Bridging the Hybrid High-Order and HybridizableDiscontinuous Galerkin Methods

B. Cockburn D. A. Di Pietro A. Ern

Institut Montpellierain Alexander Grothendieck

MAFELAP 2016

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Minimal bibliography: High-order polyhedral methods

Discontinuous Galerkin

Unified analysis [Arnold, Brezzi, Cockburn and Marini, 2002]General meshes [DP and Ern, 2012, Cangiani, Georgoulis et al. 2014]Adaptive coarsening [Bassi et al., 2012, Antonietti et al., 2013]

Hybridizable Discontinuous Galerkin

LDG framework [Castillo, Cockburn, Perugia, Schotzau, 2009]HDG for pure diffusion [Cockburn, Gopalakrishnan, Lazarov, 2009]

Weak Galerkin [Wang and Ye, 2013]

Virtual elements

Pure diffusion [Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini,Russo, 2013]Nonconforming VEM [Lipnikov and Manzini, 2014]

Hybrid High-Order (HHO)

Originally introduced for linear elasticity [DP and Ern, 2015]Pure diffusion [DP et al., 2014]Link with HDG [Cockburn, DP, Ern, M2AN, 2016]

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Outline

1 The Hybrid High-Order (HHO) method

2 Numerical trace formulation and link with HDG

3 Variations

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Outline

1 The Hybrid High-Order (HHO) method

2 Numerical trace formulation and link with HDG

3 Variations

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Model problem

Let κ : Ω Ñ Rdˆd be a SPD tensor-valued field s.t.

0 ă κ ď λpκq ď κ

We assume κ piecewise constant on a polyhedral partition PΩ of Ω

We consider the Darcy problem

´divpκ∇uq “ f in Ω

u “ 0 on BΩ

Weak formulation: Find u P U :“ H10 pΩq s.t.

apu, vq :“ pκ∇u,∇vq “ pf, vq @v P U

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Mesh I

Definition (Mesh regularity and compliance)

We consider a sequence pThqhPH of polyhedral meshes s.t., for all h P H,Th admits a simplicial submesh Th and pThqhPH is

shape-regular in the usual sense of Ciarlet;

contact-regular, i.e., every simplex S Ă T is s.t. hS « hT .

Additionally, we assume, for all h P H, Th compliant with PΩ, so that

κT :“ κ|T P P0pT qdˆd @T P Th.

Main consequences of mesh regularity:

Lp-trace and inverse inequalities

Optimal W s,p-approximation for the L2-orthogonal projector

See [DP and Ern, 2012] (p “ 2) and [DP and Droniou, 2016a] (p P r1,`8s)

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Mesh II

Figure: Examples of meshes in 2d and 3d: [Herbin and Hubert, 2008] and[DP and Lemaire, 2015] (above) and [DP and Specogna, 2016] (below)

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Local DOF space

•••

• •

•

k = 0

•

••

••

•••• ••

••

k = 1

•••

• • •• • •

• ••

•••

•••

•••

k = 2

••••• •

Figure: UkT for k P t0, 1, 2u

For all k ě 0 and all T P Th, we define the local space

UkT :“ Uk

T ˆ UkBT , Uk

T :“ PkpT q, UkBT :“ PkpFT q

For a generic element of UkT , we use the notation

vT “ pvT , vBT q

Shaded DOFs can be eliminated by static condensation

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Potential reconstruction I

Let T P Th. The local potential reconstruction operator

pk`1T : Uk

T Ñ Pk`1pT q

is s.t. ppk`1T vT ´ vT , 1qT “ 0 and, for all w P Pk`1pT q,

pκT∇pk`1T vT ,∇wqT :“ ´pvT ,divpκT∇wqqT ` pvBT ,κT∇w¨nT qBT

To compute pk`1T , we invert a small SPD matrix of size

Nk,d :“

ˆ

k ` 1` d

k ` 1

˙

´ 1

Trivially parallel task, potentially suited to GPUs!

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Potential reconstruction II

Lemma (Approximation properties of pk`1T IkT )

Define the reduction map IkT : H1pT q Ñ UkT such that

IkT v “ pπkT v, π

kBT vq.

We have, for all v P H1pT q and all w P Pk`1pT q,

pκ∇ppk`1T IkT v ´ vq,∇wqT “ 0.

Consequently, for all v P Hk`2pT q, it holds

v ´ pk`1T IkT vT ` hT ∇pv ´ pk`1

T IkT vqT ď Cκhk`2T vk`2,T ,

i.e., pk`1T IkT has optimal approximation properties in Pk`1pT q.

For the dependence of Cκ on κT see [DP et al., 2016]

W s,p-approximation properties proved in [DP and Droniou, 2016b]

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Stabilization I

The following local discrete bilinear form is in general not stable

aT puT , vT q “ pκT∇pk`1T uT ,∇pk`1

T vT qT

As a remedy, we add a local stabilization term:

aT puT , vT q :“ pκT∇pk`1T uT ,∇pk`1

T vT qT ` sT puT , vT q

We aim at expressing coercivity w.r.t. to the local (semi-)norm

vT 21,T :“ ∇vT 2T ` h´1

T vBT ´ vT 2BT

We also want to preserve the approximation properties of pk`1T

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Stabilization II

A HDG-inspired choice for the stabilization would be

shdgT puT , vT q “ pτBT puT´uBT q, vT´vBT qBT , τBT |F :“ κTnTF ¨nTF

hF

This choice is, however, suboptimal since, for all v P Hk`2pT q,

∇ppk`1T IkT v ´ vqT À hk`1vHk`2pT q,

while we only have

shdgT pIkT v, I

kT vq

12 À hkvHk`1pT q

We need to penalize higher-order differences!

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Stabilization III

Define the high-order correction of cell DOFs P k`1T : Uk

T Ñ Pk`1pT q

P k`1T vT :“ vT `

`

pk`1T vT ´ π

kT p

k`1T vT

˘

We consider the stabilization bilinear form sT s.t.

sT puT , vT q :“ pτBTπkBT pP

k`1T uT ´ uBT q, π

kBT pP

k`1T vT ´ vBT qqBT

With this choice we have stability: For all vT P UkT

vT 21,T À aT pvT , vT q À vT

21,T

Additionally, for all v P Hk`2pT q,

sT pIkT v, I

kT vq

12 À hk`1vHk`2pT q

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Discrete problem

We define the global space with single-valued interface DOFs

Ukh :“ Uk

Thˆ Uk

Fh, Uk

Th:“ PkpThq, Uk

Fh:“ PkpBThq,

We also need the subspace with strongly enforced BCs

Ukh,0 :“ Uk

Thˆ Uk

Fh,0, Uk

Fh,0:“

!

vh P Ukh | vF ” 0 @F P Fb

h

)

The discrete problem reads: Find uh P Ukh,0 such that

ahpuh, vhq :“ÿ

TPTh

aT puT , vT q “ pf, vThq @vh P U

kh,0

where vTh |T :“ vT for all T P Th

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Convergence I

Theorem (Energy-norm error estimate)

Assume u P Hk`2pΩq and define the global reduction map

Ikhu :“`

pπkTuqTPTh

, pπkFuqFPFh

˘

P Ukh,0.

Then, we have the following energy error estimate:

uh ´ Ikhu1,h À hk`1uHk`2pΩq,

where vh21,h :“

ř

TPThvT

21,T .

For the original LDG-H method, we have hk`12

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Convergence II

Theorem (L2-norm error estimate)

Further assuming elliptic regularity and f P H1pΩq if k “ 0,

uTh´ πk

hu À hk`2Bpu, kq,

with Bpu, 0q :“ fH1pΩq, Bpu, kq :“ uHk`2pΩq if k ě 1 and

uTh |T “ uT @T P Th.

For the original LDG-H method, we have hk`1

Corollary (L2-norm estimate for pk`1T uT )

Letting quh P Pk`1pThq be s.t. quh|T “ pk`1T uT for all T P Th, it holds

quh ´ u À hk`2Bpu, kq.

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Numerical example I

Figure: Triangular, Kershaw and hexagonal mesh families

We consider Le Potier’s exact solution on Ω “ p0, 1q2

upxq “ sinpπx1q sinpπx2q

The diffusion field has rotating principal axes

κpxq “

ˆ

px2 ´ x2q2` εpx1 ´ x1q

2´p1´ εqpx1 ´ x1qpx2 ´ x2q

´p1´ εqpx1 ´ x1qpx2 ´ x2q px1 ´ x1q2` εpx2 ´ x2q

2

˙

,

with anisotropy ratio ε “ 1 ¨ 10´2 and center px1, x2q “ ´p0.1, 0.1q

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Numerical example II

k “ 0 k “ 1 k “ 2 k “ 3

10´3 10´2

10´8

10´6

10´4

10´2

1000.73

2

3.02

3.96

10´2.4 10´2.2 10´2 10´1.8 10´1.610´7

10´6

10´5

10´4

10´3

10´2

10´1

100

2.02

3.3

4.35

5.58

10´2.5 10´2 10´1.5

10´8

10´6

10´4

10´2

100

1.39

2.32

3.34

4.43

10´3 10´2

10´10

10´8

10´6

10´4

10´2

100

1.8

2.76

3.83

4.85

10´2.4 10´2.2 10´2 10´1.8 10´1.6

10´9

10´7

10´5

10´3

10´1

2.17

3.53

4.8

5.61

10´2.5 10´2 10´1.510´10

10´8

10´6

10´4

10´2

1.82

2.89

3.97

4.99

Figure: Energy- (above) and L2-errors (below) for the three mesh families

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Teaser: Industrial application I

105 106 107

0.34

0.35

0.36

0.37

k = 0 unk = 0 adk = 1 unk = 1 adk = 2 unk = 2 ad

Figure: Adaptive algorithm for 3d electrostatics [DP and Specogna, 2016]

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Teaser: Industrial application II

0.5 1 1.5 2

·107

0.5

1

1.5

·104

pre-processingassemblingsolution

post-processing

(a) k “ 0

1 2 3 4 5 6 7

·106

0.5

1

1.5

·104

pre-processingassemblingsolution

post-processing

(b) k “ 1

1 2 3 4 5 6

·106

0.5

1

1.5

·104

pre-processingassemblingsolution

post-processing

(c) k “ 2

Figure: Computing wall time vs Ndof

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Outline

1 The Hybrid High-Order (HHO) method

2 Numerical trace formulation and link with HDG

3 Variations

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Numerical trace formulation I

Separating element- and face-based test functions, we have

@T P Th, aT puT , pvT , 0qq “ pf, vT qT @vT P UkT

ahpuh, p0, vFhqq “ 0 @vFh

P UkFh,0

The first set of equations define local equilibria

The second set is a global transmission condition

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Numerical trace formulation II

For T P Th, define the boundary residual rkBT : PkpFT qÑPkpFT q s.t.

@λ P PkpFT q, rkBT pλq :“ πkBT

`

λ´ pk`1T p0, λq ` πk

T pk`1T p0, λq

˘

The penalized difference rewrites: For all vT P UkT ,

πkBT pP

k`1T vT ´ vBT q “ rkBT pvT ´ vBT q

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Numerical trace formulation III

As a result, for all T P Th we have

sT puT , p0, vBT qq “ ´pτBT rkBT puT ´ uBT q, r

kBT pvBT qqBT

or, introducing the adjoint rk,˚BT of rkBT ,

sT puT , p0, vFhqq “ prk,˚BT pτBT r

kBT puT ´ uBT qq, vBT qBT

Plugging this expression into that of aT , we finally arrive at

aT puT , p0, vBT qq“ ´ pκT∇pk`1T uT ¨nT

looooooooomooooooooon

consistency

´rk,˚BT pτBT rkBT puT ´ uBT qq

loooooooooooooooomoooooooooooooooon

penalty

, vBT qBT

The term in red is the conservative normal flux trace

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Numerical trace formulation IV

Lemma (Numerical trace reformulation of HHO)

The discrete problem: Find uh P Ukh,0 s.t.

ahpuh, vhq “ pf, vThq @vh P U

kh,0

can be equivalently reformulated as follows: Find uh P Ukh,0 s.t.

@T P Th, aT puT , pvT , 0qq “ pf, vT qT @vT P UkT ,

ÿ

TPTh

ppqn,T puT q, vFhqBT “ 0 @vFh

P UkFh,0

,

with conservative normal flux trace s.t., for all T P Th,

pqn,T puT q :“ κT∇pk`1T uT ¨nT ´ r

k,˚BT pτBT r

kBT puT ´ uBT qq.

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Link with HDG methods I

Let us interpret HHO as a HDG method

HDG methods hinge on three set of spaces:

tV pT quTPTh , for the approximation of the flux

tW pT quTPTh , for the approximation of the potential

tMpF quFPFh , for the approximation of the potential traces

The definition is completed with a recipe for the normal flux trace

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Link with HDG methods II

We define the following global spaces:

Vh :“ą

TPTh

VpT q, Wh ˆMh :“

˜

ą

TPTh

W pT q

¸

ˆ

˜

ą

FPFh

MpF q

¸

We also need the subspace with strongly enforced BCs,

Mh,0 :“ t pw PMh : pw “ 0 on BΩu

The HDG method reads: Find pqh, uh, puhq P Vh ˆWh ˆMh,0 s.t.

pκ´1T qh,vqT ´ puh,div vqT ` ppuh,v¨nT qBT “ 0 @v P VpT q

´pqh,∇wqT ` ppqn,T , wqBT “ pf, wqT @w PW pT qÿ

TPTh

ppqn,T , pwqBT “ 0 @ pw PMh,0

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Link with HDG methods III

HHO corresponds to new choices for the spaces

V pT q “ κT∇Pk`1pT q, W pT q “ PkpT q, MpF q “ PkpF q

Notice that the flux is now reconstructed in a smaller space

dimpκT∇Pk`1pT qq ď dimpPkpT qdq

Another crucial novelty is the high-order normal flux trace

pqn,T “ κT∇pk`1T uT ¨nT ´ r

k,˚BT pτBT r

kBT puT ´ uBT qq

HHO can be easily adapted into existing HDG codes!

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Outline

1 The Hybrid High-Order (HHO) method

2 Numerical trace formulation and link with HDG

3 Variations

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Variations: The HHO(l) family I

Variations are possible changing the degree of element-based DOFs

Let l P tk ´ 1, k, k ` 1u and consider the local space

Uk,lT :“ PlpT q ˆ PkpFT q

The first potential reconstruction pk`1T remains formally unchanged

The second potential reconstruction used for stabilization becomes

P k`1T vT :“ vT `

`

pk`1T vT ´ π

lT p

k`1T vT

˘

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Variations: The HHO(l) family II

Convergence rates as for the original HHO method

For l “ k ´ 1 we recover a High-Order Mimetic scheme1

For l “ k we find the original HHO method

For l “ k ` 1 we have a new Lehrenfeld–Schoberl-type HDG method

k “ 0 and l “ k´ 1 on simplices yields the Crouzeix–Raviart element

The globally-coupled unknowns coincide in all the cases!

1Up to equivalent stabilization31 / 36

A nonconforming finite element interpretation I

We interpret the HHO(l) methods as nonconforming FE methods

The construction extends the ideas of [Ayuso de Dios et al., 2016]

For a fixed element T P Th, we define the local space

V k,lT :“

ϕ P H1pT q | ∇ϕ|BT ¨nT P PkpFT q and 4ϕ P Pl

pT q(

We next study the relation between V k,lT and Uk,l

T

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A nonconforming finite element interpretation II

Let ΦT : Uk,lT Ñ V k,l

T be s.t. ΦT pvT q solves the Neumann problem

4ΦT pvT q “ vT ´ |T |´1d rpvT , 1qT ´ pvBT , 1qBT s

and∇ΦT pvT q|BT ¨nT “ vBT , pΦT pvT q, 1qT “ 0

Both ΦT and Ik,lT : V k,lT Ñ Uk,l

T can be proved to be injective

Therefore, Ik,lT : V k,lT Ñ Uk,l

T is an isomorphism and we can identify

V k,lT „ Uk,l

T ,

which means that UkT contains the DOFs for V k,l

T as defined by IkT

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References I

Antonietti, P. F., Giani, S., and Houston, P. (2013).

hp-version composite discontinuous Galerkin methods for elliptic problems on complicated domains.SIAM J. Sci. Comput, 35(3):A1417–A1439.

Arnold, D. N., Brezzi, F., Cockburn, B., and Marini, L. D. (2002).

Unified analysis of discontinuous Galerkin methods for elliptic problems.SIAM J. Numer. Anal., 39(5):1749–1779.

Ayuso de Dios, B., Lipnikov, K., and Manzini, G. (2016).

The nonconforming virtual element method.ESAIM: Math. Model Numer. Anal. (M2AN), 50(3):879–904.

Bassi, F., Botti, L., Colombo, A., Di Pietro, D. A., and Tesini, P. (2012).

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations.J. Comput. Phys., 231(1):45–65.

Beirao da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L. D., and Russo, A. (2013).

Basic principles of virtual element methods.M3AS Math. Models Methods Appl. Sci., 199(23):199–214.

Cangiani, A., Georgoulis, E. H., and Houston, P. (2014).

hp-version discontinuous Galerkin methods on polygonal and polyhedral meshes.Math. Models Methods Appl. Sci., 24(10):2009–2041.

Castillo, P., Cockburn, B., Perugia, I., and Schotzau, D. (2000).

An a priori error analysis of the local discontinuous Galerkin method for elliptic problems.SIAM J. Numer. Anal., 38:1676–1706.

Cockburn, B., Di Pietro, D. A., and Ern, A. (2016).

Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin methods.ESAIM: Math. Model Numer. Anal. (M2AN), 50(3):635–650.

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References II

Cockburn, B., Gopalakrishnan, J., and Lazarov, R. (2009).

Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems.SIAM J. Numer. Anal., 47(2):1319–1365.

Di Pietro, D. A. and Droniou, J. (2016a).

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes.Math. Comp.Accepted for publication. Preprint arXiv 1508.01918.

Di Pietro, D. A. and Droniou, J. (2016b).

Ws,p-approximation properties of elliptic projectors on polynomial spaces with application to the error analysis of a HybridHigh-Order discretisation of Leray–Lions elliptic problems.Submitted.

Di Pietro, D. A. and Ern, A. (2012).

Mathematical aspects of discontinuous Galerkin methods, volume 69 of Mathematiques & Applications.Springer-Verlag, Berlin.

Di Pietro, D. A. and Ern, A. (2015).

A hybrid high-order locking-free method for linear elasticity on general meshes.Comput. Meth. Appl. Mech. Engrg., 283:1–21.

Di Pietro, D. A., Ern, A., and Lemaire, S. (2014).

An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on local reconstruction operators.Comput. Meth. Appl. Math., 14(4):461–472.

Di Pietro, D. A., Ern, A., and Lemaire, S. (2016).

Building bridges: Connections and challenges in modern approaches to numerical partial differential equations, chapter A reviewof Hybrid High-Order methods: formulations, computational aspects, comparison with other methods.Springer.To appear. Preprint hal-01163569.

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References III

Di Pietro, D. A. and Lemaire, S. (2015).

An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressible linear elasticity and Stokesflow.Math. Comp., 84(291):1–31.

Di Pietro, D. A. and Specogna, R. (2016).

An a posteriori-driven adaptive Mixed High-Order method with application to electrostatics.Submitted. Preprint hal-01310313.

Herbin, R. and Hubert, F. (2008).

Benchmark on discretization schemes for anisotropic diffusion problems on general grids.In Eymard, R. and Herard, J.-M., editors, Finite Volumes for Complex Applications V, pages 659–692. John Wiley & Sons.

Lipnikov, K. and Manzini, G. (2014).

A high-order mimetic method on unstructured polyhedral meshes for the diffusion equation.J. Comput. Phys., 272:360–385.

Wang, J. and Ye, X. (2013).

A weak Galerkin element method for second-order elliptic problems.J. Comput. Appl. Math., 241:103–115.

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