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A review of Hybrid High-Order methods: formulations, computational aspects, links with other methods Daniele A. Di Pietro, Alexandre Ern, Simon Lemaire https://sites.google.com/site/chezsimonlemaire cole des Ponts ParisTech CERMICS
A review of Hybrid High-Order methods: formulations, computational aspects, links with
Daniele A. Di Pietro, Alexandre Ern, Simon Lemaire
https://sites.google.com/site/chezsimonlemaire
École des Ponts ParisTech – CERMICS Laboratory
POEMs Workshop, Georgia Tech, USA October 28, 2015
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
Finite Volume methods
‚ Mixed/Hybrid Finite Volume (M/HFV) [Droniou and Eymard, 06 + Eymard, Gallouët, and Herbin, 10]
Mimetic/Compatible methods
‚ Mimetic Finite Difference (MFD) [Brezzi, Lipnikov, and Shashkov, 05 + Beirão da Veiga, Lipnikov, and Manzini, 14] equivalence with M/HFV [Droniou, Eymard, Gallouët, and Herbin, 10] ‚ Discrete Geometric Approach (DGA) [Codecasa, Specogna, and Trevisan, 10] ‚ Compatible Discrete Operator (CDO) [Bonelle and Ern, 14]
Non-conforming/penalized methods
‚ Cell-Centered Galerkin (CCG) [Di Pietro, 12] ‚ Generalized Crouzeix–Raviart [Di Pietro and Lemaire, 15]
Unifying frameworks
‚ Gradient Schemes [Droniou, Eymard, Gallouët, and Herbin, 13] ‚ CDO
Finite Element (FE) methods [Wachspress, 75 + Tabarraei and Sukumar, 04 +
Gillette, Rand, and Bajaj]
Virtual Element (VE) methods
‚ Conf. VE [Beirão da Veiga, Brezzi, Cangiani, Manzini, Marini, and Russo, 13] ‚ Non-conf. VE [Ayuso de Dios, Lipnikov, and Manzini] ‚ Unified framework [Cangiani, Manzini, and Sutton]
Discontinuous Galerkin (DG) methods [Arnold, Brezzi, Cockburn, and Marini,
02 + Di Pietro and Ern, 12 + Bassi, Botti, Colombo, Di Pietro, and Tesini, 12 + Antonietti, Giani, and Houston, 13]
Hybridizable DG (HDG) methods [Cockburn, Gopalakrishnan, and Lazarov, 09] Weak Galerkin (WG) methods [Wang and Ye, 13] Hybrid High-Order (HHO) methods [Di Pietro, Ern, and Lemaire, 14]
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
Let Ω Ă Rd, d ě 2, be an open, connected, bounded polytopal domain. Problem: Find a potential u : Ω Ñ R such that ✎ ✍ ☞ ✌ ´ divpM∇uq “ f in Ω u “ 0
(1) f P L2pΩq, M symmetric, piecewise Lipschitz, matrix-valued coeff. s.t. for a.e. x P Ω, and all ξ P Rd s.t. |ξ| “ 1, 0 ă µ5 ď Mpxqξ¨ξ ď µ7 ă `8
Definition
The mesh sequence pThqhPH is admissible if, for all h P H, Th is a finite collection of polygons/polyhedra T s.t. Ω “ Ť
T PTh T, and Th admits a
matching simplicial submesh Th such that pThqhPH is ‚ shape-regular in the usual sense of Ciarlet; ‚ contact-regular: every simplex S Ď T is s.t. hS « hT . Assumption: M P “ P0
dpThq
‰dˆd
sym @h P H, and @T P Th, MT :“ M|T is s.t.
µ5,T ď MT ξ¨ξ ď µ7,T (local anisotropy ratio: ρT :“ µ7,T {µ5,T q
Figure : Admissible meshes in 2D
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
‚ Di Pietro, D. A. and Ern, A., A Hybrid High-Order locking-free method for linear elasticity on general meshes, Comput. Meth.
‚ Di Pietro, D. A., Ern, A., and Lemaire, S., An arbitrary-order and compact-stencil discretization of diffusion on general meshes based
14(4):461–472, 2014. ‚ Di Pietro, D. A. and Ern, A., Hybrid High-Order methods for variable-diffusion problems on general meshes, C. R. Acad. Sci. Paris, Ser. I, 353:31–34, 2015.
‚ ‚ ‚ ‚ ‚ ‚ k “ 0 ‚ ‚‚ ‚‚ ‚‚ ‚‚ ‚‚ ‚‚ k “ 1 ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ ‚ k “ 2 ‚ ‚ ‚ ‚ ‚ ‚
Figure : DoFs associated with potential unknowns, d “ 2
Local hybrid set of potential unknowns ✎ ✍ ☞ ✌ Uk
T :“ Pk dpTq ˆ
# ą
F PFT
Pk
d´1pFq
+ Local reduction operator Ik
T : H1pTq Ñ Uk T s.t., for all v P H1pTq,
Ik
T v :“
´ Πk
T v, pΠk F vqF PFT
¯
Local potential reconstruction operator: pk`1
T
: Uk
T Ñ Pk`1 d
pTq For vT “ pvT , vFT q P Uk
T , pk`1 T
vT P Pk`1
d
pTq is s.t. ş
T pk`1 T
vT “ ş
T vT
and satisfies, for all w P Pk`1
d
pTq, ✎ ✍ ☞ ✌ pMT ∇pk`1
T
vT , ∇wqT “ ´pvT , divpMT ∇wqqT ` ÿ
F PFT
pvF , MT ∇w¨nT,F qF diffusivity included in reconstruction operator Computation Requires to invert a SPD matrix of size Npk`1q,d with Nk,l :“ dimpPk
l q
Approximation For all v P Hk`2pTq, the following holds: }v ´ pk`1
T
Ik
T v}T ` hT }∇pv ´ pk`1 T
Ik
T vq}T À ρ
1{2
T hk`2 T
}v}k`2,T
✞ ✝ ☎ ✆ aT puT , vT q :“ pMT ∇pk`1
T
uT , ∇pk`1
T
vT qT ` jT puT , vT q Local stabilization bilinear form: jT : Uk
T ˆ Uk T Ñ R
For all uT , vT P Uk
T ,
jT puT , vT q :“ ÿ
F PFT
µT,F hF pΠk
F pqk`1 T
uT ´ uF q, Πk
F pqk`1 T
vT ´ vF qqF , where µT,F :“ MT nF ¨nF , and qk`1
T
wT :“ wT ` ppk`1
T
wT ´ Πk
T pk`1 T
wT q the use of Πk
F is reminiscent of Lehrenfeld-Schöberl stabilization for HDG
[Lehrenfeld, 10]
the operator qk`1
T
is new and opens the door to lower-order cell unknowns
Approximation For all v P Hk`2pTq, the following bound holds: jT pIk
T v, Ik T vq
1{2 À µ 1{2
7,T ρ
1{2
T hk`1 T
}v}k`2,T
Global hybrid set of potential unknowns ✞ ✝ ☎ ✆ Uk
h :“ Pk dpThq ˆ Pk d´1pFhq
Discrete problem Find uh P Uk
h,0 s.t.
ahpuh, vhq “ pf, vThq for all vh P Uk
h,0
with ahpuh, vhq :“ ř
T PTh aT puT , vT q
Stability ρ´1
T }M
1{2
T ∇vT } 2 T ` ρ´1 T
ÿ
F PFT
µT,F hF }vT ´ vF }2
F À aT pvT , vT q
Theorem (Energy-norm error estimate)
Assume u P U0 X Hk`2pThq. Then, }M
1{2p∇u ´ ∇hpk`1
Th uhq} À
# ÿ
T PTh
µ7,T ρ2
T h2pk`1q T
}u}2
k`2,T
+1{2
Theorem (L2-norm error estimate)
Assume elliptic regularity under the form }zpgq}2,Th À µ´1
5 }g}. Assume
f P Hk`δpΩq, with δ “ 0 for k ě 1 and δ “ 1 for k “ 0. Then, µ5}Πk
Thu ´ uTh} À µ
1{2
7 ρ h
# ÿ
T PTh
µ7,T ρ2
T h2pk`1q T
}u}2
k`2,T
+1{2 `hk`2}f}k`δ
1 - Introduce the local bilinear form ˆ aT pwT , vT q :“ pMT ∇pk`1
T
wT , ∇pk`1
T
vT qT ` ÿ
F PFT
µT,F hF pwT ´ wF , vT ´ vF qF 2 - Define the local isomorphism ck
T : Uk T Ñ Uk T s.t.
ˆ aT pck
T wT , vT q “ ˆ
aT pwT , vT q ` jT pwT , vT q @ vT P Uk
T
3 - Define the local gradient recons. operator Gk`1
T
:“ ∇ppk`1
T
˝ ck
T q
Lemma
For all T P Th, the following local equilibrium holds: pMT Gk`1
T
uT , ∇vT qT ´ ÿ
F PFT
pΦT,F puT q, vT qF “ pf, vT qT @ vT P Pk
dpTq
with conservative numerical flux ΦT,F puT q :“ MT Gk`1
T
uT ¨nT,F ´ µT,F hF “ pck
T uT ´ uT q ´ pck F uT ´ uF q
‰
Offline step 2 fully parallelizable and f-independent substeps
‚ 1 - Compute the potential reconstruction operator pk`1
Th
invert cardpThq SPD matrices of size Npk`1q,d ‚ 2 - For all T P Th, compute the trace-based tk
T : Pk d´1pFT q Ñ Pk dpTq and
datum-based dk
T : Pk dpTq Ñ Pk dpTq lifting operators s.t.
tk
T wFT P Pk dpTq solves
aT pptk
T wFT , 0q, pvT , 0qq “ ´aT pp0, wFT q, pvT , 0qq @vT P Pk dpTq
dk
T ΨT P Pk dpTq solves
aT ppdk
T ΨT , 0q, pvT , 0qq “ pΨT , vT qT @vT P Pk dpTq
invert cardpThq SPD matrices of size Nk,d
Online step
‚ 1 - Given f P L2pΩq, compute its L2-orthogonal projection Πk
Thf onto Pk dpThq
‚ 2 - Solve the global problem: Find uFh P Pk
d´1,0pFhq s.t.
ahptk
huFh, tk hvFhq “ pΠk Thf, tk ThvFhq
@vFh P Pk
d´1,0pFhq
where tk
hwFh :“ ptk ThwFh, wFhq
solve a linear system of size « cardpFhq ˆ Nk,pd´1q ‚ 3 - Compute the discrete solution according to uh “ ptk
ThuFh ` dk ThΠk Thf, uFhq
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
Assume here M “ Idd. The HHO(l) family: Uk,l
T
:“Pl
dpTqˆPk d´1pFT q, l P tk ´ 1, k, k ` 1u
‚ The choice l “ k corresponds to the original HHO method ‚ The Non-conf. VE method is, up to equivalent stabilization, a member of the HHO family (for l “ k ´ 1) [Cockburn, Di Pietro, and Ern, 15] ‚ The final system has the same size for any choice of l
Similarities/differences among discontinuous skeletal methods
‚ HHO fits into the HDG framework [Cockburn, Di Pietro, and Ern, 15] ‚ The flux unknowns associated with HDG and WG belong to Pk
dpTqd, whereas
the flux unknowns associated with HHO belong to ∇Pk`1
d
pTq: smaller local problems must be solved to eliminate flux unknowns in HHO
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
‚ Di Pietro, D. A. and Ern, A., Arbitrary-order mixed methods for heterogeneous anisotropic diffusion on general meshes, submitted 2013, to appear 2016. ‚ Aghili, J., Boyaval, S., and Di Pietro, D. A., Hybridization of mixed high-order methods on general meshes and application to the Stokes equations, Comput. Meth. Appl. Math., 15(2):111–134, 2015.
Ò Ò Ò Ò Ò Ò k “ 0 ÒÒ ÒÒ ÒÒ ÒÒ ÒÒ ÒÒ k “ 1 ‚ ‚ ÒÒÒ ÒÒÒ ÒÒÒ ÒÒÒ ÒÒÒ ÒÒÒ k “ 2 ‚ ‚ ‚ ‚ ‚
Figure : DoFs associated with flux unknowns, d “ 2
Local thybrid set of flux unknownsu `
dpTq
( ✎ ✍ ☞ ✌ Sk
T :“ MT ∇Pk dpTq ˆ
# ą
F PFT
Pk
d´1pFq
+ Local reduction operator: S`pTq :“ tt P LqpTq | div t P L2pTqu, q ą 2 Ik
T : S`pTq Ñ Sk T s.t., @t P S`pTq,
Ik
T t :“
´ MT ∇y, pΠk
F pt¨nF qqF PFT
¯ , with y P Pk
dpTq a solution of pMT ∇y, ∇wqT “ pt, ∇wqT
@w P Pk
dpTq
Local divergence reconstruction operator: Dk
T : Sk T Ñ Pk dpTq
For all tT “ ptT , tFT q P Sk
T , Dk T tT P Pk dpTq satisfies, for all w P Pk dpTq,
✎ ✍ ☞ ✌ pDk
T tT , wqT “ ´ptT , ∇wqT `
ÿ
F PFT
ptF εT,F , wqF where εT,F :“ nF ¨nT,F Computation Requires to invert a SPD matrix of size Nk,d “ ˆ k ` d k ˙ Commuting property The following holds for all t P S`pTq: Dk
T Ik T t “ Πk T pdiv tq
this ensures inf-sup stability for the discretization
Local flux reconstruction operator: Fk`1
T
: Sk
T Ñ MT ∇Pk`1 d
pTq For all tT “ ptT , tFT q P Sk
T , Fk`1 T
tT :“ MT ∇z, where z P Pk`1
d
pTq satisfies, for all w P Pk`1
d
pTq, ✎ ✍ ☞ ✌ pMT ∇z, ∇wqT “ ´pDk
T tT , wqT `
ÿ
F PFT
ptF εT,F , wqF diffusivity included in reconstruction operator Computation Requires to invert a SPD matrix of size Npk`1q,d Approximation For all v P Hk`2pTq, letting t :“ MT ∇v, the following holds for all F P FT : }M´1{2
T
pt ´ Fk`1
T
Ik
T tq}T `h
1{2
F µ´1{2 T,F }pt ´ Fk`1 T
Ik
T tq¨nF }F À ρ
1{2
T µ
1{2
7,T hk`1 T
}v}k`2,T
✞ ✝ ☎ ✆ HT psT , tT q :“ pM´1
T Fk`1 T
sT , Fk`1
T
tT qT ` JT psT , tT q Local stabilization bilinear form: JT : Sk
T ˆ Sk T Ñ R
For all sT , tT P Sk
T ,
JT psT , tT q :“ ÿ
F PFT
hF µT,F ppFk`1
T
sT q¨nF ´ sF , pFk`1
T
tT q¨nF ´ tF qF Approximation For all v P Hk`2pTq, the following bound holds with t :“ MT ∇v: JT pIk
T t, Ik T tq
1{2 À ρ 1{2
T µ
1{2
7,T hk`1 T
}v}k`2,T
Mixed weak formulation of (1)
Let S :“ Hpdiv, Ωq, V :“ L2pΩq. Find ps, uq P S ˆ V s.t. # pM´1s, tq ` pu, div tq “ 0 @t P S ´pdivs, vq “ pf, vq @v P V
Global thybrid set of flux unknownsu `
dpThq
( ✞ ✝ ☎ ✆ Sk
h :“ M∇Pk dpThq ˆ Pk d´1pFhq
Discrete problem: Find psh, uThq P Sk
h ˆ Pk dpThq s.t.
# Hhpsh, thq ` puTh, Dk
Ththq “ 0
@th P Sk
h
´pDk
Thsh, vThq “ pf, vThq
@vTh P Pk
dpThq
with Hhpsh, thq :“ ř
T PTh HT psT , tT q
Stability µ´1
7,T }tT }2 T ` µ´1 7,T
ÿ
F PFT
hF }tF }2
F À HT ptT , tT q
` inf-sup
Theorem (Error estimate for the flux)
Assume u P V X Hk`2pThq and s P S X S`pThq. Then, }M´1{2ps ´ Fk`1
Th shq} À
# ÿ
T PTh
µ7,T ρT h2pk`1q
T
}u}2
k`2,T
+1{2
Theorem (Supercloseness of the potential)
Assume elliptic regularity under the form }zpgq}2,Th À µ´1
5 }g}. Assume
f P Hk`δpΩq, with δ “ 0 for k ě 1 and δ “ 1 for k “ 0. Then, µ5}Πk
Thu ´ uTh} À µ
1{2
7 ρ
1{2h
# ÿ
T PTh
µ7,T ρT h2pk`1q
T
}u}2
k`2,T
+1{2 `hk`2}f}k`δ
Unpatched global hybrid set of flux unknowns ˇ S
k h :“
ą
T PTh
Sk
T ,
ˇ Z
k h :“
# ˇ th P ˇ S
k h |
ÿ
T PTF
ˇ tT,F “ 0, @F P Fi
h
+ natural isomorphism Lk
h from ˇ
Z
k h to Sk h
Potential-to-flux mapping operator: ˇ ςk
h : Uk h Ñ ˇ
S
k h
For vT P Uk
T , ˇ
ςk
T vT satisfies, for all ˇ
tT P Sk
T ,
HT pˇ ςk
T vT ,ˇ
tT q “ ´pvT , Dk
Tˇ
tT qT ` ÿ
F PFT
pˇ tT,F , vF qF There holds: for all vT P Uk
T , pFk`1 T
˝ ˇ ςk
T q vT “ MT ∇pk`1 T
vT Characterization of the solution Let ˜ uh P Uk
h,0 solve Ahp˜
uh, vhq “ pf, vThq for all vh P Uk
h,0, with
Ahp˜ uh, vhq :“ ÿ
T PTh
pMT ∇pk`1
T
˜ uT , ∇pk`1
T
vT qT ` ÿ
T PTh
JT pˇ ςk
T ˜
uT , ˇ ςk
T vT q
Then, there holds ˇ ςk
h˜
uh P ˇ Z
k h and psh, uThq “ pLk hpˇ
ςk
h˜
uhq, ˜ uThq.
Offline step 4 fully parallelizable and f-independent substeps
‚ 1 - Compute the divergence reconstruction operator Dk
Th
invert cardpThq SPD matrices of size Nk,d ‚ 2 - Compute the flux reconstruction operator Fk`1
Th
invert cardpThq SPD matrices of size Npk`1q,d ‚ 3 - Compute the potential-to-flux mapping operator ˇ ςk
h
invert cardpThq SPD matrices of size Nk,d ` cardpFT qNk,pd´1q ‚ 4 - For all T P Th, compute the lifting operators tk
T : Pk d´1pFT q Ñ Pk dpTq and
dk
T : Pk dpTq Ñ Pk dpTq associated with the bilinear form AT
invert cardpThq SPD matrices of size Nk,d
Online step
‚ 1 - Given f P L2pΩq, compute its L2-orthogonal projection Πk
Thf onto Pk dpThq
‚ 2 - Solve the global coercive problem: Find ˜ uFh P Pk
d´1,0pFhq s.t.
Ahptk
h˜
uFh, tk
hvFhq “ pΠk Thf, tk ThvFhq
@vFh P Pk
d´1,0pFhq
solve a linear system of size « cardpFhq ˆ Nk,pd´1q ‚ 3 - Compute the discrete solution according to psh, uThq “ pLk
hpˇ
ςk
h˜
uhq, ˜ uThq, with ˜ uh “ ptk
Th ˜
uFh ` dk
ThΠk Thf, ˜
uFhq
Literature review Setting The HHO method in primal form Links HHO/other polytopal discretization methods The HHO method in mixed form Conclusion
‚ Capable of handling general polytopal meshes ‚ Dimension-independent construction ‚ Arbitrary approximation order (starting from k “ 0) ‚ Physical fidelity
‚ Local conservation ‚ Robustness w.r.t. physical parameters in various situations: heterogeneous/anisotropic diffusion, quasi-incompressible linear elasticity, advection-dominated transport, Stokes flow driven by large irrotational forces, Biot’s model of poroelasticity (coupled with DG). . .
‚ Reduced computational cost after static condensation N HHO
DoFs « 1
2k2cardpFhq vs. N DG
DoFs « 1
6k3cardpThq ‚ Natural offline/online solution strategy: adapted to the multi-query context
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