Hp-spectral FEMs in fast domain decomposition algorithms V . - - PowerPoint PPT Presentation

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St.-Petersburg State Polytechnical University Hp-spectral FEMs in fast domain decomposition algorithms V . KORNEEV and A . RYTOV Laboratory of New Computational Technologies

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St.-Petersburg State Polytechnical University

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Hp-spectral FEM’s in fast domain decomposition algorithms

V. KORNEEV and A. RYTOV

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An outline of the lecture

■♥tr♦❞✉❝t✐♦♥✿ t❤❡ st❛t❡ ♦❢ ❛rt ✐♥ ❞❡✈❡❧♦♣✐♥❣ ❢❛st s♦❧✈❡rs✳
❋✐♥✐t❡✲❞✐✛❡r❡♥❝❡✴❢❡♠ ♣r❡❝♦♥❞✐t✐♦♥❡rs ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ ❛♥❞ s♣❡❝tr❛❧ p

❡❧❡♠❡♥ts✳

❋❛❝t♦r✐③❡❞ ♣r❡❝♦♥❞✐t✐♦♥❡rs ❢♦r s♣❡❝tr❛❧ ❡❧❡♠❡♥ts ❛♥❞ t❤❡✐r s✐♠✐❧❛r✐t②

t♦ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡rs✲s♦❧✈❡rs ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ ❡❧❡♠❡♥ts✳

❊①❛♠♣❧❡s ♦❢ t❤❡ ❢❛❝t♦r✐③❡❞ ❢❛st s♦❧✈❡rs ❢♦r s♣❡❝tr❛❧ ❡❧❡♠❡♥ts ✿

2-d multigrid solver, 3-d fast solver based on the wavelet

multilevel decompositions,

multilevel solver for faces.

❆❧♠♦st ♦♣t✐♠❛❧ ✐♥ t❤❡ ❛r✐t❤♠❡t✐❝ ❝♦st ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥

♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ❢♦r hp s♣❡❝tr❛❧ ❡❧❡♠❡♥t ♠❡t❤♦❞s✳

❈♦♥❝❧✉s✐♦♥s✳

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Preconditioners for hierarchical elements M1,p = (Li(s) , i = 0, 1, . . . , p) ✕ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦♥ ✭✲✶✱✶✮✿ L0(s) = 1

2(1 + s) ,

L1(s) = 1

2(1 − s) ,

Li(s) := βi s

−1 Pi−1(t) dt = γi[Pi(s) − Pi−2(s)] ,

i ≥ 2 , Pi ❛r❡ ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s ❛♥❞ βi = 1 2

(2j − 3)(2j − 1)(2j + 1) , γi = 0.5
(2i − 3)(2i + 1)/(2i − 1) .

❚❤❡r❡❢♦r❡✱ Li ❛r❡ s♣❡❝✐❛❧❧② ♥♦r♠❛❧✐③❡❞ ✐♥t❡❣r❛t❡❞ ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s✳

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❇② ❤✐❡r❛r❝❤✐❝❛❧ r❡❢✳ ❡❧✳ Ehi ✐s ✉♥❞❡rst♦♦❞ r❡❢✳❡❧✳ ♦♥ t❤❡ ❝✉❜❡ τ0 = (−1, 1)d ✇✐t❤ t❤❡ ❜❛s✐s ✐♥ t❤❡ s♣❛❝❡ Qp,x Md,p =

Lα(x) = Lα1(x1)Lα2(x2)...Lαd(xd) , α ∈ ω
,

ω := (α = (α1, α2, .., αd) : 0 ≤ α1, α2, .., αd ≤ p) , ❛♥❞ ✇✐t❤ t❤❡ st✐✛♥❡ss ♠❛tr✐① A✱ ✐♥❞✉❝❡❞ ❜② Md,p ❛♥❞ ❉✐r✐❝❤❧❡t ✐♥t❡❣r❛❧ aτ0(u, v) =

∇u · ∇v dx . AI ✕ internal st✐✛✳ ♠❛tr✐①✱ ❣❡♥❡r❛t❡❞ ❜②

Md,p= (Lα, 2 ≤ αk ≤ p)✳

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■❢ t♦ r❡♦r❞❡r s❡t

Md,p✱ ♠❛tr✐❝❡s AI✱ MI ✐♥ d = 3 ❜❡❝♦♠❡ ❜❧♦❝❦

❞✐❛❣♦♥❛❧ AI = ❞✐❛❣ [Aeee, Aeeo, ..., Aooe, Aooo] , MI = ❞✐❛❣ [Meee, Meeo, ..., Mooe, Mooo] . ❆t p = 2N + 1 ❛❧❧ ✽ ❜❧♦❝❦s ❛r❡ N 3 × N 3 ♠❛tr✐❝❡s ❛♥❞✱ e.g., Aa1a2a3 = (aτ0(Lα, Lα′))N

αk,α′

k=1 ,

✇✐t❤ αk, α′

k ❡✈❡♥✴♦❞❞ r❡s♣❡❝t✐✈❡❧② t♦ ❡✈❡♥✴♦❞❞ ak✳

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❚❤❡s❡ ❜❧♦❝❦s ❛r❡ s✉♠s Aabc = K1,a ⊗ K0,b ⊗ K0,c + K0,a ⊗ K1,b ⊗ K0,c + K0,a ⊗ K0,b ⊗ K1,c , Mabc = K0,a ⊗ K0,b ⊗ K0,c , a, b, c = e, o ♦❢ ❑r♦♥❡❝❦❡r ♣r♦❞✉❝ts ♦❢ tr✐♣❧❡ts ♦❢ N × N ♠❛tr✐❝❡s✱ ✇❤✐❝❤ ♠❛② ❜❡ ♣r❡❝♦♥❞✐t✐♦♥❡❞ ❜② s✐♠♣❧❡ ♠❛tr✐❝❡s D = diag [4i2]N

i=1 ,

∆ = 1 2               2 −1 −1 2 −1 ✵ . . . . . . ✵ −1 2 −1 −1 2               .

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Lemma 1. ❋♦r ✶✲❞ ♣r❡❝♦♥❞✐t✐♦♥❡rs D, ∆ ❛♥❞ ✸✲❞ ♣r❡❝♦♥❞✐t✐♦♥❡rs Λe = ∆⊗∆⊗D+∆×D⊗∆+D⊗∆⊗∆ , M = ∆⊗∆⊗∆ , t❤❡r❡ ❤♦❧❞ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ∆ ≺ K0,a ≺ ∆ , D ≺ K1,a ≺ D , Λe ≺ Aabc ≺ Λe , M ≺ Mabc ≺ M . Pr♦♦❢✳ ■✈❛♥♦✈✴❑♦r♥❡❡✈ ❬✶✾✾✺❪ ❛♥❞ ❑♦r♥❡❡✈✴❏❡♥s❡♥ ❬✶✾✾✼❪✱ ❑♦r♥❡❡✈✴▲❛♥❣❡r✴❳❛♥t❤✐s ❬✷✵✵✸❪✳

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Finite✲difference interpretation ■♥ ✷✲❞ Λe = ∆ ⊗ D + D ⊗ ∆ ❛♥❞ ✐s t❤❡ ❋✲❉ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r Lu ≡ −2

∂2u ∂x2

+ x2

∂2u ∂x2

x ∈ π1 := (0, 1)2 , u|∂π1 = 0 , ♦♥ t❤❡ sq✉❛r❡ ♠❡s❤ ♦❢ s✐③❡ = 1/(N + 1)✳ ■♥ ✸✲❞✱ −2Λe ✐s t❤❡ ❋✲❉ ❛♣♣r♦①✐♠❛t✐♦♥ ♦♥ t❤❡ s❛♠❡ ♠❡s❤ ♦❢ t❤❡ ✹✲t❤ ♦r❞❡r ♦♣❡r❛t♦r Lu ≡ x2

3u,1,1,2,2+x2 2u,1,1,3,3+x2 1u,2,2,3,3 = f(x) ,

x ∈ π1 := (0, 1)3 , u|∂π1 = 0 , ✇❤❡r❡✱ ❡✳❣✳✱ u,1,1,2,2 = ∂4u/∂x2

1∂x2 2✳

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FEM preconditioner ❙✉♣♣♦s❡✱ d = 3✱

V (π1) ✐s t❤❡ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ♦♥ π1 ❛♥❞ ♣✐❡❝❡

✇✐s❡ tr✐❧✐♥❡❛r ♦♥ ❡❛❝❤ ❝❡❧❧ ♦❢ t❤❡ ❝✉❜✐❝ ♠❡s❤ ❢✉♥❝t✐♦♥s✱ ✈❛♥✐s❤✐♥❣ ♦♥ ∂π1✱ ❛♥❞ Λe,fem ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤✐s s♣❛❝❡ ♠❛tr✐① ♦❢ ❜✐❧✐♥❡❛r ❢♦r♠ bπ1(u, v) =

ϕku,k+1,k+2v,k+1,k+2dx , ϕk = x2

k .

Lemma 2. ❚❤❡ ♠❛tr✐① 1

ℏΛe,fem ✐s s♣❡❝tr❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ Aabc ❛♥❞

Λe ✉♥✐❢♦r♠❧② ✐♥ p✳ Pr♦♦❢✳ ❙❡❡✱ e.g✱ ❑♦r♥❡❡✈ ❬✷✵✵✷❪✳

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■♥ ✷✲❞✱ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ ❋❊ s♣❛❝❡

V△ (π1) ♦❢ ❝♦♥t✐♥✉♦✉s ❛♥❞ ♣✐❡❝❡

✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦♥ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥✱ ♦❜t❛✐♥❡❞ ❜② s✉❜❞✐✈✐s✐♦♥ ♦❢ ❡❛❝❤ sq✉❛r❡ ♥❡st ♦❢ t❤❡ ♠❡s❤ ✐♥ t✇♦ tr✐❛♥❣❧❡s✳ Pr❡❝♦♥❞✐t✐♦♥❡r Λe,fem ✐s ♠❛tr✐① ♦❢ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ bπ1(u, v) =

ϕku,3−kv,3−kdx , ♦♥ t❤❡ s♣❛❝❡

V△ (π1)✳ ❲❡ ❤❛✈❡ Λe,fem ≍ 2Aabc, 2Λe✳

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Preconditioners for the spectral elements

▲▲ ✭●❛✉ss✲▲♦❜❛tt♦✲▲❡❣❡♥❞r❡✮ ♥♦❞❡s ηi s❛t✐s❢② ❡q✉❛t✐♦♥

(1 − η2

i )P ′ p(ηi) = 0 ,

i = 0, 1, .., p , ✇❤❡r❡❛s ❢♦r ●▲❈ ✭●❛✉ss✲▲♦❜❛tt♦✲❈❤❡❜②s❤❡✈✮ ♥♦❞❡s ✇❡ ❤❛✈❡ ηi = cos (π p(p − i)) , i = 0, 1, .., p . ❖rt❤♦❣♦♥❛❧ t❡♥s♦r ♣r♦❞✉❝t ❣r✐❞ x = ηα = (ηα1, ηα2, .., ηαd), α ∈ ω , ✇✐t❤ ●▲❈ ♦r ●▲❈ ♥♦❞❡s ✐s t❡r♠❡❞ ●❛✉ss✐❛♥✱ ✇❤❡r❡❛s ❜♦t❤ t②♣❡s ♦❢ t❤❡ ▲❛❣r❛♥❣❡ r❡❢❡r❡♥❝❡ ❡❧❡♠❡♥ts ❛r❡ t❡r♠❡❞ ✭❢♦r ❜r❡✈✐t②✮ s♣❡❝tr❛❧✳ ■♥ t❤❡✐r ❝♦♦r❞✐♥❛t❡ ♣♦❧②♥♦♠✐❛❧s Lα(x) = Lα1(x1)Lα2(x2)...Lαd(xd)✱ ✶✲❞ ♣♦❧②♥♦✲ ♠✐❛❧s s❛t✐s❢② Li(ηj) = δi,j, 0 ≤ j ≤ p✱ ✇❤❡r❡ δi,j ✐s t❤❡ ❑r♦♥❡❝❦❡r✬s ❞❡❧t❛✳

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❋♦r st❡♣s ℏi := ηi − ηi−1✱ i ≤ N, ♦❢ t❤❡ ●❛✉ss✐❛♥ ♠❡s❤✱ ✇❡ ❤❛✈❡ ℏi ≍ i/p2✳ ▼❡s❤ ♦❢ ❛ ♠♦r❡ ❣❡♥❡r❛❧ ❝❧❛ss s❛t✐s❢② c1

iγ ℵ ≤ ℏi ≤ c2 iγ ℵ ,

ℵ = N

i=1 iγ,

γ ≥ 0, ♦♥ ❬✲✶✱✵❪ ❛♥❞ ✐s ❝♦♥t✐♥✉❡❞ ♦♥ ❬✵✱✶❪ ❜② s②♠♠❡tr②✳ ❆t γ = 0 ⇒ ℵ = N ✕ q✉❛s✐✉♥✐❢♦r♠ ♠❡s❤✱ ❛t γ = 1 ⇒ ℵ = N(N + 1)/2 ✕ ♠❡s❤✱ ❝❛❧❧❡❞ ♣s❡✉❞♦s♣❡❝tr❛❧✱ ❢♦r ✇❤✐❝❤ ❛t c1 = c2 = 1✱ ✇❡ ❤❛✈❡ ℏi = i/ℵ = i (N 2 + N) = β(p) i/p2 , β ∈ [4, 8] . ASp, APsp ✕ ♥♦t❛t✐♦♥s ❢♦r r❡❢✳ ❡❧✳ st✐✛♥❡ss ♠❛tr✐❝❡s ❢♦r ●❛✉ss✐❛♥ ❛♥❞ ♣s❡✉❞♦s♣❡❝tr❛❧ ♥♦❞❡s✱ r❡s♣❡❝t✐✈❡❧②✱ ASp, APsp ✕ ♥♦t❛t✐♦♥s ❢♦r ♣r❡❝♦♥❞✐t✐♦♥❡rs✱ ✇❤✐❝❤ ❛r❡ ❋❊ ♠❛tr✐❝❡s✱ ✐♥✲ ❞✉❝❡❞ ❜② t❤❡ s♣❛❝❡ H(τ0) ∩ C(τ 0) ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ❜❡❧♦♥❣✐♥❣ t♦ Q1,x ♦♥ ❡❛❝❤ sq✉❛r❡ ♥❡st ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❡s❤✳

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❙✐♠♣❧❡r ♣r❡❝♦♥❞✐t✐♦♥❡r Aℏ = ∆ℏ ⊗ Dℏ ⊗ Dℏ + Dℏ ⊗ ∆ℏ ⊗ Dℏ + Dℏ ⊗ Dℏ ⊗ ∆ℏ , ✇❤❡r❡ Dℏ = ❞✐❛❣ [ hi = 1 2(ℏi + ℏi+1)]p

i=0 ,

hi = 0 for i = 0, p + 1 ,

❛♥❞ ∆ℏ ✐s ❋❊ ♠❛tr✐①✿ (∆ℏ u)|i = − 1 ℏi ui−1 + ( 1 ℏi + 1 ℏi+1 )ui − 1 ℏi+1 ui+1 , i = 1, 2, .., p − 1 , (∆ℏ u)|i=0 = − 1 ℏ1 (u1 − u0) , (∆ℏ u)|i=p = 1 ℏp (up − up−1) .

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Lemma 3. ▲❡t Aℏ ❜❡ ♦❜t❛✐♥❡❞ ♦♥ ●❛✉ss✐❛♥ ♦r ♣s❡✉❞♦s♣❡❝tr❛❧ ✭γ = 1✮ ♠❡s❤✳ ❙t✐✛♥❡ss ♠❛tr✐① ASp ♦❢ t❤❡ s♣❡❝tr❛❧ r❡❢❡r❡♥❝❡ ❡❧❡♠❡♥t ❛♥❞ ♠❛tr✐❝❡s APsp, Aℏ ❛r❡ s♣❡❝tr❛❧❧② ❡q✉✐✈❛❧❡♥t ✉♥✐❢♦r♠❧② ✐♥ p✱ ✐✳❡✳✱ APsp, ASp, Aℏ ≺ ASp ≺ Aℏ, ASp, APsp . ▲❡t MSp ❜❡ ♠❛ss ♠❛tr✐① ♦❢ s♣❡❝tr❛❧ ❡❧❡♠❡♥t✱ MSp✱ MP/Sp ❜❡ ✐ts ❋❊ ♣r❡❝♦♥❞✐t✐♦♥❡rs✱ ❣❡♥❡r❛t❡❞ ❜② s♣❛❝❡ H(τ0) ♦♥ ●❛✉ss✐❛♥ ♦r ♣s❡✉❞♦s♣❡❝✲ tr❛❧ ♠❡s❤ ✱ ❛♥❞ Mℏ := Dℏ ⊗ Dℏ ⊗ Dℏ✳ ❚❤❡♥ ✉♥✐❢♦r♠❧② ✐♥ p MPsp, MSp, Mℏ ≺ MSp ≺ Mℏ, MSp, MPsp . Pr♦♦❢✳ ▼♦st ✐♠♣♦rt❛♥t ❝♦♥tr✐❜✉t✐♦♥ ❜② ❇❡r♥❛r❞✐✴▼❛❞❛② ❬✶✾✾✷❪✱✇❤♦ st✉❞✐❡❞ ✶✲❞ ❝❛s❡✳ ❙t❡♣ t♦ ♠♦r❡ ❞✐♠❡♥s✐♦♥s ✐♥ ❈❛♥✉t♦ ❬✶✾✾✹❪ ❈❛s❛r✐♥ ❬✶✾✾✼❪✳

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◆♦t❡ ✿ ✕ ✐♥ t❤❡ ♠✉❧t✐✲❞ ♣r❡❝♦♥❞✐t✐♦♥❡r Λe ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ r❡❢✳ ❡❧✳ Ehi✱ ♠❛tr✐❝❡s ∆ ❛♥❞ D ❛r❡ ♣r❡❝♦♥❞✐t✐♦♥❡rs ❢♦r t❤❡ mass and stiffness ♠❛tr✐❝❡s ✐♥ ✶✲❞✱ r❡s♣❡❝t✐✈❡❧②✱ ✕ ✇❤❡r❡❛s✱ ✐♥ t❤❡ ♠✉❧t✐✲❞ ♣r❡❝♦♥❞✐t✐♦♥❡r Aℏ ❢♦r s♣❡❝tr❛❧ r❡❢✳ ❡❧✳ Esp✱ ♠❛tr✐❝❡s ∆ℏ ❛♥❞ Dℏ ❛r❡ ♣r❡❝♦♥❞✐t✐♦♥❡rs ❢♦r t❤❡ stiffness and mass ♠❛tr✐❝❡s ✐♥ ✶✲❞✳

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Factored preconditioners for spectral elements ▲❡t ✉s ✐♥tr♦❞✉❝❡ (p − 1) × (p − 1) ♠❛tr✐❝❡s ∆Sp = tridiag [−1, 2, −1] , DSp = tridiag [1, 4, .., N 2, (N − 1)2, (N − 2)2, .., 4, 1] , (p − 1)3 × (p − 1)3 ♠❛tr✐❝❡s

ΛI,Sp = DSp ⊗ DSp ⊗ (∆Sp + D−1

Sp) + DSp ⊗ (∆Sp + D−1 Sp) ⊗ DSp+

(∆Sp + D−1

Sp) ⊗ DSp ⊗ DSp ,

ΛI,Sp = DSp ⊗ DSp ⊗ ∆Sp + DSp ⊗ ∆Sp ⊗ DSp + ∆Sp ⊗ DSp ⊗ DSp , diagonal transformation (p − 1)3 × (p − 1)3 matrix C = p−4 D−1/2

ℏ

⊗ D−1/2

ℏ

⊗ D−1/2

ℏ

(C = p−2 D−1/2

ℏ

⊗ D−1/2

ℏ

for 2−d) ,

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❛♥❞ ♠❛tr✐❝❡s ∆ℏ = D1/2

ℏ ∆ℏ D1/2 ℏ

❛♥❞

Aℏ := C−1AℏC−1 = p8 D1/2

ℏ

⊗ D1/2

ℏ

⊗ D1/2

ℏ Aℏ D1/2 ℏ

⊗ D1/2

ℏ

⊗ D1/2

ℏ

= p8 D2

ℏ ⊗ D2 ℏ ⊗

∆ℏ + D2

ℏ ⊗

∆ℏ ⊗ D2

ℏ + D2 ℏ ⊗ D2 ℏ ⊗

∆ℏ

.
Theorem1. ■❢ ♠❛tr✐❝❡s

AI,ℏ, ΛI,Sp, ΛI,Sp ❛r❡ ♦❜t❛✐♥❡❞ ♦♥ ●❛✉ss✐❛♥ ♦r ♣s❡✉❞♦s♣❡❝tr❛❧ ♠❡s❤ ℏi ≍ i/p2 ❢♦r 1 ≤ i ≤ N✱ t❤❡♥ t❤❡② ❛r❡ s♣❡❝tr❛❧❧② ❡q✉✐✈❛❧❡♥t ✉♥✐❢♦r♠❧② ✐♥ p✳ Pr♦♦❢✳ ❑♦r♥❡❡✈✴❘②t♦✈ ❬✷✵✵✺❪✳ ❈♦r♦❧❧❛r② ✶✳ ▲❡t ΛI,C := CΛI,SpC ❛♥❞ ΛI,C := C ΛI,SpC✳ ❯♥❞❡r ❝♦♥❞✐t✐♦♥s ♦❢ ❚❤❡♦r❡♠ ✶ ΛI,C, ΛI,C ≺ AI,Sp ≺ ΛI,C, ΛI,C .

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Finite − difference interpretation ▼❛tr✐① ΛI,Sp ✐s ✼✲♣♦✐♥t ❋✲❉ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ ❞✐✛✳ ♦♣❡r❛t♦r LSpu = −

φ2(x2)φ2(x3)u,1,1 + φ2(x1)φ2(x3)u,2,2 + φ2(x1)φ2(x2)u,3,3
,

❛t u|∂τ0 = 0 ❛♥❞ φ(x) = min(x + 1, x − 1)✳ ■♥❞❡❡❞✱ ❢♦r = 2/p✱ φi = φ(−1 + i) ❛♥❞ u = (ui)p−1

i1,i2,i3=1✱

ΛI,spu|i = − 1 2

k=1,2,3

φ2

ik+1φ2 ik+2[ui−ek−2ui+ui+ek] , 1 ≤ i1, i2, i3 ≤ (p−1) ,

✇❤❡r❡ i = (i1, i2, i3)✱ ✐♥❞✐❝❡s k, k + 1, k + 2 ❛r❡ ✉♥❞❡rst♦♦❞ ♠♦❞✉❧♦ ✸✱ ek = (δk,l)3

l=1 ✐s t❤❡ ✉♥✐t❡ ✈❡❝t♦r✳ ❋♦r d = 2✱

LSpu = −

φ2(x2)u,1,1 + φ2(x1)u,2,2
,

u|∂τ0 = 0 , ΛI,spu|i = −

k=1,2 φ2 i3−k[ui−ek − 2ui + ui+ek] ,

i = (i1, i2) .

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Finite element preconditioners ▲❡t d = 2✳ ❲❡ ❞✐✈✐❞❡ sq✉❛r❡ ♥❡sts ♦❢ s✐③❡ ✐♥ ♣❛✐rs ♦❢ tr✐❛♥❣❧❡s ❛♥❞✱ ♦♥ s✉❝❤ tr✐❛♥❣✉❧❛t✐♦♥✱ ✐♥tr♦❞✉❝❡ t❤❡ s♣❛❝❡

V△ (τ0) ∈ C(τ 0) ♦❢ ♣✐❡❝❡ ✇✐s❡

❧✐♥❡❛r ❢✉♥❝t✐♦♥s✱ ✈❛♥✐s❤✐♥❣ ♦♥ ∂τ0✳ ❚❤❡ ❋❊ ♣r❡❝♦♥❞✐t✐♦♥❡r BI,sp ✐s t❤❡ ♠❛tr✐① ♦❢ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ bτ0(u, v) =

φ2

3−ku,kv,k dx

♦♥ t❤✐s s♣❛❝❡✳ ■♥ ✷ ❛♥❞ ✸✲❞✱ BI,sp ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② t❤❡ ❋❊ s♣❛❝❡s ♦❢ ❜✐❧✐♥❡❛r ❛♥❞ tr✐❧✐♥❡❛r ❢✉♥❝t✐♦♥s✱ r❡s♣❡❝t✐✈❡❧②✳ ❲❡ ❤❛✈❡ BI,sp ≍ 4−dΛI,Sp .

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Comparison

❆t d = 2 ✐♥ ❡❛❝❤ q✉❛rt❡r ♦❢ τ0✱ ♦♣❡r❛t♦r LSp ❝♦✐♥❝✐❞❡s ✇✐t❤ L ✉♣ t♦

t❤❡ ❝♦♥st❛♥t ♠✉❧t✐♣❧✐❡r ✭❛♥❞ r♦t❛t✐♦♥ ❛♥❞ tr❛♥s♣♦rt❛t✐♦♥ ♦❢ t❤❡ ❛①❡s✮✳

❚❤❡ s❛♠❡ ✐s tr✉❡ ❢♦r ❋✲❉ ♦♣❡r❛t♦rs Λe, ΛI,sp✳
❆t d = 3✱ ❞✐✛❡r❡♥t✐❛❧ ❛♥❞ ❋✲❉ ♦♣❡r❛t♦rs ❛r❡ ❞✐✛❡r❡♥t ❡✈❡♥ ✐♥ t❤❡ ♦r❞❡r✿

L ✐s ♦❢ ✹✲t❤ ♦r❞❡r✱ ✇❤❡r❡❛s LSp ✐s ♦❢ ✷✲♥❞✳

❍♦✇❡✈❡r✱ ♠✉❧t✐♣❧✐❡rs ∆, D ❛♥❞ r❡s♣❡❝t✐✈❡❧② DSp, ∆Sp ✐♥ r❡♣r❡s❡♥t❛✲

t✐♦♥s ♦❢ Λe, ΛI,sp ❜② s✉♠s ♦❢ ❑r♦♥❡❝❦❡rs ♣r♦❞✉❝ts ❛r❡ s✐♠✐❧❛r✳

❆♥ ❛❞❞✐t✐♦♥❛❧ ❞✐✣❝✉❧t② ❢♦r ❞❡r✐✈✐♥❣ ❢❛st s♦❧✈❡rs ❢♦r ✸✲❞ ❤✐❡r❛r❝❤✐❝❛❧

❡❧❡♠❡♥ts ❞✐r❡❝t❧② ♦♥ t❤❡ ❜❛s✐s ♦❢ Λe ✐s t❤❛t ✐t ✐s ❛ ❋✲❉ ❛♥❛❧♦❣✉❡ ♦❢ ✹✲t❤ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r✳ ▼♦r❡ ♦✈❡r✱ t❤✐s ♦♣❡r❛t♦r ❝♦♥t❛✐♥s ♦♥❧② ♠✐①❡❞ ❞❡r✐✈❛t✐✈❡s✳ ❚❤❡ ✉s❡ ♦❢ t❤❡ s♣❡❝tr❛❧ ❡❧❡♠❡♥ts ❛♥❞ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡r ΛI,C s✐♠♣❧✐✜❡s t❤❡ ♣r♦❜❧❡♠ ❜② r❡❞✉❝✐♥❣ ✐t t♦ ❞❡s✐❣♥✐♥❣ ❛ ❢❛st s♦❧✈❡r ❢♦r ΛI,sp✱ ✇❤✐❝❤ ✐s t❤❡ ❋✲❉ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ✷✲♥❞ ♦r❞❡r ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r ❝♦♥t❛✐♥✐♥❣ ♦♥❧② ❞❡r✐✈❛t✐✈❡s ∂2/∂x2

k, k = 1, 2, 3.

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Conclusions ⋆All fast solvers for systems with the hier- archical reference element stiffness matrices (

r spectrally equivalent , e.g., Λe) are easily

adjusted into fast solvers for systems with the spectral reference element stiffness matrices or spectrally equivalent to them matrices like ΛI,sp ⋆The arithmetic costs of the latter and the former solvers are the same in the order. ⋆At least, these conclusions are true for the all known fast solvers see, e.g., [K1],[K2],[KA],[B],[BSS], for systems with ma- trices Λe✳

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❊①❛♠♣❧❡ 1

Algebraic multilevel solver for 2d spectral elements ❲❡ s❡t p = 2N, N = 2ℓ0−1 ❛♥❞ ✐♥tr♦❞✉❝❡ s❡q✉❡♥❝❡ ♦❢ ℓ0 ❡♠❜❡❞❞❡❞ ♠❡s❤❡s ♦❢ t❤❡ s✐③❡s l = 2−l, l = 1, 2, .., ℓ0, ✇✐t❤ t❤❡ ♥♦❞❡s x = l(i, j) − (1, 1)✱ s❡q✉❡♥❝❡ ♦❢ s♣❛❝❡s Vl(τ0) ✇✐t❤ Vℓ0(τ0) =

V△ (τ0) ❛♥❞

❋❊ ♠❛tr✐❝❡s Bl ✇✐t❤ Bℓ0 = BI,Sp✳ ❊❛❝❤ s♣❛❝❡ Vl(τ0) ❛♥❞ t❤❡ ♠❛tr✐① Bl ❛r❡ t❤❡ s♣❛❝❡

V△ (τ0) ❛♥❞ t❤❡

♠❛tr✐① BI,Sp ❢♦r t❤❡ ♠❡s❤ ♦❢ t❤❡ ❧❡✈❡❧ l✳

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❆❧s♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t❛t✐♦♥s ❛r❡ ✉s❡❞✿ ✲ Xl ✕ t❤❡ s✉❜s❡t ♦❢ ✐♥t❡r♥❛❧ ♥♦❞❡s✱ ✲ Vl ❛♥❞ Wl ✕ ✈❡❝t♦r✲s♣❛❝❡s✱ r❡❧❛t❡❞ t♦ s✉❜s❡ts ♦❢ ♥♦❞❡s Xl ❛♥❞ XW,l := Xl Xl−1, s♦ t❤❛t Vl = Vl−1 ⊕ Wl = Wl ⊕ Wl−1 ⊕ ... ⊕ W2 ⊕ V1 . ✲ Pl−1 : Vl−1 → Vl ✕ ✉s✉❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ♠❛tr✐① ❢r♦♠ t❤❡ ♠❡s❤ ✧l − 1✧ ♦♥ t❤❡ ♥❡①t ✜♥❡r ♠❡s❤ ✧l✧✳ ✲ Rl : Vl → Wl ✕ r❡str✐❝t✐♦♥ ♠❛tr✐① t♦ t❤❡ s❡t ♦❢ ♥♦❞❡s XW,l✳ ✲ BVl✱ BWl ✕ ❜❧♦❝❦s ♦♥ t❤❡ ❞✐❛❣♦♥❛❧ ♦❢ Bl r❡❧❛t❡❞ t♦ t❤❡ s✉❜s♣❛❝❡s Vl ❛♥❞ Wl✳

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One multilevel iteration ■❢ BWl ✐s ❛ ♣r❡❝♦♥❞✐t✐♦♥❡r ❢♦r BWl, ♦♥❡ ♠✉❧t✐❣r✐❞ ✐t❡r❛t✐♦♥ ❢♦r Blu = F ✱ ♣r♦❞✉❝✐♥❣ uk+1,l := ▼❣♠(l, Bl, F, uk,l) ❢♦r ❛ ❣✐✈❡♥ uk,l ✐s✿ If l > 1, then do Pr❡✲s♠♦♦t❤✐♥❣ ✐♥ t❤❡ s✉❜s♣❛❝❡ Wl✿ v := uk,l ; do ν times v := v − σ−1

l R⊤ l B−1 WlRl(Blv − F) ❀

❈♦rr❡❝t✐♦♥ ♦❢ t❤❡ s♦❧✉t✐♦♥ ♦♥ t❤❡ ❧♦✇❡r ❧❡✈❡❧ ✐♥ t❤❡ s♣❛❝❡ Vl−1✿ dl−1 := P∗

l−1(F − Blv) ; w = 0 ;

do µl−1 iterations w = Mgm(l − 1, Bl−1, dl−1, w) ❀ v := v + Pl−1w ❀ P♦st✲s♠♦♦t❤✐♥❣ ✐♥ t❤❡ s✉❜s♣❛❝❡ Wl✿ do ν times v := v − σ−1

l R⊤ l B−1 WlRl(Blv − F) ❀

uk+1,l = v else, then solve Blu = F by the exact method

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▲✐♥❡s ℑj ❛❧♦♥❣ ✇❤✐❝❤ s♠♦♦t❤✐♥❣ ✐s ♣❡r❢♦r♠❡❞ ✐♥ t❤❡ ♠✉❧t✐❣r✐❞ s♦❧✈❡rs ❢♦r s♣❡❝tr❛❧ ✭❧❡❢t✮ ❛♥❞ ❤✐❡r❛r❝❤✐❝❛❧ r❡❢❡r❡♥❝❡ ❡❧❡♠❡♥ts ✭r✐❣❤t✮

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decoupling nodes of the (l-1)-th level additional nodes of the l-th level diagonal block in the preconditioner tridiagonal block in the preconditioner

▲✐♥❡ ♣r❡❝♦♥❞✐t✐♦♥✐♥❣✳

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Two factors influence efficiency : ✶✮ ❡✣❝✐❡♥❝② ♦❢ ♣r❡❝♦♥❞✐t✐♦♥❡rs BWl✱ ✐✳❡✳✱ t❤❡ ✈❛❧✉❡s ♦❢ ck > 0 ✐♥ t❤❡ ✐♥❡q✉❛❧✐t✐❡s c1BWl ≤ BWl ≤ c2BWl , ❛♥❞ t❤❡ ❝♦st ♦❢ s♦❧✈✐♥❣ s②st❡♠s ✇✐t❤ t❤❡ ♠❛tr✐❝❡s BWl✳ ✷✮ t❤❡ ✈❛❧✉❡ ♦❢ c0 ✐♥ t❤❡ str❡♥❣t❤❡♥❡❞ ❈❛✉❝❤② ✐♥❡q✉❛❧✐t② (bτ0(u, v))2 ≤ c0 bτ0(u, u)bτ0(v, v), c0 < 1, ∀u ∈ Vl−1, ∀v ∈ Wl , ✇❤❡r❡ Wl(τ0) := Vl(τ0) ⊖ Vl−1(τ0)✳ Lemma 4✳ c1 ≥ 1 − 2/ √ 11, c2 ≤ 1 + 2/ √ 11, c0 ≤ 97/176 < 2/3 . Pr♦♦❢✳ ❘❡♣❡❛ts t❤❡ ♣r♦♦❢ ♦❢ ❇❡✉❝❤❧❡r ❬✷✵✵✷❪ ❢♦r ❤✐❡r❛r❝❤✐❝❛❧ r❡❢❡r❡♥❝❡ ❡❧❡♠❡♥t✳

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Convergence of the multigrid iterations Theorem 2 ✭❑♦r♥❡❡✈✴❘②t♦✈ ❬✷✵✵✺❪✮✳ ▲❡t Blu = F ❜❡ s♦❧✈❡❞ ❜② t❤❡ ♠✉❧t✐❣r✐❞ ♠❡t❤♦❞ ✐♥ ✇❤✐❝❤ σ = 2/(c1 + c2), µ ≥ 3 ❛♥❞ ν ❜❡ ❣r❡❛t❡r t❤❛♥ s♦♠❡ νo(c0, c1, c2)✳ ❚❤❡♥ t❤❡ ❝♦♥✈❡r❣❡♥❝❡ ❢❛❝t♦r ρl,mult := sup uk∈Ul uk+1 − u Bl/uk − u Bl ✐s ❜♦✉♥❞❡❞ ❜② t❤❡ ❝♦♥st❛♥t ρ < 1 ✐♥❞❡♣❡♥❞❡♥t ♦❢ p, l ❛♥❞ uk✳ Pr♦♦❢✳ ❋♦❧❧♦✇s ❢r♦♠ r❡s✉❧ts ♦❢ ❙❝❤✐❡✇❡❝❦ ❬✶✾✽✺❪ ❛♥❞ P✢❛✉♠ ❬✷✵✵✵❪ ❛♥❞ ▲❡♠♠❛ ✹✳

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Multigrid iteration as a preconditioner ▲❡t Mµ ❜❡ t❤❡ ❧✐♥❡❛r ❡rr♦r tr❛♥s♠✐ss✐♦♥ ♦♣❡r❛t♦r ❢♦r ♦♥❡ ♠✉❧t✐❣r✐❞ ✐t❡r❛t✐♦♥ ❢♦r s②st❡♠ BI,spu = F✳ ❚❤❡♥ κ ♠✉❧t✐❣r✐❞ ✐t❡r❛t✐♦♥s ✐♠♣❧✐❝✐t❧② ❞❡✜♥❡ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡r MgSp ❢♦r ΛI,C ❛♥❞ AI,sp✱ t❤❡ ✐♥✈❡rs❡ t♦ ✇❤✐❝❤ ✐s Mg−1

sp = −2C(I − Mκ µ)B−1 I,spC✱ C = p−2D−1/2

⊗ D−1/2
✳

Theorem 3 ✭❑♦r♥❡❡✈✴❘②t♦✈ ❬✷✵✵✺❪✮✳ ▲❡t µ = 3✱ ν ≥ 3 ❛♥❞ κ ≥ 1✳ ❚❤❡♥ c Mg−1

sp ≤ A−1 I,sp ≤ c Mg−1 sp ,

✇✐t❤ ❝♦♥st❛♥ts c, c > 0 ✐♥❞❡♣❡♥❞❡♥t ♦❢ p ✭ ❛♥❞ κ✮✳ ❚❤❡ ♣r♦❝❡❞✉r❡ ♦❢ t❤❡ ♠❛tr✐①✲✈❡❝t♦r ♠✉❧t✐♣❧✐❝❛t✐♦♥ ❜② Mg−1

sp r❡q✉✐r❡s O(p2) ❛r✐t❤♠❡t✐❝

♦♣❡r❛t✐♦♥s✳

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❊①❛♠♣❧❡ 2

Multiresolution wavelet solver for 3d spectral elements ❙✐♥❝❡✱ ❡✳❣✳✱ ΛI,Sp ✐s ❛ s✉♠ ♦❢ ❑r♦♥❡❝❦❡r ♣r♦❞✉❝ts ♦❢ ♠❛tr✐❝❡s ∆Sp, DSp r❡❧❛t❡❞ t♦ ✶✲❞ ✐♥t❡❣r❛❧s✱ ❢❛st s♦❧✈❡r ❢♦r ΛI,Sp ✐s ❝♦♥str✉❝t❡❞ ❜② ❞❡r✐✈✐♥❣ ♠✉❧t✐❧❡✈❡❧ ♣r❡❝♦♥❞✐t✐♦♥❡rs ❢♦r t❤❡s❡ ♠❛tr✐❝❡s✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ s❡t ❛❣❛✐♥ p = 2N, N = 2ℓ0−1, ❛♥❞ ❢♦r l = 1, 2, ..., l0 ✐♥tr♦❞✉❝❡

✉♥✐❢♦r♠ ♠❡s❤ ♦❢ s✐③❡ ℏl = 21−l ♦♥ t❤❡ ✐♥t❡r✈❛❧ (−1, 1)

xl

i = −1 + iℏl, i = 0, 1, 2, .., 2Nl,

x0 = −1, x2Nl = 1, Nl = 2l−1

s♣❛❝❡ Vl(−1, 1) ♦❢ ❝♦♥t✐♥✉♦✉s ♣✐❡❝❡ ✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s✱ ✈❛♥✐s❤✐♥❣ ❛t

x = −1, 1✱

♥♦❞❛❧❂❤❛t ❜❛s✐s ❢✉♥❝t✐♦♥ σl

i ∈ Vl(−1, 1)✱ s✉❝❤ t❤❛t σl i(xl j) = δi,j ❛♥❞

Vl(−1, 1) = span

pl−1

i=1 ,

pl = 2l,

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●r❛♠ ♠❛tr✐❝❡s ✐♥ t❤❡ ♥♦❞❛❧ ❜❛s✐s

∆l = ℏl 1

−1

(σl

i)′, (σl j)′

pl−1

i,j=1

, Ml = ℏ−1

1

−1

φ2 σl

i, σl j

pl−1

i,j=1

,

s✐♥❣❧❡ s❝❛❧❡ ✇❛✈❡❧❡t ❜❛s✐s (ψl

k)pl−1 k=1 ✐♥ t❤❡ s♣❛❝❡ Wl := Vl ⊖ Vl−1✱ s♦

t❤❛t Wl = span [ ψl

k ]pl−1 k=1✱

♠✉❧t✐s❝❛❧❡ ✇❛✈❡❧❡t ❜❛s✐s (ψl

k )pl−1, l0 k,l=1 ✱ ❝♦♠♣♦s❡❞ ♦❢ s✐♥❣❧❡ s❝❛❧❡ ❜❛s❡s ❛❝✲

❝♦r❞✐♥❣ t♦ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ V = W1 ⊕ W2 ⊕ .. ⊕ Wl1, where V = Vl0, W1 = V1,

●r❛♠ ♠❛tr✐❝❡s ✐♥ t❤❡ ♠✉❧t✐s❝❛❧❡ ✇❛✈❡❧❡t ❜❛s✐s

∆wlet =

(ℏkℏl)1/2 1

−1(ψk i )′, (ψl j)′ dx

pl−1; l0

i,j=1; k,l=1 ,

Mwlet =

(ℏkℏl)−1/2 1

−1 φ2 ψk i , ψl j dx

pl−1; l0

i,j=1; k,l=1 ,

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❞✐❛❣♦♥❛❧ ♠❛tr✐❝❡s ✇✐t❤ t❤❡ ♠❛✐♥ ❞✐❛❣♦♥❛❧s ❢r♦♠ ∆wlet ❛♥❞ Mwlet

D1 = diag [ℏl 1

−1

((ψl

i)′)2 dx] pl−1,l0 i,l=1 ,

D0 = diag [ℏ−1

1

−1

φ2(ψl

i)2 dx] pl−1,l0 i,l=1 .

❚❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ♠❛tr✐① ❢r♦♠ t❤❡ ♠✉❧t✐s❝❛❧❡ ✇❛✈❡❧❡t ❜❛s✐s t♦ t❤❡ ❜❛✲ s✐s (σl0

k )p−1 k=1 ✐s ❞❡♥♦t❡❞ ❜② Q✳ ■❢ v ❛♥❞ vwavelet ❛r❡ t❤❡ ✈❡❝t♦rs ♦❢ t❤❡

❝♦❡✣❝✐❡♥ts ♦❢ ❛ ❢✉♥❝t✐♦♥ ❢r♦♠ V(0, 1) ✐♥ t❤❡ ♦♥❡ s❝❛❧❡ ♥♦❞❛❧ ❛♥❞ t❤❡ ♠✉❧t✐s❝❛❧❡ ✇❛✈❡❧❡t ❜❛s❡s✱ r❡s♣❡❝t✐✈❡❧②✱ t❤❡♥ v = Q vwavelet✳ Theorem 4. ❚❤❡r❡ ❡①✐st ✇❛✈❡❧❡t ❜❛s❡s (ψl

k)pl−1,l0 k,l=1 s✉❝❤ t❤❛t ♠❛tr✐❝❡s

∆wlet ❛♥❞ Mwlet ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② s♣❡❝tr❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡✐r ❞✐✲ ❛❣♦♥❛❧s D1 ❛♥❞ D0✱ r❡s♣❡❝t✐✈❡❧②✱ ✭✉♥✐❢♦r♠❧② ✐♥ p✮ ❛♥❞ ♠✉❧t✐♣❧✐❝❛t✐♦♥s Q vwlet ❛♥❞ QT v r❡q✉✐r❡ O(p) ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s✳ Pr♦♦❢✳ ❇❛s✐❝❛❧❧② ✐t ✐s t❤❡ s❛♠❡ ❛s t❤❡ ♣r♦♦❢ ♦❢ ❛ s✐♠✐❧❛r r❡s✉❧t ❜② ❇❡✉❝❤❧❡r✴❙❝❤♥❡✐❞❡r✴❙❝❤✇❛❜ ❬✷✵✵✹❪ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❤✐❡r❛r❝❤✐❝❛❧ ❡❧❡♠❡♥t✳

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Theorem 5. ▲❡t A−1

I,sp←w =

         (QT ⊗ QT )[ D0 ⊗ D1 + D1 ⊗ D0 ]−1(Q ⊗ Q), d = 2, (QT ⊗ QT ⊗ QT)[D0 ⊗ D1 ⊗ D1 + D1 ⊗ D0 ⊗ D1+ D1 ⊗ D1 ⊗ D0]−1(Q ⊗ Q ⊗ Q), d = 3 , t❤❡♥ AI,sp←w ≍ AI ❛♥❞ t❤❡r❡❢♦r❡ cond [A−1

I,sp←wAI] ≺ 1 .

❚❤❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❝♦st ♦❢ t❤❡ ♦♣❡r❛t✐♦♥ A−1

I,sp←wv ❢♦r ❛♥② v ✐s

ps [A−1

I,sp←wv] = O(pd)✳

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Laboratory of New Computational Technologies

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❊①❛♠♣❧❡ 3

Multiresolution wavelet solver for faces

♦♦❞ ♠❛st❡r ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ❢♦r ♦♥❡ ❢❛❝❡ s✉❜♣r♦❜❧❡♠ ♠❛② ❜❡

♠❛tr✐① s♣❡❝tr❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ♠❛tr✐① ♦❢ t❤❡ ♥♦r♠

00| v |2 1/2,F0 = | v |2 1/2,F0 +

|v(x)|2 dist [x, ∂F0] dx , ∀ v ∈

Qp,x ,

❢♦r ❛ t②♣✐❝❛❧ ❢❛❝❡ F0 = (−1, 1) × (−1, 1) ♦❢ t❤❡ r❡❢❡r❡♥❝❡ ❡❧❡♠❡♥t✳ ❇② ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s d0,i, d1,i ♦❢ D0, D1✱ r❡s♣❡❝t✐✈❡❧②✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❞✐❛❣♦♥❛❧ (2N − 1)2 × (2N − 1)2 ♠❛tr✐① D1/2 ✇✐t❤ ❞✐❛❣♦♥❛❧ ❡♥tr✐❡s d(1/2)

i,j

= d0,id0,j

d1,i/d0,i + d1,j/d0,j .

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St.-Petersburg State Polytechnical University

Laboratory of New Computational Technologies

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Theorem 6 ✭❑♦r♥❡❡✈✴❘②t♦✈ ❬✷✵✵✺❪✮✳ ▲❡t S−1

0 = (Q⊤ ⊗ Q⊤) D−1 1/2 (Q ⊗ Q) ,

S0 = C S0 C . ❚❤❡♥ ❢♦r ❛❧❧ v ∈

Qp,x ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡❝t♦rs v✱ t❤❡ ♥♦r♠s

00| v |1/2,τ0 ❛♥❞ ||v||S 0✱ r❡s♣❡❝t✐✈❡❧②✱ ❛r❡ ❡q✉✐✈❛❧❡♥t ✉♥✐❢♦r♠❧② ✐♥ p✱ ✐✳❡✳✱ 00| v |1/2,τ0 ≍ ||v||S 0 .

Pr♦♦❢✳ ❇❛s✐s t♦♦❧ ✐s P❡❡tr❡✬s ❑✲✐♥t❡r♣♦❧❛t✐♦♥ ♠❡t❤♦❞✳ S0 ✐s ❛ ♠✉❧t✐s❝❛❧❡ ✇❛✈❡❧❡t ♣r❡❝♦❞✐t✐♦♥❡r ❢♦r ✇❤✐❝❤ ops [S−1

0 v]

= O(p2), ∀v✱ ❛♥❞✱ t❤❡r❡❢♦r❡✱ ops [S−1

0 v] = O(p2) ❛s ✇❡❧❧✳

❙✐♠✐❧❛r ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ❢♦r ❢❛❝❡s ♦❢ ❤✐❡r❛r❝❤✐❝❛❧ ❡❧❡♠❡♥ts ✇❛s ❛♣♣r♦✈❡❞ ✐♥ ❑♦r♥❡❡✈✴▲❛♥❣❡r✴❳❛♥t❤✐s ❬✷✵✵✸❪✳

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St.-Petersburg State Polytechnical University

Laboratory of New Computational Technologies

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DOMAIN DECOMPOSITION ALGORITHM ❚❤❡ ♣r♦❜❧❡♠ t♦ ❜❡ s♦❧✈❡❞ aΩ(u, v) :=

̺(x)∇u · ∇v dx = (f, v)Ω , ∀ v ∈

H 1(Ω) ,

✐♥ t❤❡ ❞♦♠❛✐♥ Ω = ∪R

r=1τ r , ✇❤✐❝❤ ✐s ❛♥ ❛ss❡♠❜❧❛❣❡ ♦❢ ❝♦♠♣❛t✐❜❧❡ ❛♥❞ ✐♥

❣❡♥❡r❛❧ ❝✉r✈✐❧✐♥❡❛r ✜♥✐t❡ ❡❧❡♠❡♥ts ♦❝❝✉♣②✐♥❣ ❞♦♠❛✐♥s τr✳ ■t ✐s ❛ss✉♠❡❞ t❤❛t ✜♥✐t❡ ❡❧❡♠❡♥ts s❛t✐s❢② t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥❞✐t✐♦♥s ♦❢ s❤❛♣❡ r❡❣✉❧❛r✐t②✳ ❚❤❡ ♣♦s✐t✐✈❡ ❝♦❡✣❝✐❡♥t ̺(x) ✐s ❛ss✉♠❡❞ t♦ ❜❡ ♣✐❝❡ ✇✐s❡ ❝♦♥st❛♥t✱ i.e.✱ ̺(x) = ̺r ❢♦r x ∈ τr✳

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St.-Petersburg State Polytechnical University

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❚❤❡ ✜♥✐t❡ ❡❧❡♠❡♥t st✐✛♥❡ss ♠❛tr✐① ♠❛② ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❜❧♦❝❦ ❢♦r♠s K =

KIB KBI KB

  KI KIF KIW KFI KF KFW KWI KWF KWW   =    KI KIF KIE KIV KFI KF KFE KFV KEI KEF KE KEV KV I KV F KV E KV    , where ■ ✕ st❛♥❞s ❢♦r ✐♥t❡r♥❛❧ ❞✳♦✳❢✳✱ ❋ ✕ ❢❛❝❡s✱ ❊ ✕ ❡❞❣❡s✱ ❱ ✕ ✈❡rt✐❝❡s✱ ❇ ✕ ✐♥t❡r❢❛❝❡ ❜♦✉♥❞❛r②✱ ❲ ✕ ✇✐r❡ ❜❛s❦❡t✳

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St.-Petersburg State Polytechnical University

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❲❡ ❝♦♥s✐❞❡r t❤❡ ❉❉ ❉✐r✐❝❤❧❡t✲❉✐r✐❝❤❧❡t ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r K K−1 = K +

I + PVB→V S−1 B P⊤ VB→V ,

S−1

B = S + F + PVW →VB(SB W)−1P⊤ VW →VB .

✭✵✳✶✮ i✮ ❚❤❡ ❜❧♦❝❦ ❞✐❛❣♦♥❛❧ ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ❢♦r t❤❡ ✐♥t❡r♥❛❧ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠s ♦♥ ✜♥✐t❡ ❡❧❡♠❡♥ts ❤❛s t❤❡ ❢♦r♠ K +

I :=

K−1

where KI = ❞✐❛❣ [h1̺1BI,sp, h2̺2BI,sp, . . . , hR̺RBI,sp] BI,sp = AI,sp←w ✕ ♠✉❧t✐r❡s♦❧✉t✐♦♥ ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ♦❢ ❚❤❡♦r❡♠ ✺✳

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St.-Petersburg State Polytechnical University

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ii) ❇❧♦❝❦ ❞✐❛❣♦♥❛❧ ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ❢♦r ✐♥t❡r♥❛❧ ♣r♦❜❧❡♠s ♦♥ ❢❛❝❡s S +

F =

S−1

where SF = ❞✐❛❣ [κ1S0, κ2S0, . . . , κQS0] , ✕ Q ✐s t❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝❡s Fk ⊂ Ω✱ ✕ κk ❛r❡ ♠✉❧t✐♣❧✐❡rs κk = (hr1(k)̺r1(k) + hr2(k)̺r2(k)) , ✇✐t❤ r1(k), r2(k) ❜❡✐♥❣ ♥✉♠❜❡rs ♦❢ t✇♦ ❡❧❡♠❡♥ts τ r1(k) ❛♥❞ τ r2(k)✱ s❤❛r✐♥❣ t❤❡ ❢❛❝❡ Fk✱ ✕ hr ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ s✐③❡ ♦❢ ❛♥ ❡❧❡♠❡♥t✱ ✕ S0 ✐s t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r ❢♦r ♦♥❡ ❢❛❝❡✱ ❞❡✜♥❡❞ ✐♥ ❚❤❡♦r❡♠ ✹✳

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St.-Petersburg State Polytechnical University

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iii) Pr❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r SB

W ❢♦r ✇✐r❡ ❜❛s❦❡t s✉❜♣r♦❜❧❡♠ ♦❢ r❡❧❛t✐✈❡❧②

s♠❛❧❧ ❞✐♠❡♥s✐♦♥ O(Rp) × O(Rp)✳ ❲❡ ❜♦rr♦✇ ✐t ❢r♦♠ ❈❛s❛r✐♥ ❬✶✾✾✼❪ ❛♥❞ P❛✈❛r✐♥♦✴❲✐❞❧✉♥❞ ❬✶✾✾✻❪✱ ❛ss✉♠✐♥❣ t❤❛t ✐ts ❛r✐t❤♠❡t✐❝❛❧ ❝♦st ❞♦❡s ♥♦t ❞✐st✉r❜ ♦♣t✐♠❛❧✐t② ♦❢ ❉❉ s♦❧✈❡r✱ i.e.✱ ♦♣s[(SB

W)−1v] = O(Rp3)✳

The prolongation operations include : iv) ♣r♦❧♦♥❣❛t✐♦♥ PVB→V ❢r♦♠ ✐♥t❡r❡❧❡♠❡♥t ❜♦✉♥❞❛r② ♦♥ t❤❡ ✇❤♦❧❡ ❝♦♠✲ ♣✉t❛t✐♦♥❛❧ ❞♦♠❛✐♥ Ω✱ ❝♦♠♣❧❡t❡❞ ❜② ♠❡❛♥s ♦❢ ✐♥❡①❛❝t s♦❧✈❡r ✇✐t❤ t❤❡ ♣r❡❝♦♥❞✐t✐♦♥❡r BI,sp✱ v) s✐♠♣❧❡ ♣r♦❧♦♥❣❛t✐♦♥ PVW →VB ❢r♦♠ ✇✐r❡ ❜❛s❦❡t ♦♥ ✐♥t❡r❡❧❡♠❡♥t ❜♦✉♥❞✲ ❛r②✱ ♥♦t r❡q✉✐r✐♥❣ s♦❧✉t✐♦♥ ♦❢ ❛♥② s②st❡♠s✱ ✇❤✐❝❤ ✐s t❤❡ s❛♠❡ ❛s ✐♥ P❛✈❛r✐♥♦✴❲✐❞❧✉♥❞ ❬✶✾✾✻❪ ❛♥❞ ❈❛s❛r✐♥ ❬✶✾✾✼❪✳

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St.-Petersburg State Polytechnical University

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Theorem 7✳ ❙✉♣♣♦s❡✱ t❤❡ ❣❡♥❡r❛❧✐③❡❞ ❝♦♥❞✐t✐♦♥s ♦❢ s❤❛♣❡ r❡❣✉❧❛r✐t② ❛r❡ ❢✉❧✜❧❧❡❞ ❛♥❞ t❤❡ ❝♦❡✣❝✐❡♥t ρ > 0 ✐s ♣✐❡❝❡ ✇✐s❡ ❝♦♥st❛♥t✳ ❚❤❡♥ t❤❡ ❜♦✉♥❞ ❢♦r t❤❡ r❡❧❛t✐✈❡ ❝♦♥❞✐t✐♦♥ ♥✉♠❜❡r ♦❢ ❉❉ ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r K ✐s cond [K−1K] ≤ c(1 + log p)2 . ❙✉♣♣♦s❡ ❛❞❞✐t✐♦♥❛❧❧② t❤❛t t❤❡ ✇✐r❡ ❜❛s❦❡t s♦❧✈❡r s❛t✐s❢② t❤❡ ❛❜♦✈❡ ❛s✲ s✉♠♣t✐♦♥ iii)✳ ❚❤❡♥ t❤❡ ♥✉♠❜❡r ♦❢ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s ♥❡❡❞❡❞ ❢♦r s♦❧✈✐♥❣ t❤❡ s②st❡♠ K−1v = f ❤❛s t❤❡ ♠❛❥♦r❛♥t

ps [K−1f] ≤ O(p3(1 + log p)R) ,

∀f .

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CONCLUSIONS ❋❛❝t♦r❡❞ ♣r❡❝♦♥❞t✐♦♥❡rs✱ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ❧❡❝t✉r❡ ❢♦r t❤❡ s♣❡❝tr❛❧ r❡❢❡r❡♥❝❡ ❡❧❡♠❡♥t st✐✛♥❡ss ❛♥❞ ♠❛ss ♠❛tr✐❝❡s✱ ❛❧❧♦✇ t♦ ❞❡s✐❣♥ ❛❧♠♦st ♦♣t✐♠❛❧ ✐♥ ❝♦♠♣✉t❛t✐♦♥❧ ✇♦r❦ ♣r❡❝♦♥❞✐t✐♦♥❡rs✲s♦❧✈❡rs ❢♦r t❤r❡❡ ♠♦st ✐♠✲ ♣♦rt❛♥t s✉❜♣r♦❜❧❡♠s✱ ❛r✐s✐♥❣ ✐♥ ❉❉ ❛❧❣♦r✐t❤♠s ❢♦r ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s ✐♥ ✸❞ ❞♦♠❛✐♥s✳ ■♥❞❡❡❞✱ t✇♦ ♦❢ t❤❡s❡ ♣r❡❝♦♥❞✐t✐♦♥❡rs✲s♦❧✈❡rs ❛r❡ ♦♣t✐♠❛❧✳ ■♥ t❤❡ ♣r❡s❡♥t❡❞ ❉❉ ♣r❡❝♦♥❞✐t✐♦♥❡r✲s♦❧✈❡r✱ ♦♥❧② ♦♥❡ s♣❛rs❡ s✉❜s②st❡♠ ♦❢ t❤❡ r❡❧❛t✐✈❡❧② s♠❛❧❧ ❞✐♠❡♥s✐♦♥ O(R) × O(R)✱ ✇❤✐❝❤ ✐s ❛ ♣❛rt ♦❢ t❤❡ ✇✐r❡ ❜❛s❦❡t s✉❜♣r♦❜❧❡♠✱ ✇❛s ♥♦t s✉♣♣❧✐❡❞ ✇✐t❤ t❤❡ s♦❧✈❡r ♦♣t✐♠❛❧ ✇✐t❤ t❤❡ r❡s♣❡❝t t♦ ✐ts ❞✐♠❡♥s✐♦♥ O(R)✳

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Related publications [AK] Anufriev I and Korneev V. ❋❛st ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♠❡t❤♦❞ ❢♦r ❞❡t❡rr✐♦r❛t✐♥❣ ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥ ♦❢ ✹✲t❤ ♦r❞❡r ✐♥ ✸✲❞✳ ■♥ ♣r♦❝✳ ♦❢ ✺✲t❤ ❆❧❧✲❘✉ss✐❛♥ s❡♠✐♥❛r ▼❡s❤ ♠❡t❤♦❞s ❢♦r ❜♦✉♥❞❛r② ✈❛❧✉❡s ♣r♦❜✲ ❧❡♠s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✱ ❑❛③❛♥✱ ✶✼✲✷✶ ❙❡♣t✳ ✷✵✵✹✳ ❑❛③❛♥ ❙t❛t❡ ❯♥✐✈✳✱ ✷✵✵✹✱ ✽ ✕✶✹✳✭■♥ ❘✉ss✐❛♥✮ [BM1] Bernardi Ch and Maday Y. P♦❧②♥♦♠✐❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ r❡✲ s✉❧ts ✐♥ ❙♦❜♦❧❡✈ s♣❛❝❡s✳ ❏♦✉r♥❛❧ ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡✲ ♠❛t✐❝s✳ ✶✾✾✷❀ ✹✸✿✺✸✕✽✵✳ [BM2] Bernardi Ch and Maday Y. ❆♣♣r♦①✐♠❛t✐♦♥s s♣❡❝tr❛❧❡s ❞❡ ♣r♦❜❧è♠❡s ❛✉① ❧✐♠✐t❡s ❡❧❧✐♣t✐q✉❡s ✭▼❛t❤✳ ❆♣♣❧✳ ✶✵✮✱ ❙♣r✐♥❣❡r✲❱❡r❧❛❣✿ P❛r✐s✱ ✶✾✾✷✳ [B] Beuchler S. ▼✉❧t✐❣r✐❞ s♦❧✈❡r ❢♦r t❤❡ ✐♥♥❡r ♣r♦❜❧❡♠ ✐♥ ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ♠❡t❤♦❞s ❢♦r p✲❋❊▼✳ ❙■❆▼ ❏✳ ◆✉♠✳ ❆♥❛❧✳ ✷✵✵✷❀ ✹✵✭✹✮✿ ✾✷✽✲✾✹✹✳

SLIDE 44

St.-Petersburg State Polytechnical University

Laboratory of New Computational Technologies

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[BSS] Beuchler S, Schneider R and SchwabC. ▼✉❧t✐r❡s♦❧✉t✐♦♥ ✇❡✐❣❤t❡❞ ♥♦r♠ ❡q✉✐✈❛❧❡♥❝❡ ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s✳ ◆✉♠✳ ▼❛t❤✳ ✷✵✵✹❀ ✾✽✭✶✮✿ ✻✼✲✾✼✳ [Ca] Canuto C. t❡①t❝♦❧♦r❝②❛♥ ❙t❛❜✐❧✐③❛t✐♦♥ ♦❢ s♣❡❝tr❛❧ ♠❡t❤♦❞s ❜② ✜✲ ♥✐t❡ ❡❧❡♠❡♥t ❜✉❜❜❧❡ ❢✉♥❝t✐♦♥s✳ ❈♦♠♣✉t❡r ▼❡t❤♦❞s ✐♥ ❆♣♣❧✐❡❞ ▼❡❝❤❛♥✐❝s ❛♥❞ ❊♥❣✐♥❡❡r✐♥❣✳ ✶✾✾✹❀ ✶✶✻✭✮✿ ✶✸ ✕ ✷✻✳ [Cas] Casarin MA. ◗✉❛s✐✲♦♣t✐♠❛❧ ❙❝❤✇❛r③ ♠❡t❤♦❞s ❢♦r t❤❡ ❝♦♥✲ ❢♦r♠✐♥❣ s♣❡❝tr❛❧ ❡❧❡♠❡♥t ❞✐s❝r❡t✐③❛t✐♦♥✳ ❙■❆▼ ❏✳ ◆✉♠❡r✳ ❆♥❛❧✳ ✶✾✾✼❀ ✸✹✭✻✮✿ ✷✹✽✷✕✷✺✵✷✳ [K1] Korneev VG. ❆❧♠♦st ♦♣t✐♠❛❧ ♠❡t❤♦❞ ❢♦r s♦❧✈✐♥❣ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠s ♦♥ s✉❜❞♦♠❛✐♥s ♦❢ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❤✐❡r❛r❝❤✐❝❛❧ hp✕✈❡rs✐♦♥✳ ❉✐✛❡r❡ts✐❛❧✬♥②❡ ✉r❛✈♥❡♥✐❛ ✭❉✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s✮✳ ✷✵✵✶❀ ✸✼✭✼✮ ✿ ✶✕✶✵✳✭✐♥ ❘✉ss✐❛♥✮ [K2] Korneev VG. ▲♦❝❛❧ ❉✐r✐❝❤❧❡t ♣r♦❜❧❡♠s ♦♥ s✉❜❞♦♠❛✐♥s ♦❢ ❞❡✲ ❝♦♠♣♦s✐t✐♦♥ ✐♥ hp ❞✐s❝r❡t✐③❛t✐♦♥s✱ ❛♥❞ ♦♣t✐♠❛❧ ❛❧❣♦r✐t❤♠s ❢♦r t❤❡✐r s♦❧✉✲ t✐♦♥✳ ▼❛t❡♠❛t✐❝❤❡s❦♦✐❡ ♠♦❞❡❧✐r♦✈❛♥✐❡ ✭▼❛t❤❡♠❛t✐❝❛❧ ♠♦❞❡❧❧✐♥❣✮✳ ✷✵✵✷❀ ✶✹✭✺✮✿ ✺✶ ✕ ✼✹✳

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St.-Petersburg State Polytechnical University

Laboratory of New Computational Technologies

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[KLX1] Korneev V, Langer U and Xanthis; L. ❖♥ ❢❛st ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ s♦❧✈✐♥❣ ♣r♦❝❡❞✉r❡s ❢♦r hp✲❞✐s❝r❡t✐③❛t✐♦♥s ♦❢ ✸✲d ❡❧❧✐♣t✐❝ ♣r♦❜❧❡♠s✳ ❈♦♠♣✉t✳ ▼❡t❤✳ ✐♥ ❆♣♣❧✳ ▼❛t❤✳ ✷✵✵✸❀ ✸✭✹✮✿ ✺✸✻✕✺✺✾✳ [KLX2] Korneev V, Langer U and Xanthis L. ❋❛st ❛❞❛♣t✐✈❡ ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ ❛❧❣♦r✐t❤♠s ❢♦r hp✲❞✐s❝r❡t✐③❛t✐♦♥s ♦❢ ✷✲d ❛♥❞ ✸✲d ❡❧❧✐♣t✐❝ ❡q✉❛t✐♦♥s✿ r❡❝❡♥t ❛❞✈❛♥❝❡s✳ ❍❡r♠✐s✲µπ✿ ❆♥ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r✲ ♥❛❧ ♦❢ ❈♦♠♣✉t❡r ▼❛t❤✳ ❛♥❞ ❆♣♣❧✳ ✷✵✵✸❀ ✹✿ ✷✼✕ ✹✹✳ [KXA] Korneev VG, Xanthis L and Anufriev IE. ❍✐❡r❛r❝❤✐❝❛❧ ❛♥❞ ▲❛❣r❛♥❣❡ hp ❞✐s❝r❡t✐③❛t✐♦♥s ❛♥❞ ❢❛st ❞♦♠❛✐♥ ❞❡❝♦♠♣♦s✐t✐♦♥ s♦❧✈❡rs ❢♦r t❤❡♠✳ ❙❋❇ ❘❡♣♦rt ✵✷✲✶✽✱ ❏♦❤❛♥♥❡s ❑❡♣❧❡r ❯♥✐✈❡rs✐t② ▲✐♥③✿▲✐♥③✱ ✷✵✵✷✳ [PW] Pavarino LF and WidlundOB. ■t❡r❛t✐✈❡ s✉❜str✉❝t✉r✐♥❣ ♠❡t❤♦❞s ❢♦r s♣❡❝tr❛❧ ❡❧❡♠❡♥t ❞✐s❝r❡t✐③❛t✐♦♥s ♦❢ ❡❧❧✐♣t✐❝ s②st❡♠s ■❀ ❈♦♠✲ ♣r❡ss✐❜❧❡ ❧✐♥❡❛r ❡❧❛st✐❝✐t②✳ ❏✳ ◆✉♠✳ ❆♥❛❧✳ ✶✾✾✾❀ ✸✼✭✷✮✿ ✸✺✸✕✸✼✹✳ [SWX] Shen J, Wang F, Xu J. A finite element multigrid preconditioner for Chebyshev − collocation methods. ❆♣♣❧✳ ◆✉♠✳ ▼❛t❤✳ ✷✵✵✵❀ ✭✸✸✮✿ ✹✼✶✕✹✼✼✳