bridging the hybrid high-order and hybridizable ...di-pietro/presentations/... · k;d: k 1 d k 1 1...
TRANSCRIPT
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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
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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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