1976),

R. Hartmann; P. Houston An optimal order penalty discontinuous Galerkin discretization of

the compressible Navier-Stokes equations (2008)

I Bassi-Rebay 2 (BR2)

F. Bassi; S. Rebay A high-order accurate discontinuous finite element method for thenumerical solution of the compressible Navier-Stokes equations (1997),

F. Bassi; S. Rebay; G. Mariotti; S. Pedinotti; M. Savini A high-order accurate discontinuous

finite element method for invisid turbomachinery flows (1997)

I Local Discontinuous Galerkin (LDG) !!!

B. Cockburn; C.-W. Shu The Local Discontinuous Galerkin method for time-dependent

convection-diffusion problems (1998)

I Compact Discontinuous Galerkin (CDG)

J. Peraire; P.-O. Persson The Compact Discontinuous Galerkin (CDG) method for elliptic

problems (2008)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 5 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Stencils of DG methods

I 1st order PDEs: compact stencil even for higher order

I 2nd order PDEs: compact stencil for higher order?

I Interior Penalty (IP)

J. Douglas; T. Dupont Interior penalty procedures for elliptic and parabolic Galerkin methods(1976),

R. Hartmann; P. Houston An optimal order penalty discontinuous Galerkin discretization of

the compressible Navier-Stokes equations (2008)

I Bassi-Rebay 2 (BR2)

F. Bassi; S. Rebay A high-order accurate discontinuous finite element method for thenumerical solution of the compressible Navier-Stokes equations (1997),

F. Bassi; S. Rebay; G. Mariotti; S. Pedinotti; M. Savini A high-order accurate discontinuous

finite element method for invisid turbomachinery flows (1997)

I Local Discontinuous Galerkin (LDG) !!!

B. Cockburn; C.-W. Shu The Local Discontinuous Galerkin method for time-dependent

convection-diffusion problems (1998)

I Compact Discontinuous Galerkin (CDG)

J. Peraire; P.-O. Persson The Compact Discontinuous Galerkin (CDG) method for elliptic

problems (2008)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 5 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Stencils of DG methods

I 1st order PDEs: compact stencil even for higher order

I 2nd order PDEs: compact stencil for higher order?I Interior Penalty (IP)

J. Douglas; T. Dupont Interior penalty procedures for elliptic and parabolic Galerkin methods(1976),

R. Hartmann; P. Houston An optimal order penalty discontinuous Galerkin discretization of

the compressible Navier-Stokes equations (2008)

I Bassi-Rebay 2 (BR2)

F. Bassi; S. Rebay A high-order accurate discontinuous finite element method for thenumerical solution of the compressible Navier-Stokes equations (1997),

F. Bassi; S. Rebay; G. Mariotti; S. Pedinotti; M. Savini A high-order accurate discontinuous

finite element method for invisid turbomachinery flows (1997)

I Local Discontinuous Galerkin (LDG) !!!

B. Cockburn; C.-W. Shu The Local Discontinuous Galerkin method for time-dependent

convection-diffusion problems (1998)

I Compact Discontinuous Galerkin (CDG)

J. Peraire; P.-O. Persson The Compact Discontinuous Galerkin (CDG) method for elliptic

problems (2008)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 5 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Stencils of DG methods

I 1st order PDEs: compact stencil even for higher order

I 2nd order PDEs: compact stencil for higher order?I Interior Penalty (IP)

J. Douglas; T. Dupont Interior penalty procedures for elliptic and parabolic Galerkin methods(1976),

R. Hartmann; P. Houston An optimal order penalty discontinuous Galerkin discretization of

the compressible Navier-Stokes equations (2008)

I Bassi-Rebay 2 (BR2)F. Bassi; S. Rebay A high-order accurate discontinuous finite element method for thenumerical solution of the compressible Navier-Stokes equations (1997),

F. Bassi; S. Rebay; G. Mariotti; S. Pedinotti; M. Savini A high-order accurate discontinuous

finite element method for invisid turbomachinery flows (1997)

I Local Discontinuous Galerkin (LDG) !!!

B. Cockburn; C.-W. Shu The Local Discontinuous Galerkin method for time-dependent

convection-diffusion problems (1998)

I Compact Discontinuous Galerkin (CDG)

J. Peraire; P.-O. Persson The Compact Discontinuous Galerkin (CDG) method for elliptic

problems (2008)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 5 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Stencils of DG methods

I 1st order PDEs: compact stencil even for higher order

I 2nd order PDEs: compact stencil for higher order?I Interior Penalty (IP)

J. Douglas; T. Dupont Interior penalty procedures for elliptic and parabolic Galerkin methods(1976),

R. Hartmann; P. Houston An optimal order penalty discontinuous Galerkin discretization of

the compressible Navier-Stokes equations (2008)

I Bassi-Rebay 2 (BR2)F. Bassi; S. Rebay A high-order accurate discontinuous finite element method for thenumerical solution of the compressible Navier-Stokes equations (1997),

F. Bassi; S. Rebay; G. Mariotti; S. Pedinotti; M. Savini A high-order accurate discontinuous

finite element method for invisid turbomachinery flows (1997)

I Local Discontinuous Galerkin (LDG) !!!

B. Cockburn; C.-W. Shu The Local Discontinuous Galerkin method for time-dependent

convection-diffusion problems (1998)

I Compact Discontinuous Galerkin (CDG)

J. Peraire; P.-O. Persson The Compact Discontinuous Galerkin (CDG) method for elliptic

problems (2008)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 5 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Stencils of DG methods

I 1st order PDEs: compact stencil even for higher order

I 2nd order PDEs: compact stencil for higher order?I Interior Penalty (IP)

J. Douglas; T. Dupont Interior penalty procedures for elliptic and parabolic Galerkin methods(1976),

R. Hartmann; P. Houston An optimal order penalty discontinuous Galerkin discretization of

the compressible Navier-Stokes equations (2008)

I Bassi-Rebay 2 (BR2)F. Bassi; S. Rebay A high-order accurate discontinuous finite element method for thenumerical solution of the compressible Navier-Stokes equations (1997),

F. Bassi; S. Rebay; G. Mariotti; S. Pedinotti; M. Savini A high-order accurate discontinuous

finite element method for invisid turbomachinery flows (1997)

I Local Discontinuous Galerkin (LDG) !!!

B. Cockburn; C.-W. Shu The Local Discontinuous Galerkin method for time-dependent

convection-diffusion problems (1998)

I Compact Discontinuous Galerkin (CDG)

J. Peraire; P.-O. Persson The Compact Discontinuous Galerkin (CDG) method for elliptic

problems (2008)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 5 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Elliptic case.

The Poisson equation

−∆u = s in Ω,

u = gD on ∂Ω,

Ω ⊂ Rd, d ∈ N a bounded polygon, s ∈ Ld(Ω), gD ∈ Ld(∂Ω), andu : Rd → R.

Let ϕ and ψ be test functions and K ⊂ Ω

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 6 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Elliptic case.

The Poisson equation

σ = ∇u, −∇ · σ = s in Ω,

u = gD on ∂Ω,

Ω ⊂ Rd, d ∈ N a bounded polygon, s ∈ Ld(Ω), gD ∈ Ld(∂Ω), andu : Rd → R.

Let ϕ and ψ be test functions and K ⊂ Ω

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 6 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Elliptic case.

The Poisson equation

σ = ∇u, −∇ · σ = s in Ω,

u = gD on ∂Ω,

Ω ⊂ Rd, d ∈ N a bounded polygon, s ∈ Ld(Ω), gD ∈ Ld(∂Ω), andu : Rd → R.

Let ϕ and ψ be test functions and K ⊂ Ω

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 6 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG flux formulation

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

For a given partition Th into polygons we define

Pmk (Th) = v : v|K ∈ [Pk(K)]m, K ∈ Th for m ∈ N.

DG flux formulation

Find uh ∈ Pk(Th) and σh ∈ Pdk (Th) so that

(σh, ψ)K = −(uh,∇ · ψ)K + 〈uh,nK · ψ〉∂K ,(σh,∇ϕ)K = (s, ϕ)K + 〈σh · nK , ϕ〉∂K

for all ϕ ∈ Pk(K), ψ ∈ Pdk (K), and K ∈ Th.

Can we avoid computation of the intermediate variable σ?

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 7 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG flux formulation

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

For a given partition Th into polygons we define

Pmk (Th) = v : v|K ∈ [Pk(K)]m, K ∈ Th for m ∈ N.

DG flux formulation

Find uh ∈ Pk(Th) and σh ∈ Pdk (Th) so that

(σh, ψ)K = −(uh,∇ · ψ)K + 〈uh,nK · ψ〉∂K ,(σh,∇ϕ)K = (s, ϕ)K + 〈σh · nK , ϕ〉∂K

for all ϕ ∈ Pk(K), ψ ∈ Pdk (K), and K ∈ Th.

Can we avoid computation of the intermediate variable σ?

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 7 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG flux formulation

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

For a given partition Th into polygons we define

Pmk (Th) = v : v|K ∈ [Pk(K)]m, K ∈ Th for m ∈ N.

DG flux formulation

Find uh ∈ Pk(Th) and σh ∈ Pdk (Th) so that

(σh, ψ)K = −(∇uh, ψ)K + 〈uh − u,nK · ψ〉∂K ,(σh,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

for all ϕ ∈ Pk(K), ψ ∈ Pdk (K), and K ∈ Th.

Can we avoid computation of the intermediate variable σ?

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 7 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG flux formulation

(σ, ψ)K = −(u,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σ,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

For a given partition Th into polygons we define

Pmk (Th) = v : v|K ∈ [Pk(K)]m, K ∈ Th for m ∈ N.

DG flux formulation

Find uh ∈ Pk(Th) and σh ∈ Pdk (Th) so that

(σh, ψ)K = −(∇uh, ψ)K + 〈uh − u,nK · ψ〉∂K ,(σh,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

for all ϕ ∈ Pk(K), ψ ∈ Pdk (K), and K ∈ Th.

Can we avoid computation of the intermediate variable σ?

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 7 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG primal formulation 1

DG flux formulation

Find uh ∈ Pk(Th) and σh ∈ Pdk (Th) so that

(σh, ψ)K = −(uh,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σh,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

for all ϕ ∈ Pk(K), ψ ∈ Pdk (K), and K ∈ Th.

Can we avoid computation of the intermediate variable σ?Yes, if σ = σ(uh) (i.e. Arnold et al 2002).

DG primal formulation

Find uh ∈ Pk(Th) so that B(uh, ϕ) = (s, ϕ)Ω for all ϕ ∈ Pk(Th) with

B(uh, ϕ) = (∇uh,∇ψ)Ω −Xe∈Γ

〈[[uh]], ∇ϕ〉e + 〈σ, [[ϕ]]〉e

+Xe∈Γi

〈u− uh, [∇ϕ]〉e − 〈[σ], ϕ〉e .

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 8 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG primal formulation 1

DG flux formulation

Find uh ∈ Pk(Th) and σh ∈ Pdk (Th) so that

(σh, ψ)K = −(uh,∇ · ψ)K + 〈u,nK · ψ〉∂K ,(σh,∇ϕ)K = (s, ϕ)K + 〈σ · nK , ϕ〉∂K

for all ϕ ∈ Pk(K), ψ ∈ Pdk (K), and K ∈ Th.

Can we avoid computation of the intermediate variable σ?Yes, if σ = σ(uh) (i.e. Arnold et al 2002).

DG primal formulation

Find uh ∈ Pk(Th) so that B(uh, ϕ) = (s, ϕ)Ω for all ϕ ∈ Pk(Th) with

B(uh, ϕ) = (∇uh,∇ψ)Ω −Xe∈Γ

〈[[uh]], ∇ϕ〉e + 〈σ, [[ϕ]]〉e

+Xe∈Γi

〈u− uh, [∇ϕ]〉e − 〈[σ], ϕ〉e .

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 8 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG primal formulation 2

DG primal formulation

Find uh ∈ Pk(Th) so that B(uh, ϕ) = (s, ϕ)Ω for all ϕ ∈ Pk(Th) with

B(uh, ϕ) = (∇uh,∇ψ)Ω −Xe∈Γ

〈[[uh]], ∇ϕ〉e + 〈σ, [[ϕ]]〉e

+Xe∈Γi

〈u− uh, [∇ϕ]〉e − 〈[σ], ϕ〉e .

Jump and average. Let K−e ,K+e ∈ Th and e ∈ Γi

[[φ]]e = (ϕ|K−e− ϕ|

K+e

)nK−e

, [ξ]e = (ξ|K−e− ξ|

K+e

) · nK−e

φe = 1/2 (ϕ|K−e

+ ϕ|K+

e), ξe = 1/2 (ξ|

K−e+ ξ|

K+e

) .

If e ∈ Γ \ Γi

[[φ]]e = (ϕ|Ke − g′D)nKe , [ξ]e = ξ|Ke · nKe

φe = ϕ|Ke , ξe = ξ|Ke .

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 9 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG primal formulation 2

DG primal formulation

Find uh ∈ Pk(Th) so that B(uh, ϕ) = (s, ϕ)Ω for all ϕ ∈ Pk(Th) with

B(uh, ϕ) = (∇uh,∇ψ)Ω −Xe∈Γ

〈[[uh]], ∇ϕ〉e + 〈σ, [[ϕ]]〉e

+Xe∈Γi

〈u− uh, [∇ϕ]〉e − 〈[σ], ϕ〉e .

Jump and average. Let K−e ,K+e ∈ Th and e ∈ Γi

[[φ]]e = (ϕ|K−e− ϕ|

K+e

)nK−e

, [ξ]e = (ξ|K−e− ξ|

K+e

) · nK−e

φe = 1/2 (ϕ|K−e

+ ϕ|K+

e), ξe = 1/2 (ξ|

K−e+ ξ|

K+e

) .

If e ∈ Γ \ Γi

[[φ]]e = (ϕ|Ke − g′D)nKe , [ξ]e = ξ|Ke · nKe

φe = ϕ|Ke , ξe = ξ|Ke .

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 9 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG methods

Method u σ

IP uh ∇uh − η[[uh]]

BR2 uh ∇uh+χre([[uh]])

CDG uh − βe · [[uh]] ∇uh − η[[uh]] + βe[∇uh]+χ`Le(uh)+ βe[Le(uh)]

´CDG2 uh ∇uh − η[[uh]]

+χ`Le(uh)+ βe[Le(uh)]

´Parameter-free. Determine the penalty factor η and the lifting factor χ.

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 10 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

DG methods

Method u σ

IP uh ∇uh − η[[uh]]

BR2 uh ∇uh+χre([[uh]])

CDG uh − βe · [[uh]] ∇uh − η[[uh]] + βe[∇uh]+χ`Le(uh)+ βe[Le(uh)]

´CDG2 uh ∇uh − η[[uh]]

+χ`Le(uh)+ βe[Le(uh)]

´Lifting. re, Le : [L2(e)]d → Pmk (Th) and le : L2(e)→ Pmk (Th) given as

(re(ξ), τ )Ω = −〈ξ, τ〉e, (le(φ), τ )Ω = −〈ψ, [τ ]〉e,Le(ξ) = re(ξ) + le(βe · ξ)

for all ξ, τ ∈ Pmk (Th) and φ ∈ Pk(Th).

Switch. βe induces ordering between adjacent grid elements.

βe = nK−e

/2 = −nK+

e/2

For CDG choose any w ∈ Rd such that w ·ne 6= 0, ∀ e the upwind switchis determined by w · n

K−e> 0.

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 10 / 27

Motivation

Theoretical results

Numerical results

Summary

Motivation

Theoreticalresults

Numericalresults

Summary

Theory for linear elliptic problem.

A-priori estimate:For stable, bounded, consistent, and adjoint consistent methods thediscrete solution uh ∈ Vh of a linear elliptic problem (−∆u = 0) isestimated by

‖u− uh‖Ω ≤ Chk+1|u|k+1,Ω,

I C is a constantI k is the polynomial degree of the basis functions from Vh.I All the methods studied here fall into this category

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 12 / 27

Motivation

Theoreticalresults

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Summary

Coercivity for linear elliptic problem

Theorem (Coercivity)Let each K ∈ Th be an image of a fixed reference element K under anaffine mapping.The BR2, CDG, CDG2 method are coercive if one of the followingconditions is fulfilled:

a) η is chosen sufficiently large and χ ≥ 0.

b) η ≥ 0 and χ > χ0, whereI χ0 = NTh for BR2,I χ0 = Nout

Thfor CDG,

I and for CDG2

χ0 =NTh

4

`1 + ν(β)

´with ν(β) = maxe∈Γi|K−e |/|K+

e | and K−e , K+e determined

by β.

NTh is the maximal number of interfaces one element can have.NoutTh

is the maximal number of outflow interfaces one element can havew.r.t. the upwind vector.

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 13 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Area switch

For CDG2 choose area switch to fullfill |nK−e| ≤ |n

K+e|.

1e-05

0.0001

0.001

0.01

0.1 1 10 100

L2-e

rror

computational time

IPBR2

CDG (upwind)CDG2 (upwind)

CDG2 (area)

factor of cpu time spent

1 1.12

1.37

1.42

1e-05

0.0001

0.001

0.01

1 10 100 1000 10000

L2-e

rror

computational time

IPBR2

CDG (upwind)CDG2 (upwind)

CDG2 (area)

factor of cpu time spent

1 1.141.52

2.61

5.82

structured grid unstructured grid

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 14 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

BR2 vs CDG2 in linear case

Lemma (BR2 and CDG2 on special grids)Under the conditions of the Coercivity Theorem BR2 and CDG2 coincideon grids Th where each element is isometric to another if

χBR2 = 2 · χCDG2 .

Remark (BR2 and CDG2 on general grids)Consider the conditions of the Coercivity Theorem.

χBR2 = 2 · χCDG2 +NTh

2(1− ν(βe)) .

Remark (Evaluation of lifting operators)BR2 requires evaluation of the lifting operator re on both elementscontaining the edge e, whereas CDG2 requires evaluation of Le on onlyone element.

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 15 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

BR2 vs CDG2 for non-affine grid mappings

1e-06

1e-05

10 100 1000

L2 -err

or

computation time

CDG2 (A)BR2 (A)

CDG2 (B)BR2 (B)

AScheme CPU time L2-errorCDG2 305 3.91e-07BR2 329 3.92e-07

BScheme CPU time L2-errorCDG2 1790 5.47e-07BR2 2469 5.46e-07

Figure: Comparison on affine (A) and non-affine (B) quadrilateral grids.Problem is on the quadrilateral domain with corners(0, 0), (1, 0), (1, 1), (0, 1) (A) and (0.4, 0), (1, 0), (1, 1.4), (0.1) (B). Thegraph (left) contains level 4 and 5 of the simulation cycle and the table(right) contains the numbers of the level 5 run.

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 16 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

CDG coercivity for χ = 1?

In Peraire and Persson (2008) CDG has χ = 1.

Counterexample in 3D

In

Eymard, Henry, Herbin, Hubert, Klofkorn, and Manzini3D Benchmark on Discretization Schemes for Anisotropic Diffusion Problems onGeneral Grids (2011)

we get for Test Case 1 using grid tet.0.msh that the minimal eigenvalueof the stiffness matrix of the bilinear form B(uh, uh) is −12.167.

With theoretical values minimal eigenvalue is positive.

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 17 / 27

Motivation

Theoretical results

Numerical results

Summary

Motivation

Theoreticalresults

Numericalresults

Summary

Elliptic problem

Herbin, HubertBenchmark on discretization schemes for anisotropic diffusion problems ongeneral grids, FVCA, London (2008)

−∇ · (A∇u) = f on Ω = [0, 1]2,

u = g on ∂Ω,

A =

„1 00 ε

«, ε = 103,

with analytical solution u = g, where

g(x, y) = sin(2πx)e−2π√

1/εy.

I mesh1 of the benchmark,I CG solver is used,I IP uses η of Ainsworth, Rankin (2009).

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 19 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Efficiency

k = 1, 2, 3, χBR2 = 3, χCDG = 2, χCDG2 = 1.5

efficiency condition number

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 20 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Linear advection-diffusion.

∂tu+∇ · (uv)− ε∆u = 0 on Ω = (0, 1)2 × (0, 0.1),

u = gD on ∂Ω,

where ε = 0.1 and

v = (0.1, 0.2)

gD = gD,1 + gD,2

gD,1 = 0.6 cos(2πx) + 0.8 sin(2πx) + 1.2 cos(πx) + 0.4 sin(πx)

gD,2 = 0.9 cos(0.7πx) + 0.2 sin(0.7πx) + 0.3 cos(0.5πx) + 0.1 sin(0.5πx)

u(x, y, t) = e−3πεtgD,1(x, y) + e−1.2πεtgD,2(x, y)

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 21 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Linear advection-diffusion.

k = 2 with 3rd order semi-implicit YZ(3,3) of Yoh and Zhong

1e-05

0.0001

0.001

0.01

0.1 1 10 100

L2-e

rror

computational time

IPBR2

CDG (upwind)CDG2 (upwind)

CDG2 (area)

factor of cpu time spent

1 1.12

1.37

1.42

1e-05

0.0001

0.001

0.01

1 10 100 1000 10000

L2-e

rror

computational time

IPBR2

CDG (upwind)CDG2 (upwind)

CDG2 (area)

factor of cpu time spent

1 1.141.52

2.61

5.82

structured grid unstructured grid

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 22 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Nonlinear convection-diffusion systems

Navier-Stokes equations,Analytical solution: C∞ solution from Lorcher, Gassner, Munz (2008),Grid: Unstructured triangular

1e-5

1e-4

1e-3

1e-2

1e-1

1e+0

1e-1 1e-0 1e+1 1e+2 1e+3 1e+4

L2-e

rror

computation time

k=1

k=2

k=3

BR2CDG

CDG2

1 1.12 1.58

factor of cpu time spent

1e+3

1e+4

1e+5

1e+6

96 384 1536 6144

GM

RE

S it

erat

ions

grid elements

BR2CDG

CDG2

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 23 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

CDG2 for an atmospheric test case

Navier-Stokes equations in θ-form (pot. temperature),Reference solution: Straka et al. (1993),

5 km

4 km

1 km

g = 9.81ms2

0 km

3 km

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 24 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

CDG2 for an atmospheric test case

Navier-Stokes equations in θ-form (pot. temperature),Reference solution: Straka et al. (1993),

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 24 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Convergence study: Straka’s test case

Navier-Stokes equations in θ-form (pot. temperature),Reference solution: Straka et al. (1993),

-5

-4

-3

-2

-1

0

1

0 2000 4000 6000 8000 10000 12000 14000 16000

50m200m -5

-4

-3

-2

-1

0

1

0 2000 4000 6000 8000 10000 12000 14000 16000

50m400m

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 25 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Convergence study: Straka’s test case

Navier-Stokes equations in θ-form (pot. temperature),Reference solution: Straka et al. (1993),

-5

-4

-3

-2

-1

0

1

0 2000 4000 6000 8000 10000 12000 14000 16000

50m200m -5

-4

-3

-2

-1

0

1

0 2000 4000 6000 8000 10000 12000 14000 16000

50m100m

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 25 / 27

Motivation

Theoretical results

Numerical results

Summary

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problems

I new parameter free method (CDG2) presentedI stability proven for linear heat equationI improved stability of CDG in the linear caseI numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problemsI new parameter free method (CDG2) presented

I stability proven for linear heat equationI improved stability of CDG in the linear caseI numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problemsI new parameter free method (CDG2) presentedI stability proven for linear heat equation

I improved stability of CDG in the linear caseI numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problemsI new parameter free method (CDG2) presentedI stability proven for linear heat equationI improved stability of CDG in the linear case

I numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problemsI new parameter free method (CDG2) presentedI stability proven for linear heat equationI improved stability of CDG in the linear caseI numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problemsI new parameter free method (CDG2) presentedI stability proven for linear heat equationI improved stability of CDG in the linear caseI numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27

Motivation

Theoreticalresults

Numericalresults

Summary

Summary

I overview on DG method for convection-diffusion problemsI new parameter free method (CDG2) presentedI stability proven for linear heat equationI improved stability of CDG in the linear caseI numerical results show the performance of the method

B., Dedner, Klofkorn.Compact and stable Discontinuous Galerkin methods for convection-diffusionproblems, Preprint 2010, Math. Institute, Uni Freiburg

Thank you for your attention!

Brdar, Dedner, Klofkorn Compact and Stable DG methods MetStrom 27 / 27