Parallel Space-Time Methods M. Neum uller Special Semester - PowerPoint PPT Presentation

Institute of Computational Mathematics Parallel Space-Time Methods M. Neum¨ uller Special Semester Space-Time Methods for PDEs Nov. 7 - 11, 2016 M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 1 / 29

Institute of Computational Mathematics Outline Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 2 / 29

Institute of Computational Mathematics Model problem Heat equation: ∂ t u − ∆ u = f in Q := Ω × (0 , T ) , u = g D on Σ := ∂ Ω × (0 , T ) , u = u 0 on Σ 0 := Ω × { 0 } . t Σ T T Q Σ Σ x Σ 0 U. Langer, S.E. Moore and M.N., Space–time isogeometric analysis of parabolic evolution problems (2016) M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 3 / 29

Institute of Computational Mathematics Outline Model problem Space-time method Numerical analysis Numerical examples Solvers Standard solvers Space-time multigrid method Conclusions and outlook M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 4 / 29

Institute of Computational Mathematics IgA function space ◮ Parameter space-time domain: � Q := (0 , 1) d +1 ◮ Geometrical mapping: Φ : � Q → Q ˆ t Q t ˆ t t Q � T Q ˆ � n x � Q n t K 1 Φ − 1 T � K Φ − 1 1 x 2 ˆ x 2 Ω Φ Φ Ω � 0 0 0 0 x Ω 0 � 1 x ˆ Ω x 1 a 0 x 1 ˆ b 0 1 IgA function space: V p ϕ h , k = � h := span { ϕ h , k } k ⊂ C p − 1 ( Q ) R k , p ◦ Φ − 1 , with V p 0 h := { v h ∈ V p h : v h = 0 on Σ ∪ Σ 0 } . M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 5 / 29

Institute of Computational Mathematics Variational formulation Let v h ∈ V p 0 h for p ≥ 2 and w h := v h + θ h ∂ t v h with θ > 0 . M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics Variational formulation Let v h ∈ V p 0 h for p ≥ 2 and w h := v h + θ h ∂ t v h with θ > 0 . � � � f w h dxdt = ∂ t u w h dxdt − ∆ u w h dxdt Q Q Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics Variational formulation Let v h ∈ V p 0 h for p ≥ 2 and w h := v h + θ h ∂ t v h with θ > 0 . � � � f w h dxdt = ∂ t u w h dxdt − ∆ u w h dxdt Q Q Q � � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt − n x · ∇ x u w h ds Q Q ∂ Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics Variational formulation Let v h ∈ V p 0 h for p ≥ 2 and w h := v h + θ h ∂ t v h with θ > 0 . � � � f w h dxdt = ∂ t u w h dxdt − ∆ u w h dxdt Q Q Q � � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt − n x · ∇ x u w h ds Q Q ∂ Q � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt . Q Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics Variational formulation Let v h ∈ V p 0 h for p ≥ 2 and w h := v h + θ h ∂ t v h with θ > 0 . � � � f w h dxdt = ∂ t u w h dxdt − ∆ u w h dxdt Q Q Q � � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt − n x · ∇ x u w h ds Q Q ∂ Q � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt . Q Q Bilinear form: � � a h ( u h , v h ) := ∂ t u h ( v h + θ h ∂ t v h ) dxdt + ∇ x u h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics Variational formulation Let v h ∈ V p 0 h for p ≥ 2 and w h := v h + θ h ∂ t v h with θ > 0 . � � � f w h dxdt = ∂ t u w h dxdt − ∆ u w h dxdt Q Q Q � � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt − n x · ∇ x u w h ds Q Q ∂ Q � � = ∂ t u w h dxdt + ∇ x u · ∇ x w h dxdt . Q Q Bilinear form: � � a h ( u h , v h ) := ∂ t u h ( v h + θ h ∂ t v h ) dxdt + ∇ x u h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q Linear form: � l h ( v h ) := f ( v h + θ h ∂ t v h ) dxdt Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 6 / 29

Institute of Computational Mathematics Variational formulation Find u h ∈ V g := g + V p 0 h with g = u 0 on Σ 0 and g = g D on Σ, such that for all v h ∈ V p a h ( u h , v h ) = l h ( v h ) 0 h . Discrete problem: Find u h = g + u 0 h with u 0 h ∈ V p 0 h , such that for all v h ∈ V p a h ( u 0 h , v h ) = l h ( v h ) − a h ( g , v h ) 0 h . M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 7 / 29

Institute of Computational Mathematics Variational formulation Find u h ∈ V g := g + V p 0 h with g = u 0 on Σ 0 and g = g D on Σ, such that for all v h ∈ V p a h ( u h , v h ) = l h ( v h ) 0 h . Discrete problem: Find u h = g + u 0 h with u 0 h ∈ V p 0 h , such that for all v h ∈ V p a h ( u 0 h , v h ) = l h ( v h ) − a h ( g , v h ) 0 h . Bilinear form: � � a h ( u h , v h ) := ∂ t u h ( v h + θ h ∂ t v h ) dxdt + ∇ x u h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q Linear form: � l h ( v h ) := f ( v h + θ h ∂ t v h ) dxdt Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 7 / 29

Institute of Computational Mathematics Coercivity For v h ∈ V p 0 h with p ≥ 2 we have � � a h ( v h , v h ) = ∂ t v h ( v h + θ h ∂ t v h ) dxdt + ∇ x v h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics Coercivity For v h ∈ V p 0 h with p ≥ 2 we have � � a h ( v h , v h ) = ∂ t v h ( v h + θ h ∂ t v h ) dxdt + ∇ x v h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + ∂ t v h v h dxdt + θ h ∇ x v h · ∇ x ∂ t v h dxdt Q Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics Coercivity For v h ∈ V p 0 h with p ≥ 2 we have � � a h ( v h , v h ) = ∂ t v h ( v h + θ h ∂ t v h ) dxdt + ∇ x v h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + ∂ t v h v h dxdt + θ h ∇ x v h · ∇ x ∂ t v h dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + 1 ∂ t ( v h ) 2 dxdt + θ h ∂ t |∇ x v h | 2 dxdt 2 2 Q Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics Coercivity For v h ∈ V p 0 h with p ≥ 2 we have � � a h ( v h , v h ) = ∂ t v h ( v h + θ h ∂ t v h ) dxdt + ∇ x v h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + ∂ t v h v h dxdt + θ h ∇ x v h · ∇ x ∂ t v h dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + 1 ∂ t ( v h ) 2 dxdt + θ h ∂ t |∇ x v h | 2 dxdt 2 2 Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + 1 n t ( v h ) 2 ds + θ h n t |∇ x v h | 2 dxdt 2 2 ∂ Q ∂ Q M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics Coercivity For v h ∈ V p 0 h with p ≥ 2 we have � � a h ( v h , v h ) = ∂ t v h ( v h + θ h ∂ t v h ) dxdt + ∇ x v h · ∇ x ( v h + θ h ∂ t v h ) dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + ∂ t v h v h dxdt + θ h ∇ x v h · ∇ x ∂ t v h dxdt Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + 1 ∂ t ( v h ) 2 dxdt + θ h ∂ t |∇ x v h | 2 dxdt 2 2 Q Q = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 L 2 ( Q ) � � + 1 n t ( v h ) 2 ds + θ h n t |∇ x v h | 2 dxdt 2 2 ∂ Q ∂ Q L 2 ( Q ) + 1 L 2 (Σ T ) + θ h = θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 2 � v h � 2 2 �∇ x v h � 2 L 2 (Σ T ) . M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 8 / 29

Institute of Computational Mathematics Numerical analysis Coercivity: For v h ∈ V p 0 h we have L 2 ( Q ) + 1 a h ( v h , v h ) ≥ θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 2 � v h � 2 L 2 (Σ T ) := � v h � 2 h . M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 9 / 29

Institute of Computational Mathematics Numerical analysis Coercivity: For v h ∈ V p 0 h we have L 2 ( Q ) + 1 a h ( v h , v h ) ≥ θ h � ∂ t v h � 2 L 2 ( Q ) + �∇ x v h � 2 2 � v h � 2 L 2 (Σ T ) := � v h � 2 h . Boundedness: For u ∈ H 2 , 1 ( Q ) + V p 0 h and v h ∈ V p 0 h we have a h ( u , v h ) ≤ µ b � u � h , ∗ � v h � h , with � u � 2 h , ∗ := � u � 2 h + ( θ h ) − 1 � v h � 2 L 2 ( Q ) . M. Neum¨ uller Parallel Space-Time Methods Linz, Nov. 7, 2016 9 / 29