O . MODELING AND SCIENTIFIC COMPUTING . MODELLISTICA E CALCOLO SCIENTIFICO . . . M . X Multigrid algorithms for high–order Discontinuous Galerkin methods on polygonal and polyhedral meshes Paola F. Antonietti and Marco Verani Politecnico di Milano MOX-Dipartimento di Matematica Joint work with: P. Houston (Nottingham), M. Sarti (PoliMi) POEMS - GEORGIA TECH, 27 th OCTOBER 2015
. M O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING . Aims AIM Design and analysis of efficient solution techniques for A h u h = F h when A h results from hp -DG approximations of: � in Ω ∈ R d , d = 2, 3 − ∆ u = f u =0 on ∂ Ω on polytopic grids. Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 1 of 14
. M . MODELING AND SCIENTIFIC COMPUTING . MODELLISTICA E CALCOLO SCIENTIFICO . X . . . O DG formulation DG space: V hp = { v ∈ L 2 (Ω) : v | κ ∈ P p ( κ ) ∀ κ ∈ T h } . Weak formulation: � ∀ v h ∈ V hp , A h ( u h , v h ) = fv h dx Ω with A h ( · , · ) defined as: � � � � A h ( u h , v h ) = ∇ u h · ∇ v h dx − { {∇ u h } } · � v h � ds κ F κ ∈T h F ∈F h � � � � − � u h � · { {∇ v h } } ds + σ F � u h � · � v h � ds F F F ∈F h F ∈F h with σ F = σ F ( 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
. M . O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING Multigrid methods on agglomerated grids The set of (nested) partitions {T j } 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
. M . . . MODELLISTICA E CALCOLO SCIENTIFICO MODELING AND SCIENTIFIC COMPUTING X . . . O Multigrid methods: the idea The solution u h is approximated by a sequence of u ( k ) h , k = 0, 1, 2, . . . . u ( k ) u ( k +1) u ( k +1) = MG( J , F h , u ( k ) h , m ): h h h level J {T J , V J } • m pre-smoothing steps; level j {T j , V j } • recursive correction of the residual on level j ; m post-smoothing steps; • level 1 {T 1 , V 1 } h j h , p j h j , p j h j − 1 h , p j − 1 h j − 1 , p j − 1 h -multigrid p -multigrid hp -multigrid Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 4 of 14
. M . O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING Multigrid methods: the idea The solution u h is approximated by a sequence of u ( k ) h , k = 0, 1, 2, . . . . u ( k ) u ( k +1) u ( k +1) = MG( J , F h , u ( k ) h , m ): h h h level J {T J , V J } • m pre-smoothing steps; level j {T j , V j } • recursive correction of the residual on level j ; m post-smoothing steps; • level 1 {T 1 , V 1 } Subspaces defined as V j = { v ∈ L 2 (Ω) : v | κ ∈ P p j ( κ ) ∀ κ ∈ T j } , j = 1, . . . , J , and we assume h j − 1 � h j ≤ h j − 1 , p j − 1 ≤ p j � p j − 1 . Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 4 of 14
. M . . MODELING AND SCIENTIFIC COMPUTING MODELLISTICA E CALCOLO SCIENTIFICO . X . . . O Multigrid methods: the j th level iteration if j = 1 then • MG( j , g , z 0 , m ) ≈ z Set MG(1, g , z 0 , m ) := A − 1 g . approximate solution of j else A j z = g , Pre-smoothing: (i=1,. . . ,m) z ( i ) = z ( i − 1) + B j ( g − A j z ( i − 1) ); obtained from the j th level Error correction: iteration with i.g. z 0 . e (0) j − 1 = 0; • A j =matrix representation of for r = 1, . . . , s do A h ( · , · ) on level j . e ( r ) j − 1 = MG( j − 1, I j − 1 ( g − A j z ( m ) ), e ( r − 1) j − 1 , m ); j end for • B j = your favorite smoother. Set z ( m +1) = z ( m ) + e ( s ) j − 1 ; Post-smoothing: (i=m+2,. . . ,2m+1) z ( i ) = z ( i − 1) + B j ( g − A j z ( i − 1) ); Set MG( j , g , z 0 , m ) := z (2 m +1) ; end if Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 5 of 14
. M . O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING Convergence W-cycle algorithm (standard grids) Error propagation operator ( G j = Id j − B j A j ) � E 1, m v = 0, E j , m v = G m j ((Id j − P j − 1 ) − E 2 j − 1, m P j − 1 ) G m j v , j > 1. THEOREM - [A., Sarti, Verani, SINUM, 2015] B j = Richardson smoother. It holds that p 2 j � E j , m � A j ≤ C ∀ j = 1, . . . , J . 1 + m Therefore, if m is chosen large enough ( i.e. , m ≈ O ( p 2 j )), and then u ( k ) � E j , m � A j < 1 − k →∞ u h . − − → h Proof: Multigrid framework of [Brenner & Zhao, 2005]. Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 6 of 14
. M . MODELING AND SCIENTIFIC COMPUTING . MODELLISTICA E CALCOLO SCIENTIFICO . X . . . O Convergence W-cycle algorithm (polytopic grids) THEOREM - [A., Houston, Sarti, Verani. Submitted.] p 2 j � E j , m � A j ≤ C P ∀ j = 1, . . . , J . 1 + m Therefore, if m is chosen large enough and then u ( k ) � E j , m � A j < 1 − k →∞ u h . − − → h C P = C P ( C INV , quality of agglomerated meshes ) • 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
. M . . . MODELLISTICA E CALCOLO SCIENTIFICO MODELING AND SCIENTIFIC COMPUTING X . . . O Numerical evaluation of C and C P Triangular grids Polytopic grids 1 1 0 . 8 0 . 8 ˜ C ( p j , p j − 1 ) ≈ O (1) SIPG, J = 2 SIPG, J = 3 0 . 6 LDG, J = 2 Two-level 0 . 6 LDG, J = 3 W-cycle, 3 levels 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 p p Numerical evaluation of C: Numerical evaluation of C P : h -multigrid scheme with m = 2 p 2 . h -multigrid scheme with m = 2 p 2 . Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 8 of 14
. M O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING . Numerical experiments • Convergence factor � 1 N ln � r N � 2 � ρ = exp � r 0 � 2 • 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
. M O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING . h -multigrid, p = 1 (polytopic grids) N elem = 512 N elem = 1024 N elem = 2048 N elem = 4096 W-cycle W-cycle W-cycle W-cycle TL TL TL TL 3 lvl 4 lvl 3 lvl 4 lvl 3 lvl 4 lvl 3 lvl 4 lvl m =3 133 160 167 121 191 188 140 188 192 162 198 198 m =5 95 113 113 88 121 125 99 124 128 112 131 131 m =8 72 82 81 67 86 88 74 89 91 83 94 94 N CG N CG N CG N CG iter = 445 iter = 633 iter = 946 iter = 1234 Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 10 of 14
. M . O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING p -multigrid: 2D tests J = 2 J = 3 J = 4 p J = 2 0.62 - - Convergence factor as a function of J and p J ( m = 6). p J = 3 0.77 0.77 - p J = 4 0.79 0.80 0.86 p J = 5 0.83 0.82 0.87 p J = 6 0.86 0.86 0.86 J = 2 J = 3 J = 4 m = 2 0.91 0.91 0.94 Convergence factor as a function J and m ( p J = 5). m = 4 0.85 0.85 0.90 m = 10 0.78 0.77 0.80 Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 11 of 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
. M . O . . . X . MODELLISTICA E CALCOLO SCIENTIFICO . MODELING AND SCIENTIFIC COMPUTING hp -multigrid (polytopic grids) p = 2 p = 3 p = 4 p = 5 W-cycle W-cycle W-cycle TL TL TL TL 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 N CG N CG N CG N CG iter = 1701 iter = 2809 iter = 4574 iter = 6796 Antonietti-Verani | MOX-PoliMi Multigrid methods for hp -DG on polygons | 13 of 14
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