St.-Petersburg State Polytechnical University Hp-spectral FEM’s in fast domain decomposition algorithms V . KORNEEV and A . RYTOV ✵✴✸✹ ✵✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University 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✳ ✶✴✸✹ ✶✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University Preconditioners for hierarchical elements M 1 ,p = ( L i ( s ) , i = 0 , 1 , . . . , p ) ✕ s❡t ♦❢ ♣♦❧②♥♦♠✐❛❧s ♦♥ ✭✲✶✱✶✮✿ L 0 ( s ) = 1 L 1 ( s ) = 1 2 (1 + s ) , 2 (1 − s ) , � s L i ( s ) := β i − 1 P i − 1 ( t ) dt = γ i [ P i ( s ) − P i − 2 ( s )] , i ≥ 2 , P i ❛r❡ ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s ❛♥❞ � � β i = 1 (2 j − 3)(2 j − 1)(2 j + 1) , γ i = 0 . 5 (2 i − 3)(2 i + 1) / (2 i − 1) . 2 ❚❤❡r❡❢♦r❡✱ L i ❛r❡ s♣❡❝✐❛❧❧② ♥♦r♠❛❧✐③❡❞ ✐♥t❡❣r❛t❡❞ ▲❡❣❡♥❞r❡✬s ♣♦❧②♥♦♠✐❛❧s✳ ✷✴✸✹ ✷✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University ❇② ❤✐❡r❛r❝❤✐❝❛❧ r❡❢✳ ❡❧✳ E hi ✐s ✉♥❞❡rst♦♦❞ r❡❢✳❡❧✳ ♦♥ t❤❡ ❝✉❜❡ τ 0 = ( − 1 , 1) d ✇✐t❤ t❤❡ ❜❛s✐s ✐♥ t❤❡ s♣❛❝❡ Q p,x � � M d,p = L α ( x ) = L α 1 ( x 1 ) L α 2 ( x 2 ) ... L α d ( x d ) , α ∈ ω , ω := ( α = ( α 1 , α 2 , .., α d ) : 0 ≤ α 1 , α 2 , .., α d ≤ p ) , ❛♥❞ ✇✐t❤ t❤❡ st✐✛♥❡ss ♠❛tr✐① A ✱ ✐♥❞✉❝❡❞ ❜② M d,p ❛♥❞ ❉✐r✐❝❤❧❡t ✐♥t❡❣r❛❧ � a τ 0 ( u, v ) = ∇ u · ∇ v d x . τ 0 ◦ A I ✕ internal st✐✛✳ ♠❛tr✐①✱ ❣❡♥❡r❛t❡❞ ❜② M d,p = ( L α , 2 ≤ α k ≤ p ) ✳ ✸✴✸✹ ✸✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University ◦ ■❢ t♦ r❡♦r❞❡r s❡t M d,p ✱ ♠❛tr✐❝❡s A I ✱ M I ✐♥ d = 3 ❜❡❝♦♠❡ ❜❧♦❝❦ ❞✐❛❣♦♥❛❧ A I = ❞✐❛❣ [ A eee , A eeo , ..., A ooe , A ooo ] , M I = ❞✐❛❣ [ M eee , M eeo , ..., M ooe , M ooo ] . ❆t p = 2 N + 1 ❛❧❧ ✽ ❜❧♦❝❦s ❛r❡ N 3 × N 3 ♠❛tr✐❝❡s ❛♥❞✱ e.g., A a 1 a 2 a 3 = ( a τ 0 ( L α , L α ′ )) N k =1 , α k ,α ′ ✇✐t❤ α k , α ′ k ❡✈❡♥✴♦❞❞ r❡s♣❡❝t✐✈❡❧② t♦ ❡✈❡♥✴♦❞❞ a k ✳ ✹✴✸✹ ✹✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University ❚❤❡s❡ ❜❧♦❝❦s ❛r❡ s✉♠s A abc = K 1 ,a ⊗ K 0 ,b ⊗ K 0 ,c + K 0 ,a ⊗ K 1 ,b ⊗ K 0 ,c + K 0 ,a ⊗ K 0 ,b ⊗ K 1 ,c , M abc = K 0 ,a ⊗ K 0 ,b ⊗ K 0 ,c , a, b, c = e, o ♦❢ ❑r♦♥❡❝❦❡r ♣r♦❞✉❝ts ♦❢ tr✐♣❧❡ts ♦❢ N × N ♠❛tr✐❝❡s✱ ✇❤✐❝❤ ♠❛② ❜❡ ♣r❡❝♦♥❞✐t✐♦♥❡❞ ❜② s✐♠♣❧❡ ♠❛tr✐❝❡s 2 − 1 ✵ − 1 2 − 1 ∆ = 1 D = diag [4 i 2 ] N i =1 , . . . . . . . 2 ✵ − 1 2 − 1 − 1 2 ✺✴✸✹ ✺✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University Lemma 1 . ❋♦r ✶✲❞ ♣r❡❝♦♥❞✐t✐♦♥❡rs D , ∆ ❛♥❞ ✸✲❞ ♣r❡❝♦♥❞✐t✐♦♥❡rs Λ e = ∆ ⊗ ∆ ⊗ D + ∆ × D ⊗ ∆ + D ⊗ ∆ ⊗ ∆ , M = ∆ ⊗ ∆ ⊗ ∆ , t❤❡r❡ ❤♦❧❞ t❤❡ ✐♥❡q✉❛❧✐t✐❡s ∆ ≺ K 0 ,a ≺ ∆ , D ≺ K 1 ,a ≺ D , Λ e ≺ A abc ≺ Λ e , M ≺ M abc ≺ M . Pr♦♦❢✳ ■✈❛♥♦✈✴❑♦r♥❡❡✈ ❬✶✾✾✺❪ ❛♥❞ ❑♦r♥❡❡✈✴❏❡♥s❡♥ ❬✶✾✾✼❪✱ ❑♦r♥❡❡✈✴▲❛♥❣❡r✴❳❛♥t❤✐s ❬✷✵✵✸❪✳ ✻✴✸✹ ✻✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University Finite ✲ difference interpretation ■♥ ✷✲❞ Λ e = ∆ ⊗ D + D ⊗ ∆ ❛♥❞ ✐s t❤❡ ❋✲❉ ❛♣♣r♦①✐♠❛t✐♦♥ ♦❢ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ♦♣❡r❛t♦r � � ∂ 2 u ∂ 2 u x ∈ π 1 := (0 , 1) 2 , x 2 + x 2 Lu ≡ − 2 , u | ∂π 1 = 0 , 1 2 ∂x 2 ∂x 2 2 1 ♦♥ t❤❡ sq✉❛r❡ ♠❡s❤ ♦❢ s✐③❡ � = 1 / ( N + 1) ✳ ■♥ ✸✲❞✱ � − 2 Λ e ✐s t❤❡ ❋✲❉ ❛♣♣r♦①✐♠❛t✐♦♥ ♦♥ t❤❡ s❛♠❡ ♠❡s❤ ♦❢ t❤❡ ✹✲t❤ ♦r❞❡r ♦♣❡r❛t♦r x ∈ π 1 := (0 , 1) 3 , Lu ≡ x 2 3 u , 1 , 1 , 2 , 2 + x 2 2 u , 1 , 1 , 3 , 3 + x 2 1 u , 2 , 2 , 3 , 3 = f ( x ) , u | ∂π 1 = 0 , ✇❤❡r❡✱ ❡✳❣✳✱ u , 1 , 1 , 2 , 2 = ∂ 4 u/∂x 2 2 ✳ 1 ∂x 2 ✼✴✸✹ ✼✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University FEM preconditioner ◦ ❙✉♣♣♦s❡✱ d = 3 ✱ V ( π 1 ) ✐s t❤❡ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ♦♥ π 1 ❛♥❞ ♣✐❡❝❡ ✇✐s❡ tr✐❧✐♥❡❛r ♦♥ ❡❛❝❤ ❝❡❧❧ ♦❢ t❤❡ ❝✉❜✐❝ ♠❡s❤ ❢✉♥❝t✐♦♥s✱ ✈❛♥✐s❤✐♥❣ ♦♥ ∂π 1 ✱ ❛♥❞ Λ e , fem ✐s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤✐s s♣❛❝❡ ♠❛tr✐① ♦❢ ❜✐❧✐♥❡❛r ❢♦r♠ � 3 � ϕ k = x 2 b π 1 ( u, v ) = ϕ k u ,k +1 ,k +2 v ,k +1 ,k +2 dx , k . π 1 k =1 Lemma 2 . ❚❤❡ ♠❛tr✐① 1 ℏ Λ e , fem ✐s s♣❡❝tr❛❧❧② ❡q✉✐✈❛❧❡♥t t♦ A abc ❛♥❞ Λ e ✉♥✐❢♦r♠❧② ✐♥ p ✳ Pr♦♦❢✳ ❙❡❡✱ e.g ✱ ❑♦r♥❡❡✈ ❬✷✵✵✷❪✳ ✽✴✸✹ ✽✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
St.-Petersburg State Polytechnical University ◦ ■♥ ✷✲❞✱ ♦♥❡ ❝❛♥ ✉s❡ t❤❡ ❋❊ s♣❛❝❡ V △ ( π 1 ) ♦❢ ❝♦♥t✐♥✉♦✉s ❛♥❞ ♣✐❡❝❡ ✇✐s❡ ❧✐♥❡❛r ❢✉♥❝t✐♦♥s ♦♥ t❤❡ tr✐❛♥❣✉❧❛t✐♦♥✱ ♦❜t❛✐♥❡❞ ❜② s✉❜❞✐✈✐s✐♦♥ ♦❢ ❡❛❝❤ sq✉❛r❡ ♥❡st ♦❢ t❤❡ ♠❡s❤ ✐♥ t✇♦ tr✐❛♥❣❧❡s✳ Pr❡❝♦♥❞✐t✐♦♥❡r Λ e , fem ✐s ♠❛tr✐① ♦❢ t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ � 2 � b π 1 ( u, v ) = ϕ k u , 3 − k v , 3 − k dx , π 1 k =1 ◦ ♦♥ t❤❡ s♣❛❝❡ V △ ( π 1 ) ✳ ❲❡ ❤❛✈❡ Λ e , fem ≍ � 2 A abc , � 2 Λ e ✳ ✵✴✴✸✹ ✵✴✴✸✹ Laboratory of New Computational Technologies ◭ ◭ ◭ ◭ � ◮ ◮ ◭ ◭ � ◮ ◮ ◮ ◮
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