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Hybrid High-Order (HHO) methods on generalmeshes
Daniele A. Di Pietro
University of MontpellierInstitut Montpellierain Alexander Grothendieck
Paris, 19th June, 2015
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µBibliography: Lowest-order polyhedral methods
Mimetic Finite Differences
Extension to polyhedral meshes [Kuznetsov et al., 2004]Convergence analysis [Brezzi et al., 2005]
Mixed/Hybrid Finite Volumes
Pure diffusion (mixed) [Droniou and Eymard, 2006]Pure diffusion (primal) [Eymard et al., 2010]Link with MFD [Droniou et al., 2010]
More recently
Compatible Discrete Operators [Bonelle and Ern, 2014]Generalized Crouzeix–Raviart [DP and Lemaire, 2015]
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µBibliography: High-order polyhedral methods
Discontinuous Galerkin
General meshes [DP and Ern, 2012]Adaptive coarsening [Bassi et al., 2012, Antonietti et al., 2013]
Hybridizable Discontinuous Galerkin
Pure diffusion [Cockburn et al., 2009]
Virtual elements
Pure diffusion [Beirao da Veiga et al., 2013a]Nonconforming VEM [Ayuso de Dios et al., 2014]Linear elasticity [Beirao da Veiga et al., 2013b]
Hybrid High-Order
Pure diffusion [DP and Ern, 2014b]Linear elasticity [DP and Ern, 2015]Bridge between HHO and HDG [Cockburn, DP and Ern, 2015]
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Features of HHO
Capability of handling general polyhedral meshes
Construction valid for arbitrary space dimensions
Arbitrary approximation order (including k “ 0)
Reproduction of desirable continuum properties
Integration by parts formulasKernels of operatorsSymmetries
Reduced computational cost after hybridization
Nhhodof «
1
2k2 cardpFhq Ndg
dof «1
6k3 cardpThq
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Outline
1 Poisson
2 Variable diffusion and local conservation
3 Linear elasticity
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Outline
1 Poisson
2 Variable diffusion and local conservation
3 Linear elasticity
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Mesh regularity I
Definition (Mesh regularity)
We consider a sequence pThqhPH of polyhedral meshes s.t., for allh P H, Th admits a simplicial submesh Th and pThqhPH is
shape-regular in the sense of Ciarlet;
contact-regular: every simplex S Ă T is s.t. hS « hT .
Main consequences:
Trace and inverse inequalities
Optimal approximation for broken polynomial spaces
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Mesh regularity II
Figure: Admissible meshes in 2d and 3d: [Herbin and Hubert, 2008, FVCA5] and[Di Pietro and Lemaire, 2015] (above) and [Eymard et al., 2011, FVCA6] (below)
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Model problem
Let Ω denote a bounded, connected polyhedral domain
For f P L2pΩq, we consider the Poisson problem
´4u “ f in Ω
u “ 0 on BΩ
In weak form: Find u P H10 pΩq s.t.
apu, vq :“ p∇u,∇vq “ pf, vq @v P H10 pΩq
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Key ideas
DOFs: polynomials of degree k ě 0 at elements and faces
Differential operators reconstructions taylored to the problem:
a|T pu, vq « p∇pk`1T uT ,∇pk`1
T vT q ` stab.
with
high-order reconstruction pk`1T from local Neumann solves
stabilization via face-based penalty
Construction yielding superconvergence on general meshes
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DOFs
••
•• •
•
k = 0
•
••
••
•••• ••
••
k = 1
•••
• • •
• • •
• ••
•••
•••
•••
k = 2
••••• •
Figure: UkT for k P t0, 1, 2u
For k ě 0 and all T P Th, we define the local space of DOFs
UkT :“ PkdpT q ˆ#
ą
FPFT
Pkd´1pF q+
The global space has single-valued interface DOFs
Ukh :“#
ą
TPThPkdpT q
+
ˆ#
ą
FPFh
Pkd´1pF q+
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Local potential reconstruction I
Let T P Th. The local potential reconstruction operator
pk`1T : UkT Ñ Pk`1
d pT q
is s.t. @vT P UkT , ppk`1T vT , 1qT “ pvT , 1qT and @w P Pk`1
d pT q,
p∇pk`1T vT ,∇wqT :“ ´pvT ,4wqT `
ÿ
FPFT
pvF ,∇w¨nTF qF
To compute pk`1T , we solve a small SPD linear system of size
Nk,d :“ˆ
k ` 1` dk ` 1
˙
Perfectly suited to GPU computing!
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Local potential reconstruction II
Lemma (Approximation properties for pk`1T IkT )
Define the local reduction map IkT : H1pT q Ñ UkT s.t.
IkT : v ÞÑ `
πkT v, pπkF vqFPFT
˘
.
Then, for all T P Th and all v P Hk`2pT q,v ´ pk`1
T IkT vT ` hT ∇pv ´ pk`1T IkT vqT À hk`2
T vk`2,T .
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Local potential reconstruction III
Since 4w P Pk´1d pT q and ∇w|F ¨nTF P Pkd´1pF q,
p∇pk`1T IkT v,∇wqT “ ´pπk
T v,4wqT `ÿ
FPFT
pπkF v,∇w¨nTF qF
“ ´pv,4wqT `ÿ
FPFT
pv,∇w¨nTF qF “ p∇v,∇wqT
This shows that pk`1T IkT is the elliptic projector on Pk`1
d pT q:
p∇pk`1T IkT v ´∇v,∇wqT “ 0 @w P Pk`1
d pT q
The approximation properties follow
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Stabilization I
The following naive choice is not stable
a|T pu, vq « p∇pk`1T uT ,∇pk`1
T vT qTTo remedy, we add a local stabilization term
p∇pk`1T uT ,∇pk`1
T vT qT ` sT puT , vT q
Coercivity and boundedness are expressed w.r.t. to
vT 21,T :“ ∇vT 2T `ÿ
FPFT
1
hFvF ´ vT 2F
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Stabilization II
Define, for T P Th, the stabilization bilinear form sT as
sT puT , vT q :“ÿ
FPFT
h´1F pπk
F pppk`1T uT ´ uF q, πk
F pppk`1T vT ´ vF qqF ,
with ppk`1T high-order correction of cell DOFs based on pk`1
T
ppk`1T vT :“ vT ` ppk`1
T vT ´ πkT pk`1T vT q
With this choice, aT satisfies, for all vT P UkT ,
vh21,T À aT pvT , vT q À vT 21,T
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Stabilization III
Lemma (High-order consistency of sT )
sT preserves the approximation properties of ∇pk`1T .
For all u P Hk`2pT q, letting puT :“ IkTu “`
πkTu, pπkFuqFPFT
˘
,
πkF pppk`1
T puT ´ puF qF “ πkF
`
πkTu` pk`1
T puT ´ πkT p
k`1T puT ´ πk
Fu˘F
ď πkF
`
pk`1T puT ´ u
˘F ` πkT
`
u´ pk`1T puT
˘FÀ h
´12
T pk`1T puT ´ uT
Recalling the approximation properties of pk`1T , this yields
!
∇ppk`1T puT ´ uq2T ` sT ppuT , puT q
)12 À hk`1T uk`2,T
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Discrete problem
We enforce boundary conditions strongly considering the space
Ukh,0 :“!
vh P Ukh | vF ” 0 @F P Fbh
)
The discrete problem reads: Find uh P Ukh,0 s.t.
ahpuh, vhq :“ÿ
TPThaT puT , vT q “
ÿ
TPThpf, vT qT @vh P Ukh,0
Well-posedness follows from the coercivity of ah
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Convergence I
Theorem (Energy-norm error estimate)
Assume u P Hk`2pThq and let
puh :“ `pπkTuqTPTh , pπkFuqFPFh
˘ P Ukh,0.Then, we have the following energy error estimate:
max puh ´ puh1,h, uh ´ puha,hq À hk`1uHk`2pΩq,
withvh21,h :“
ÿ
TPThvT 21,T .
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Convergence II
Theorem (L2-norm error estimate)
Further assuming elliptic regularity and f P H1pΩq if k “ 0,
max pquh ´ u, puh ´ uhq À hk`2Nk,
with N0 :“ fH1pΩq, Nk :“ uHk`2pThq if k ě 1, and, @T P Th,
quh|T :“ pk`1T uT , puh|T :“ pk`1
T IkTu, uh|T :“ uT .
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Convergence for a smooth 2d solution I
10´3 10´2
10´12
10´9
10´6
10´3
100
1.99
2.94
3.994.02
4.81
k “ 0k “ 1k “ 2k “ 3k “ 4
10´3 10´210´14
10´11
10´8
10´5
10´2
1.99
3.02
3.99
4.99
5.53
k “ 0k “ 1k “ 2k “ 3k “ 4
10´2.5 10´2 10´1.510´11
10´9
10´7
10´5
10´3
10´1
1.31
2.63
3.11
4.09
5.19
k “ 0k “ 1k “ 2k “ 3k “ 4
10´2.5 10´2 10´1.5
10´12
10´9
10´6
10´3 1.9
2.98
4.02
4.99
5.94
k “ 0k “ 1k “ 2k “ 3k “ 4
Figure: Energy (left) and L2-norm (right) of the error vs. h for uniformly refinedtriangular (top) and hexagonal (bottom) mesh families, upxq “ sinpπx1q sinpπx2q
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Convergence for a smooth 2d solution II
103 10410´1
100
101
k “ 0k “ 1k “ 2k “ 3k “ 4k “ 5
103 10410´1
100k “ 0k “ 1k “ 2k “ 3k “ 4k “ 5
Figure: Assembly/solution time for triangular (left) and hexagonal (right) meshfamilies, sequential implementation
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Mesh adaptivity: Fichera’s 3d test case I
Let Ω :“ p´1, 1q3zr0, 1s3We consider the following exact solution:
upxq “ px21 ` x2
2 ` x23q
14
corresponding to the forcing term
fpxq “ ´3
4px2
1 ` x22 ` x2
3q´34
We consider an adaptive procedure driven by guaranteedresidual-based a posteriori estimators [DP & Specogna, 2015]
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Mesh adaptivity: Fichera’s 3d test case II
Figure: HHO solution on a sequence of adaptively refined meshes
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Mesh adaptivity: Fichera’s 3d test case III
103 104 105 10610−3
10−2
10−1
k = 0 un.k = 0 ad.k = 1 un.k = 1 ad.k = 2 un.k = 2 ad.
Figure: Energy error vs. dimpUkhq
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Mesh adaptivity: Fichera’s 3d test case IV
Figure: Estimated (left) and true (right) error distribution
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Outline
1 Poisson
2 Variable diffusion and local conservation
3 Linear elasticity
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Variable diffusion I
Let ν : Ω Ñ Rdˆd be a SPD tensor-valued field s.t.
@T P Th, 0 ă νT ď λpνq ď νT
Consider the variable diffusion problem
´∇¨pν∇uq “ f in Ω
u “ 0 on BΩ
We confer built-in homogeneization features to pk`1T
pν∇pk`1T vT ,∇wqT “ pν∇vT ,∇wqT`
ÿ
FPFT
pvF´vT ,ν∇w¨nTF qF
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Variable diffusion II
Lemma (Approximation properties of pk`1T IkT )
There is C independent of hT and ν s.t., for all v P Hk`2pT q, itholds with α “ 1
2 if ν is piecewise constant and α “ 1 otherwise:
v ´ pk`1T IkT vT ` hT ∇pv ´ pk`1
T IkT vqT ď CραThk`2T vk`2,T ,
with local heterogeneity/anisotropy ratio
ρT :“ νTνTě 1.
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Variable diffusion III
Theorem (Energy-error estimate)
Assume that u P Hk`2pThq and modify the bilinear form as
aν ,T puT , vT q :“ pν∇pk`1T uT ,∇pk`1
T vT qT ` sν ,T puT , vT q
where, setting νTF :“ nTF ¨ν |T ¨nTF L8pF q,
sν ,T puT , vT q :“ÿ
FPFT
νTF
hFpπk
F pppk`1T uT ´ uF q, πk
F pppk`1T vT ´ vF qqF .
Then, with puh and α as above,
puh ´ uhν ,h À#
ÿ
TPThνTρ
1`2αT h
2pk`1qT u2k`2,T
+12.
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Local conservation and numerical fluxes I
A highly prized property in practice is local conservation
At the discrete level, we wish to mimick the local balance
pν∇u,∇vqT´ÿ
FPFT
pν |T∇u¨nTF , vqF “ pf, vqT @v P H1pT q
where, for all interface F P FT1 X FT2 ,
ν |T1∇u¨nT1F ` ν |T2∇u¨nT2F “ 0
This requires to identify numerical fluxes
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Local conservation and numerical fluxes II
Define the face residual operator RkT : PkdpFT q Ñ PkdpFT q s.t.
RkTϕ|F “ πk
F
`
ϕ|F ´ pk`1T p0, ϕq ` πk
T pk`1T p0, ϕq˘
Denote by R˚,kT its adjoint and let τBT and uBT be s.t.
τBT |F “ νTFhF
and uBT |F “ uF @F P FT
The penalty term can be rewritten in conservative form as
sT puT , vT q “ÿ
FPFT
pR˚,kT pτBTRkT puBT ´ uT qq, vF ´ vT qqF
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Local conservation and numerical fluxes III
Lemma (Flux formulation)
The HHO solution uh P Ukh,0 satisfies, for all T P Th and all vT P Pk
dpT q
pν∇pk`1T uT ,∇vT qT ´
ÿ
FPFT
pΦTF puT q, vT qF “ pf, vT qT ,
with numerical flux
ΦTF puT q :“ ν |T∇pk`1T uT ¨nTF ´R˚,kT pτBTRk
T puBT ´ uT qq,s.t., for all interface F P FT1
X FT2,
ΦT1F puT1q ` ΦT2F puT2
q “ 0.
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Link with HDG
The flux formulation shows that HHO = HDG on steroids
Smaller local problems to eliminate flux unknowns:
∇Pk`1d pT q vs. PkdpT qd
Superconvergence of the potential in the L2-norm
hk`2 vs. hk`1
HHO can be adapted into existing HDG codes!
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Outline
1 Poisson
2 Variable diffusion and local conservation
3 Linear elasticity
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Continuous setting
Consider the linear elasticity problem: Find u : Ω Ñ Rd s.t.
´∇¨σpuq “ f in Ω,
u “ 0 on BΩwith real Lame parameters λ ě 0 and µ ą 0 and
σpuq “ 2µ∇su` λp∇¨uqIdWhen λÑ `8 we need to approximate nontrivialincompressible displacement fields
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Rigid body motions
Applied to vector fields, the operator ∇s yields strains
Its kernel RMpΩq contains rigid-body motions
RMpΩq :“
v P H1pΩq3 | Dα,ω P R3, vpxq “ α` ω b x(
We note for further use that
P0dpΩqd Ă RMpΩq Ă P1
dpΩqd
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DOFs and reduction map I
•• ••
•• ••
••••
••••
••••
••••
k = 1
•••• ••
•• •• ••
•• •• ••
••••••
••••••
••••••
••••••
k = 2
••••••••••••••
Figure: UkT for k P t1, 2u
For k ě 1 and all T P Th, we define the local space of DOFs
UkT :“ Pk
dpT qd ˆ#
ą
FPFT
Pkd´1pF qd
+
The global space has single-valued interface DOFs
Ukh :“
#
ą
TPTh
PkdpT qd
+
ˆ#
ą
FPFh
Pkd´1pF qd
+
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Displacement reconstruction I
Let T P Th. The local displacement reconstruction operator
pk`1T : Uk
T Ñ Pk`1d pT qd
is s.t., for all vT “`
vT , pvF qFPFT
˘ P UkT and w P Pk`1
d pT qd,
p∇spk`1T vT ,∇swqT “ ´pvT ,∇¨∇swqT `
ÿ
FPFT
pvF ,∇swnTF qF
Rigid-body motions are prescribed from vT settingż
T
pk`1T vT “
ż
T
vT ,
ż
T
∇sspk`1T vT “
ÿ
FPFT
ż
F
1
2pnTFbvF´vFbnTF q
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Displacement reconstruction II
Lemma (Approximation properties for pk`1T IkT )
There exists C ą 0 independent of hT s.t., for all v P Hk`2pT qd,
v ´ pk`1T IkTvT ` hT ∇pv ´ pk`1
T IkTvqT ď Chk`2T vHk`2pT qd .
Proceeding as for Poisson, one can show that
p∇spk`1T IkTv ´∇sv,∇swqT “ 0 @w P Pk`1
d pT qd,
and the approximation properties follow.
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Stabilization I
Define, for T P Th, the stabilization bilinear form sT as
sT puT ,vT q :“ÿ
FPFT
h´1F pπkF pppk`1
T uT´uF q, πkF pppk`1T vT´vF qqF ,
with displacement reconstruction ppk`1T : Uk
T Ñ Pk`1d pT qd s.t.
@vT P UkT , ppk`1
T vT :“ vT ` ppk`1T vT ´ πkTpk`1
T vT q
Stability can be proved in terms of the discrete strain norm
vT 2ε,T :“ ∇svT 2T `ÿ
FPFT
h´1F vF 2F
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Stabilization II
Lemma (Stability and approximation)
Let T P Th and assume k ě 1. Then,
vT 2ε,T À ∇spk`1T vT 2T ` sT pvT ,vT q À vT 2ε,T .
Moreover, for all v P Hk`2pT qd, we have
!
∇spIkTv ´ vq2T ` sT pIkTv, IkTvq)12 À hk`1
T vHk`2pT qd .
Generalization of a classical result: Crouzeix–Raviart does notmeet Korn!
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Stabilization III
For all F P FT one has, inserting ˘πkF ppk`1T vT ,
vF´vT F À πkF pvF´ppk`1
T vT qF`h´12
F pk`1T vT´πk
Tpk`1T vT T
For any function w P H1pT qd with rigid-body motions wRM,
w ´ πkTwT “ pw ´wRMq ´ πk
T pw ´wRMqT À hT ∇swTwhere πkTwRM “ wRM requires k ě 1 to have
RMpT q Ă PkdpT qd
Clearly, this reasoning breaks down for k “ 0
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Divergence reconstruction
We define the local local discrete divergence operator
DkT : Uk
T Ñ PkdpT qs.t., for all vT “
`
vT , pvF qFPFT
˘ P UkT and all q P PkdpT q,
pDkTvT , qqT :“ ´pvT ,∇qqT `
ÿ
FPFT
pvF ¨nTF , qqF
By construction, we have the following commuting diagram:
H1pT q L2pT q
UkT PkdpT q
∇¨
πkTDkT
IkT
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Discrete problem
We define the local bilinear form aT on UkT ˆUk
T as
aT puT ,vT q :“ 2µp∇spk`1T uT ,∇sp
k`1T vT qT
` λpDkTuT , D
kTvT q ` p2µqsT puT ,vT q
The discrete problem reads: Find uh P Ukh,0 s.t.
ahpuh,vhq :“ÿ
TPThaT puT ,vT q “
ÿ
TPThpf ,vT qT @vh P Ukh,0
with Ukh,0 incorporating boundary conditions
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Convergence I
Theorem (Energy-norm error estimate)
Assume k ě 1 and the additional regularity
u P Hk`2pThqd and ∇¨u P Hk`1pThq.Then, there exists C ą 0 independent of h, µ, and λ s.t.
p2µq12uh ´ puha,h ď Chk`1Bpu, kq,with
Bpu, kq :“ p2µquHk`2pThqd ` λ∇¨uHk`1pThq.
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Convergence II
Locking-free if Bpu, kq is bounded uniformly in λ
For d “ 2 and Ω convex, one has using Cattabriga’s regularity
Bpu, 0q “ uH2pΩqd ` λ∇¨uH1pΩq ď Cµf
More generally, for k ě 1, we need the regularity shift
Bpu, kq ď CµfHkpΩqd
Key point: commuting property for DkT
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Convergence III
Theorem (L2-error estimate for the displacement)
Let eh P PkdpThqd be s.t.
eh|T :“ uT ´ πkTu @T P Th.Then, assuming elliptic regularity for Ω and provided that
u P Hk`2pThqd and ∇¨u P Hk`1pThq,it holds with Cą0 independent of λ and h,
eh ď Chk`2Bpu, kq.
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Numerical examples I
We consider the following exact solution:
upxq “ `
sinpπx1q sinpπx2q`p2λq´1x1, cospπx1q cospπx2q`p2λq´1x2˘
The solution u has vanishing divergence in the limit λÑ `8:
∇¨upxq “ 1
λ
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Numerical examples II
k “ 1 k “ 2 k “ 3 k “ 4
10´3 10´2
10´8
10´7
10´6
10´5
10´4
10´3
10´2
10´1
1.89
2.97
3.93
4.95
10´2.5 10´2 10´1.5
10´9
10´7
10´5
10´3
10´1
2.65
3.02
4.22
5.17
10´3 10´2
10´8
10´7
10´6
10´5
10´4
10´3
10´2
10´1
1.93
2.98
3.94
4.97
10´2.5 10´2 10´1.5
10´9
10´7
10´5
10´3
10´1
2.69
3.04
4.26
5.15
Figure: Energy error with λ “ 1 (above) and λ “ 1000 (below) vs. h for the triangular(left) and hexagonal (right) mesh families
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Numerical examples III
10´1 100 101 102
10´9
10´7
10´5
10´3
10´1
´0.84
´1.24
´1.73
´2.29
k “ 1k “ 2k “ 3k “ 4
10´2 10´1 100 101 102
10´9
10´7
10´5
10´3
10´1
´1.04
´1.09
´1.64
´1.99
k “ 1k “ 2k “ 3k “ 4
10´1 100 101 10210´12
10´10
10´8
10´6
10´4
10´2
´1.3
´1.66
´2.19´2.77
k “ 1k “ 2k “ 3k “ 4
10´2 10´1 100 101 10210´12
10´10
10´8
10´6
10´4
10´2
´1.17
´1.46
´1.97´2.32
k “ 1k “ 2k “ 3k “ 4
Figure: Energy (above) and displacement (below) error vs. τtot (s) for the triangularand hexagonal mesh families
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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.
Ayuso de Dios, B., Lipnikov, K., and Manzini, G. (2014).
The nonconforming virtual element method.Submitted. ArXiV preprint arXiv:1405.3741.
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. (2013a).
Basic principles of virtual element methods.Math. Models Methods Appl. Sci., 23:199–214.
Beirao da Veiga, L., Brezzi, F., and Marini, L. (2013b).
Virtual elements for linear elasticity problems.SIAM J. Numer. Anal., 51(2):794–812.
Bonelle, J. and Ern, A. (2014).
Analysis of compatible discrete operator schemes for elliptic problems on polyhedral meshes.M2AN Math. Model. Numer. Anal., 48:553–581.
Brezzi, F., Lipnikov, K., and Shashkov, M. (2005).
Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes.SIAM J. Numer. Anal., 43(5):1872–1896.
52 / 55
References II
Cockburn, B., Di Pietro, D. A., and Ern, A. (2015).
Bridging the Hybrid High-Order and Hybridizable Discontinuous Galerkin methods.Submitted. Preprint hal-01115318.
Cockburn, B., Gopalakrishnan, J., and Lazarov, R. (2009).
Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second orderelliptic problems.SIAM J. Numer. Anal., 47(2):1319–1365.
Di Pietro, D. A., Droniou, J., and Ern, A. (2014a).
A discontinuous-skeletal method for advection-diffusion-reaction on general meshes.Preprint arXiv:1411.0098.
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. Methods Appl. Mech. Engrg., 283:1–21.DOI: 10.1016/j.cma.2014.09.009.
Di Pietro, D. A., Ern, A., and Lemaire, S. (2014b).
An arbitrary-order and compact-stencil discretization of diffusion on general meshes based on localreconstruction operators.Comput. Methods Appl. Math., 14(4):461–472.DOI: 10.1515/cmam-2014-0018.
53 / 55
References III
Di Pietro, D. A. and Lemaire, S. (2015).
An extension of the Crouzeix–Raviart space to general meshes with application to quasi-incompressiblelinear elasticity and Stokes flow.Math. Comp., 84(291):1–31.
Droniou, J. and Eymard, R. (2006).
A mixed finite volume scheme for anisotropic diffusion problems on any grid.Numer. Math., 105:35–71.
Droniou, J., Eymard, R., Gallouet, T., and Herbin, R. (2010).
A unified approach to mimetic finite difference, hybrid finite volume and mixed finite volume methods.M3AS Mathematical Models and Methods in Applied Sciences, 20(2):1–31.
Eymard, R., Gallouet, T., and Herbin, R. (2010).
Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes.SUSHI: a scheme using stabilization and hybrid interfaces.IMA J. Numer. Anal., 30(4):1009–1043.
Eymard, R., Henry, G., Herbin, R., Hubert, F., Klofkorn, R., and Manzini, G. (2011).
3D benchmark on discretization schemes for anisotropic diffusion problems on general grids.In Finite Volumes for Complex Applications VI - Problems & Perspectives, volume 2, pages 95–130.Springer.
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. JohnWiley & Sons.
54 / 55
References IV
Kuznetsov, Y., Lipnikov, K., and Shashkov, M. (2004).
Mimetic finite difference method on polygonal meshes for diffusion-type problems.Comput. Geosci., 8:301–324.
55 / 55