Domain Decomposition for Multiscale PDEs Robert Scheichl Bath Institute for Complex Systems Department of Mathematical Sciences University of Bath in collaboration with Clemens Pechstein (Linz, AUT), Ivan Graham & Jan Van lent (Bath), Eero Vainikko (Tartu, EST) Scaling Up & Modelling for Transport and Flow in Porous Media Dubrovnik, Wednesday, October 15th 2008 R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 1 / 24
Motivation: Groundwater Flow Safety assessment for radioactive waste disposal at Sellafield � NIREX UK Ltd. c EDZ CROWN SPACE WASTE VAULTS FAULTED GRANITE GRANITE q + A ( x ) ∇ p = f Darcy’s Law: DEEP SKIDDAW N-S SKIDDAW DEEP LATTERBARROW N-S LATTERBARROW FAULTED TOP M-F BVG Incompressibility: ∇ · q = 0 TOP M-F BVG FAULTED BLEAWATH BVG BLEAWATH BVG FAULTED F-H BVG F-H BVG FAULTED UNDIFF BVG + UNDIFF BVG Boundary Conditions FAULTED N-S BVG N-S BVG FAULTED CARB LST CARB LST FAULTED COLLYHURST COLLYHURST (More generally: Multiphase Flow in Porous Media, e.g. FAULTED BROCKRAM BROCKRAM SHALES + EVAP Oil Reservoir Modelling or CO 2 Sequestration) FAULTED BNHM BOTTOM NHM FAULTED DEEP ST BEES DEEP ST BEES FAULTED N-S ST BEES N-S ST BEES FAULTED VN-S ST BEES VN-S ST BEES FAULTED DEEP CALDER DEEP CALDER FAULTED N-S CALDER N-S CALDER FAULTED VN-S CALDER VN-S CALDER MERCIA MUDSTONE QUATERNARY R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 2 / 24
Model Problem Elliptic PDE in 2D or 3D bounded domain Ω −∇ · ( α ∇ u ) = f + u = 0 on ∂ Ω R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 3 / 24
Model Problem Elliptic PDE in 2D or 3D bounded domain Ω −∇ · ( α ∇ u ) = f + u = 0 on ∂ Ω Highly variable (discontinuous) coefficients α ( x ) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 3 / 24
Model Problem Elliptic PDE in 2D or 3D bounded domain Ω −∇ · ( α ∇ u ) = f + u = 0 on ∂ Ω Highly variable (discontinuous) coefficients α ( x ) FE discretisation (p.w. linears V h ) on mesh T h : A u = b ( n × n linear system) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 3 / 24
Model Problem Elliptic PDE in 2D or 3D bounded domain Ω −∇ · ( α ∇ u ) = f + u = 0 on ∂ Ω Highly variable (discontinuous) coefficients α ( x ) FE discretisation (p.w. linears V h ) on mesh T h : A u = b ( n × n linear system) Aim: Find efficient & robust preconditioner for A (i.e. independent of variations in h and in α ( x )) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 3 / 24
Heterogeneous multiscale deterministic media Society of Petroleum Engineers (SPE) Benchmark SPE10 Multiscale stochastic media ( λ = 5 h , 10 h , 20 h ) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 4 / 24
Difficulties Requires very fine mesh resolution: h ≪ diam(Ω) A very large and very ill-conditioned, i.e. � α τ � h − 2 κ ( A ) � max α τ ′ τ,τ ′ ∈T h Variation of α ( x ) on many scales (often anisotropic) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 5 / 24
Difficulties Requires very fine mesh resolution: h ≪ diam(Ω) A very large and very ill-conditioned, i.e. � α τ � h − 2 κ ( A ) � max α τ ′ τ,τ ′ ∈T h Variation of α ( x ) on many scales (often anisotropic) Homogenisation or Scaling Up − → “cell problem” in each cell: Cost ≥ O ( n ) ( n = #DOFs on fine grid) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 5 / 24
Difficulties Requires very fine mesh resolution: h ≪ diam(Ω) A very large and very ill-conditioned, i.e. � α τ � h − 2 κ ( A ) � max α τ ′ τ,τ ′ ∈T h Variation of α ( x ) on many scales (often anisotropic) Homogenisation or Scaling Up − → “cell problem” in each cell: Cost ≥ O ( n ) ( n = #DOFs on fine grid) Alternative: Multilevel Iterative Solution on fine grid (directly!) − → Cost ≈ O ( n ) as well!! R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 5 / 24
Difficulties Requires very fine mesh resolution: h ≪ diam(Ω) A very large and very ill-conditioned, i.e. � α τ � h − 2 κ ( A ) � max α τ ′ τ,τ ′ ∈T h Variation of α ( x ) on many scales (often anisotropic) Homogenisation or Scaling Up − → “cell problem” in each cell: Cost ≥ O ( n ) ( n = #DOFs on fine grid) Alternative: Multilevel Iterative Solution on fine grid (directly!) − → Cost ≈ O ( n ) as well!! Meaning of � R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 5 / 24
Goals Efficient, scalable & parallelisable method, ◮ robust w.r.t. problem size n and mesh resolution h ◮ robust w.r.t. coefficients α ( x ) ! with underpinning theory = ⇒ “handle” for choice of components R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 6 / 24
Goals Efficient, scalable & parallelisable method, ◮ robust w.r.t. problem size n and mesh resolution h ◮ robust w.r.t. coefficients α ( x ) ! with underpinning theory = ⇒ “handle” for choice of components Possible Methods & Existing Theory Standard Domain Decomposition and Multigrid robust if coarse grid(s) resolve(s) coefficients [Chan, Mathew, Acta Numerica, 94] , [J. Xu, Zhu, Preprint, 07] Otherwise: coefficient-dependent coarse spaces [Alcouffe, Brandt, Dendy et al, SISC, 81] , [Sarkis, Num Math, 97] R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 6 / 24
Practically most successful: Algebraic Multigrid No theory explaining coefficient robustness for standard AMG ! First attempts in [Aksoylu, Graham, Klie, Sch., Comp.Visual.Sci. 08] R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 7 / 24
Practically most successful: Algebraic Multigrid No theory explaining coefficient robustness for standard AMG ! First attempts in [Aksoylu, Graham, Klie, Sch., Comp.Visual.Sci. 08] Two-Level Overlapping Schwarz ◮ [Sarkis, Num Math, 97] ◮ [Graham, Lechner, Sch., Num Math, 07] ◮ [Sch., Vainikko, Computing, 07] ◮ [Graham, Sch., Vainikko, NMPDE, 07] ◮ [Van lent, Sch., Graham, submitted, 08] Th m . κ ( M − 1 A ) � max j δ 2 � α |∇ Ψ j | 2 � L ∞ (Ω) (1 + H /δ ) − → low energy coarse spaces! R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 7 / 24
Practically most successful: Algebraic Multigrid No theory explaining coefficient robustness for standard AMG ! First attempts in [Aksoylu, Graham, Klie, Sch., Comp.Visual.Sci. 08] Two-Level Overlapping Schwarz ◮ [Sarkis, Num Math, 97] ◮ [Graham, Lechner, Sch., Num Math, 07] ◮ [Sch., Vainikko, Computing, 07] ◮ [Graham, Sch., Vainikko, NMPDE, 07] ◮ [Van lent, Sch., Graham, submitted, 08] Th m . κ ( M − 1 A ) � max j δ 2 � α |∇ Ψ j | 2 � L ∞ (Ω) (1 + H /δ ) − → low energy coarse spaces! FETI (Finite Element Tearing & Interconnecting) ◮ [Pechstein, Sch., Num Math, 08] ← − Today! R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 7 / 24
Finite Element Tearing & Interconnecting (non-overlapping dual substructuring techniques) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 8 / 24
FETI methods – Idea Domain decomposition Ω = � N Ω i =1 Ω i Γ i := ∂ Ω i H i := diam Ω i Ω Ω i j R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 9 / 24
FETI methods – Idea Domain decomposition Ω = � N Ω i =1 Ω i Γ i := ∂ Ω i H i := diam Ω i Conforming FE mesh on Ω (p.w. linear FEs) Mesh size on subdomain Ω i : h i R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 9 / 24
FETI methods – Idea Domain decomposition Ω = � N Ω i =1 Ω i Γ i := ∂ Ω i H i := diam Ω i Conforming FE mesh on Ω Ω i (p.w. linear FEs) Mesh size on subdomain Ω i : h i Subdomain stiffness matrix A i (including boundary, i.e. Neumann) R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 9 / 24
FETI methods – Idea Tearing: Introduce local soln u i , Ω i.e. > 1 dofs per interface node Interconnecting: Enforce conti- nuity by pointwise constraints: u i ( x h ) − u j ( x h ) = 0 , x h ∈ Γ i ∩ Γ j or compactly written, B u := � i B i u i = 0 where u := [ u ⊤ 1 u ⊤ 2 . . . u ⊤ N ] ⊤ R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 9 / 24
FETI methods – Idea Tearing: Introduce local soln u i , Ω i.e. > 1 dofs per interface node Interconnecting: Enforce conti- nuity by pointwise constraints: Introduce Lagrange multipliers to obtain the new global system: � � � � � � A B ⊤ u f = B 0 λ 0 with A := diag ( A i ) & f := [ f ⊤ 1 . . . f ⊤ N ] ⊤ R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 9 / 24
FETI methods – Idea Tearing: Introduce local soln u i , Ω i.e. > 1 dofs per interface node Interconnecting: Enforce conti- nuity by pointwise constraints: Eliminate u & solve dual problem ′′ B A − 1 B ⊤ λ = B A − 1 f ′′ � � ′′ � ′′ (Fully parallel!) 0 0 B ⊤ with preconditioner i B i i 0 S i where S i := A i , ΓΓ − A i , Γ I A − 1 i , II A i , I Γ (Schur complement) . R. Scheichl (Bath) DD for Multiscale PDEs Dubrovnik, Wed 10-15-08 9 / 24
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