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

4 / 55

Outline

1 Poisson

2 Variable diffusion and local conservation

3 Linear elasticity

5 / 55

Outline

1 Poisson

2 Variable diffusion and local conservation

3 Linear elasticity

6 / 55

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.

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References II

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