Paola F. Antonietti and Marco Verani · Multigridalgorithmsforhigh–order...

Multigrid algorithms for high–orderDiscontinuous Galerkin methods

on polygonal and polyhedral meshesPaola F. Antonietti and Marco Verani

Politecnico di MilanoMOX-Dipartimento di Matematica

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MODELLISTICA E CALCOLO SCIENTIFICO

.MODELING AND SCIENTIFIC COMPUTING

Joint work with: P. Houston (Nottingham), M. Sarti (PoliMi)

POEMS - GEORGIA TECH, 27th OCTOBER 2015

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.MODELING AND SCIENTIFIC COMPUTINGAims

Design and analysis of efficient solution techniques for

Ahuh = Fh

when Ah results from hp-DG approximations of:−∆u =f in Ω ∈ Rd , d = 2, 3

u =0 on ∂Ω

on polytopic grids.

AIM

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 1 of 14

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.MODELING AND SCIENTIFIC COMPUTINGDG formulation

DG space:V hp = v ∈ L2(Ω) : v |κ ∈ Pp(κ) ∀κ ∈ Th.

Weak formulation:

Ah(uh, vh) =∫

Ωfvh dx ∀vh ∈ V hp ,

with Ah(·, ·) defined as:

Ah(uh, vh) =∑κ∈Th

∫κ

∇uh · ∇vh dx −∑

F∈Fh

∫F∇uh · JvhK ds

−∑

F∈Fh

∫FJuhK · ∇vh ds +

∑F∈Fh

∫FσF JuhK · JvhK ds

with σF = σF (p,κ±,F ,C±INV,α) ∈ L∞(F )

[Cangiani, Georgoulis, Houston, 2014]


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.MODELING AND SCIENTIFIC COMPUTINGMultigrid methods on agglomerated grids

The set of (nested) partitions TjJj=1 is obtained by agglomeration

hj, pj

hj−1, pj−1

hj−2, pj−2

hp-DG methods on polytopic grids

[Cangiani, Georgoulis, Houston, 2014] [Cangiani, Dong, Georgoulis, Houston, 2015][A., Giani, Houston, 2013],[Bassi, Botti, Colombo, Di Pietro, Tesini, 2012]


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.MODELING AND SCIENTIFIC COMPUTINGMultigrid methods: the idea

The solution uh is approximated by a sequence of u(k)h , k = 0, 1, 2, . . . .

u(k+1)h = MG(J ,Fh, u(k)

h ,m):• m pre-smoothing steps;• recursive correction of the residual

on level j ;• m post-smoothing steps;

u(k)h u

(k+1)h

level J TJ , VJ

level j Tj, Vj

level 1 T1, V1

hj

hj−1

h-multigrid

h, pj

h, pj−1

p-multigrid

hj , pj

hj−1, pj−1

hp-multigrid


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.MODELING AND SCIENTIFIC COMPUTINGMultigrid methods: the idea

The solution uh is approximated by a sequence of u(k)h , k = 0, 1, 2, . . . .

u(k+1)h = MG(J ,Fh, u(k)

h ,m):• m pre-smoothing steps;• recursive correction of the residual

on level j ;• m post-smoothing steps;

u(k)h u

(k+1)h

level J TJ , VJ

level j Tj, Vj

level 1 T1, V1

Subspaces defined as

Vj = v ∈ L2(Ω) : v |κ ∈ Ppj (κ) ∀κ ∈ Tj, j = 1, . . . , J ,

and we assume

hj−1 . hj ≤ hj−1, pj−1 ≤ pj . pj−1.


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.MODELING AND SCIENTIFIC COMPUTINGMultigrid methods: the jth level iteration

• MG(j, g , z0, m) ≈ zapproximate solution of

Ajz = g ,

obtained from the jth leveliteration with i.g. z0.

• Aj=matrix representation ofAh(·, ·) on level j.

• Bj = your favorite smoother.

if j = 1 thenSet MG(1, g , z0, m) := A−1

j g .else

Pre-smoothing: (i=1,. . . ,m)z (i) = z (i−1) + Bj(g − Ajz (i−1));Error correction:e(0)

j−1 = 0;for r = 1, . . . , s do

e(r)j−1 = MG(j − 1, I j−1

j (g − Ajz (m)), e(r−1)j−1 , m);

end forSet z (m+1) = z (m) + e(s)

j−1;Post-smoothing: (i=m+2,. . . ,2m+1)z (i) = z (i−1) + Bj(g − Ajz (i−1));

Set MG(j, g , z0, m) := z (2m+1);end if


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.MODELING AND SCIENTIFIC COMPUTINGConvergence W-cycle algorithm (standard grids)

Error propagation operator (Gj = Idj − BjAj)E1,mv = 0,Ej,mv = Gm

j ((Idj − Pj−1)− E2j−1,mPj−1)Gm

j v , j > 1.

Bj= Richardson smoother. It holds that

‖Ej,m‖Aj ≤ Cp2

j1 + m ∀j = 1, . . . , J .

Therefore, if m is chosen large enough (i.e., m ≈ O(p2j )),

‖Ej,m‖Aj < 1 and then u(k)h −−−→

k→∞uh.

THEOREM - [A., Sarti, Verani, SINUM, 2015]

Proof: Multigrid framework of [Brenner & Zhao, 2005].


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.MODELING AND SCIENTIFIC COMPUTINGConvergence W-cycle algorithm (polytopic grids)

‖Ej,m‖Aj ≤ CPp2

j1 + m ∀j = 1, . . . , J .

Therefore, if m is chosen large enough

‖Ej,m‖Aj < 1 and then u(k)h −−−→

k→∞uh.

CP = CP(CINV, quality of agglomerated meshes)

THEOREM - [A., Houston, Sarti, Verani. Submitted.]

• Mild geometrical assumptions onagglomerated meshes (only for the theory)

• The above result holds provided thenumber of levels is kept limited!


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.MODELING AND SCIENTIFIC COMPUTINGNumerical evaluation of C and CP

Triangular grids

1 2 3 4 5 6 7 8 9 10

0.6

0.8

1

p

SIPG, J = 2

SIPG, J = 3

LDG, J = 2

LDG, J = 3

Numerical evaluation of C:h-multigrid scheme with m = 2p2.

Polytopic grids

1 2 3 4 5 6

0.6

0.8

1

C(pj , pj1) O(1)

p

Two-level

W-cycle, 3 levels

Numerical evaluation of CP :h-multigrid scheme with m = 2p2.


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.MODELING AND SCIENTIFIC COMPUTINGNumerical experiments

• Convergence factor

ρ = exp(1N ln ‖rN‖2

‖r0‖2

)• N iterations required to attain convergence up to a (relative) tol. of 10−8.• Agglomerated grids obtained with MGridGen [Moulitsas, Karypis, 01]


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.MODELING AND SCIENTIFIC COMPUTINGh-multigrid, p = 1 (polytopic grids)

m=3

m=5

m=8

TLW-cycle

3 lvl 4 lvlTL

W-cycle

3 lvl 4 lvlTL

W-cycle

3 lvl 4 lvlTL

W-cycle

3 lvl 4 lvl

133 160 167 121 191 188 140 188 192 162 198 198

95 113 113 88 121 125 99 124 128 112 131 131

72 82 81 67 86 88 74 89 91 83 94 94

Nelem = 512 Nelem = 1024 Nelem = 2048 Nelem = 4096

NCGiter = 445 NCG

iter = 633 NCGiter = 946 NCG

iter = 1234


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.MODELING AND SCIENTIFIC COMPUTINGp-multigrid: 2D tests

J = 2 J = 3 J = 4

pJ = 2 0.62 - -

pJ = 3 0.77 0.77 -

pJ = 4 0.79 0.80 0.86

pJ = 5 0.83 0.82 0.87

pJ = 6 0.86 0.86 0.86

J = 2 J = 3 J = 4

m = 2 0.91 0.91 0.94

m = 4 0.85 0.85 0.90

m = 10 0.78 0.77 0.80

Convergence factor as afunction of J and pJ (m = 6).

Convergence factor as afunction J and m (pJ = 5).


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.MODELING AND SCIENTIFIC COMPUTINGh-multigrid: 3D tests (tetrahedral grids)

p = 1 p = 2 p = 3

J = 2 J = 3 J = 4 J = 2 J = 3 J = 4 J = 2 J = 3

Richardson smoother

m = 2 0.57 0.55 0.53 0.82 0.81 0.80 0.90 0.90

m = 4 0.71 0.71 0.69 0.91 0.90 0.90 0.95 0.95

m = 10 0.46 0.44 0.41 0.79 0.78 0.77 0.88 0.88

Gauss-Seidel smoother

m = 2 0.57 0.55 0.53 0.82 0.81 0.79 0.89 0.89

m = 4 0.35 0.33 0.30 0.68 0.67 0.65 0.81 0.80

m = 10 0.13 0.15 0.12 0.43 0.41 0.40 0.61 0.60

symmetric Gauss-Seidel smoother

m = 2 0.35 0.33 0.30 0.68 0.67 0.65 0.81 0.80

m = 4 0.17 0.19 0.16 0.50 0.48 0.46 0.67 0.66

m = 10 0.05 0.08 0.07 0.22 0.22 0.20 0.41 0.39


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.MODELING AND SCIENTIFIC COMPUTINGhp-multigrid (polytopic grids)

p = 2 p = 3 p = 4 p = 5

TL TL W-cycle TL W-cycle TL W-cycle

3 lvl 3 lvl 4 lvl 3 lvl 4 lvl

m = 12 334 631 1528 860 1028 1051 890 1197 1418

m = 14 292 550 607 748 889 908 772 1033 1220

NCGiter = 1701 NCG

iter = 2809 NCGiter = 4574 NCG

iter = 6796


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.MODELING AND SCIENTIFIC COMPUTINGFuture developments

• 3D/parallel coding on polyhedral grids• Theoretical analysis for complex smoothers• Strongly heterogeneous/anisotropic diffusion• Extension to VEM• . . . . . .

Acknowledgments

SIR project n. RBSI14VT0S:PolyPDEs: Non-conforming polyhedral finite elementmethods for the approximation of partial differentialequations.


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.MODELING AND SCIENTIFIC COMPUTINGGeometric assumptions (for any level j = 1, . . . , J)

• The number of faces is uniformly bounded• For any κ ∈ Tj we assume that

hdκ ≥ |κ| & hd

κ, d = 2, 3

.• For any κ ∈ Tj , there exists K ∈ T ]j (covering of Ω) such that κ ⊂ K and

cardκ′ ∈ Tj : κ′ ∩ K 6= ∅, K ∈ T ]j such that κ ⊂ K

. 1.

Consequently, for each pair κ, K ∈ T ]j , with κ ⊂ K,

diam(K) . hκ.


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Paola F. Antonietti and Marco Verani · Multigridalgorithmsforhigh–order...

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