Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Incomplete Factorization by Local Exact Factorization (ILUE) Johannes Kraus and Maria Lymbery "Modelling 2014" June 2–6, 2014, Roznov Pod Radhostem In honor of Professor Owe Axelsson’s 80th birthday Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results • The ILUE preconditioner ◦ General setting and definition ◦ Properties • Application within ASMG preconditioning ◦ Problem formulation ◦ ASMG method for the weighted H ( div ) -norm • Numerical results Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results General setting and definition Consider the linear system of algebraic equations A y = f where A = � N i = 1 R T i A i R i and A i , i = 1 , . . . N , are SP(S)D matrices. Incomplete factorization using exact local factorization (ILUE) B ILUE := LU where N � R T L := U T diag ( U ) − 1 , U := i U i R i , A i = L i U i , diag ( L i ) = I i . i = 1 Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties Aim: to estimate κ ( B − 1 ILUE A ) Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties Aim: to estimate κ ( B − 1 ILUE A ) w T B ILUE w ≤ w T A w A − B ILUE ≥ 0 ∀ w or Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties Aim: to estimate κ ( B − 1 ILUE A ) w T B ILUE w ≤ w T A w A − B ILUE ≥ 0 ∀ w or w T A w ≤ c w T B ILUE w ∀ w Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties Aim: to estimate κ ( B − 1 ILUE A ) w T B ILUE w ≤ w T A w A − B ILUE ≥ 0 ∀ w or w T A w ≤ c w T B ILUE w ∀ w c := λ max n color where Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties Aim: to estimate κ ( B − 1 ILUE A ) w T B ILUE w ≤ w T A w A − B ILUE ≥ 0 ∀ w or w T A w ≤ c w T B ILUE w ∀ w c := λ max n color where n color is the coloring constant of the adjacency graph of subgraphs G i color ( v i ) � = color ( v j ) ⇔ G ( A i ) ∩ G ( A j ) � = ∅ Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Properties Aim: to estimate κ ( B − 1 ILUE A ) w T B ILUE w ≤ w T A w A − B ILUE ≥ 0 ∀ w or w T A w ≤ c w T B ILUE w ∀ w c := λ max n color where n color is the coloring constant of the adjacency graph of subgraphs G i color ( v i ) � = color ( v j ) ⇔ G ( A i ) ∩ G ( A j ) � = ∅ R T i A i R i v = λ i U T diag ( U ) − 1 U v λ max := max 1 ≤ i ≤ N { λ i , max } Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • Ω is a domain in R 2 ; Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • n is the outward unit vector normal to the boundary ∂ Ω ; Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • ∂ Ω = Γ D ∪ Γ N ; Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • f ∈ L 2 is the forcing term; Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • K ( x ) : R 2 �→ R 2 × 2 SPD is the permeability tensor; Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • p ∈ H 1 0 is the fluid pressure; Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Second order elliptic boundary value problem in mixed form u + K ( x ) ∇ p = 0 in Ω , div u = f in Ω , p = 0 on Γ D , u · n = 0 on Γ N . • u ∈ H ( div ) is the velocity. Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation Dual mixed weak form: Find u ∈ V and p ∈ W such that A DM ( u , p ; v , q ) = − ( f , q ) , for all ( v , q ) ∈ V × W , where A DM ( u , p ; v , q ) : ( V , W ) × ( V , W ) → R is defined as A DM ( u , p ; v , q ) := ( α u , v ) − ( p , div v ) − ( div u , q ) , where α ( x ) = K − 1 ( x ) and V ≡ H N ( div ; Ω) = { v ∈ L 2 (Ω) : div v ∈ L 2 (Ω) , and v · n = 0 on Γ N } � W ≡ { q ∈ L 2 (Ω) and q dx = 0 if Γ N = ∂ Ω } . Ω Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results Problem formulation • piecewise constant functions for the pressure variable • lowest order Raviart-Thomas functions for the velocity Discrete system � � � � � � v T M α u = ( α u h , v h ) M α B div u 0 = where B T v T B T 0 p f div p = ( p h , div v h ) div Arnold-Falk-Winther A 0 B h = 0 I u T A v = ( α u h , v h ) + ( ∇ · u h , ∇ · v h ) = Λ α ( u h , v h ) Efficient preconditioning of the system u , b ∈ R N . A u = b , Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H ( div ) -norm n n � � R T Ω = Ω i A = i A i R i , D = D f ⊕ D c , i = 1 i = 1 � A 11 � � A i : 11 � A 12 A i : 12 A = A i = i = 1 , . . . , n . , , A 21 A 22 A i : 21 A i : 22 A 1 : 11 A 1 : 12 R 1 : 2 A 2 : 11 A 2 : 12 R 2 : 2 . ... . . � A = A n : 11 A n : 12 R n : 2 n � R T R T R T R T . . . 1 : 2 A 1 : 21 2 : 2 A 2 : 21 n : 2 A n : 21 i : 2 A i : 22 R i : 2 i = 1 A 22 = � n Setting � A 11 = diag { A 1 : 11 , . . . , A n : 11 } , � i = 1 R T i : 2 A i : 22 R i : 2 we have Incomplete Factorization by Local Exact Factorization (ILUE)
Outline The ILUE preconditioner Application within ASMG preconditioning Numerical results ASMG method for the weighted H ( div ) -norm n n � � R T Ω = Ω i A = i A i R i , D = D f ⊕ D c , i = 1 i = 1 � A 11 � � A i : 11 � A 12 A i : 12 A = A i = i = 1 , . . . , n . , , A 21 A 22 A i : 21 A i : 22 A 1 : 11 A 1 : 12 R 1 : 2 A 2 : 11 A 2 : 12 R 2 : 2 . ... . . � A = A n : 11 A n : 12 R n : 2 n � R T R T R T R T . . . 1 : 2 A 1 : 21 2 : 2 A 2 : 21 n : 2 A n : 21 i : 2 A i : 22 R i : 2 i = 1 A 22 = � n Setting � A 11 = diag { A 1 : 11 , . . . , A n : 11 } , � i = 1 R T i : 2 A i : 22 R i : 2 we have Incomplete Factorization by Local Exact Factorization (ILUE)
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