O . MODELING AND SCIENTIFIC COMPUTING . MODELLISTICA E CALCOLO - - PowerPoint PPT Presentation

▶
o
SMART_READER_LITE
LIVE PREVIEW

O . MODELING AND SCIENTIFIC COMPUTING . MODELLISTICA E CALCOLO - - PowerPoint PPT Presentation

Feb 05, 2023 169 likes •357 views

O . MODELING AND SCIENTIFIC COMPUTING . MODELLISTICA E CALCOLO SCIENTIFICO . . . M . X Multigrid algorithms for highorder Discontinuous Galerkin methods on polygonal and polyhedral meshes Paola F. Antonietti and Marco Verani

slide-1
SLIDE 1

Multigrid algorithms for high–order Discontinuous Galerkin methods

  • n polygonal and polyhedral meshes

Paola F. Antonietti and Marco Verani

Politecnico di Milano MOX-Dipartimento di Matematica

. .M .

O

.

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

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

POEMS - GEORGIA TECH, 27th OCTOBER 2015

slide-2
SLIDE 2

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Aims

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

  • n ∂Ω
  • n polytopic grids.

AIM

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

slide-3
SLIDE 3

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

DG formulation

DG space: V hp = {v ∈ L2(Ω) : v|κ ∈ Pp(κ) ∀κ ∈ Th}. Weak formulation: Ah(uh, vh) =

  • Ω

fvh dx ∀vh ∈ V hp, with Ah(·, ·) defined as: Ah(uh, vh) =

  • κ∈Th
  • κ

∇uh · ∇vh dx −

  • F∈Fh
  • F

{ {∇uh} } · vh ds −

  • F∈Fh
  • F

uh · { {∇vh} } ds +

  • F∈Fh
  • F

σF uh · vh ds with σF = σF(p, κ±, F, C ±

INV, α) ∈ L∞(F)

[Cangiani, Georgoulis, Houston, 2014]

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 2 of 14

slide-4
SLIDE 4

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Multigrid methods on agglomerated grids

The set of (nested) partitions {Tj}J

j=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]

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 3 of 14

slide-5
SLIDE 5

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Multigrid 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
  • n 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

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 4 of 14

slide-6
SLIDE 6

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Multigrid 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
  • n 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.

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 4 of 14

slide-7
SLIDE 7

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Multigrid methods: the jth level iteration

  • MG(j, g, z0, m) ≈ z

approximate solution of Ajz = g,

  • btained from the jth level

iteration with i.g. z0.

  • Aj=matrix representation of

Ah(·, ·) on level j.

  • Bj = your favorite smoother.

if j = 1 then Set 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, Ij−1 j

(g − Ajz(m)), e(r−1)

j−1 , m);

end for Set 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

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 5 of 14

slide-8
SLIDE 8

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Convergence W-cycle algorithm (standard grids)

Error propagation operator (Gj = Idj − BjAj)

  • E1,mv = 0,

Ej,mv = Gm

j ((Idj − Pj−1) − E2 j−1,mPj−1)Gm j v,

j > 1. Bj= Richardson smoother. It holds that Ej,mAj ≤ C p2

j

1 + m ∀j = 1, . . . , J. Therefore, if m is chosen large enough (i.e., m ≈ O(p2

j )),

Ej,mAj < 1 and then u(k)

h

− − − →

k→∞ uh.

THEOREM - [A., Sarti, Verani, SINUM, 2015] Proof: Multigrid framework of [Brenner & Zhao, 2005].

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 6 of 14

slide-9
SLIDE 9

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Convergence W-cycle algorithm (polytopic grids)

Ej,mAj ≤ CP p2

j

1 + m ∀j = 1, . . . , J. Therefore, if m is chosen large enough Ej,mAj < 1 and then u(k)

h

− − − →

k→∞ uh.

CP = CP(CINV, quality of agglomerated meshes) THEOREM - [A., Houston, Sarti, Verani. Submitted.]

  • Mild geometrical assumptions on

agglomerated meshes (only for the theory)

  • The above result holds provided the

number of levels is kept limited!

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 7 of 14

slide-10
SLIDE 10

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Numerical 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, pj−1) ≈ O(1) p

Two-level W-cycle, 3 levels

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

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 8 of 14

slide-11
SLIDE 11

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Numerical experiments

  • Convergence factor

ρ = exp 1 N ln rN2 r02

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

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 9 of 14

slide-12
SLIDE 12

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

h-multigrid, p = 1 (polytopic grids)

m=3 m=5 m=8 TL W-cycle 3 lvl 4 lvl TL W-cycle 3 lvl 4 lvl TL W-cycle 3 lvl 4 lvl TL 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 NCG

iter = 445

NCG

iter = 633

NCG

iter = 946

NCG

iter = 1234 Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 10 of 14

slide-13
SLIDE 13

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

p-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 a function of J and pJ (m = 6). Convergence factor as a function J and m (pJ = 5).

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 11 of 14

slide-14
SLIDE 14

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

h-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

Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 12 of 14

slide-15
SLIDE 15

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

hp-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 NCG

iter = 1701

NCG

iter = 2809

NCG

iter = 4574

NCG

iter = 6796 Antonietti-Verani | MOX-PoliMi Multigrid methods for hp-DG on polygons | 13 of 14

slide-16
SLIDE 16

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Future 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 element methods for the approximation of partial differential equations.

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

slide-17
SLIDE 17

.M .

O

. . .

X

. MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING

Geometric assumptions (for any level j = 1, . . . , J)

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

hd

κ ≥ |κ| hd κ,

d = 2, 3 .

  • For any κ ∈ Tj, there exists K ∈ T ♯

j (covering of Ω) such that κ ⊂ K and

card

  • κ′ ∈ Tj : κ′ ∩ K = ∅, K ∈ T ♯

j

such that κ ⊂ K

  • 1.

Consequently, for each pair κ, K ∈ T ♯

j , with κ ⊂ K,

diam(K) hκ.

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