An abstract two-level Schwarz method for systems with high contrast - PowerPoint PPT Presentation

An abstract two-level Schwarz method for systems with high contrast coefficients Clemens Pechstein Johannes Kepler University Linz (A) joint work with Nicole Spillane , Fr´ ed´ eric Nataf (Univ. Paris VI) Victorita Dolean (Univ. Nice Sophia-Antipolis) Patrice Hauret (Michelin, Clermont-Ferrand) Robert Scheichl (Univ. Bath) RICAM Special Semester Multiscale Simulation & Analysis in Energy & Environment October 8, 2011

Motivation Large discretized system of PDEs Applications: flow in heterogeneous / strongly heterogeneous coefficients stochastic / layered media (high contrast, nonlinear, multiscale) structural mechanics electromagnetics E.g. Darcy pressure equation, etc. P 1 -finite elements: Au = f cond ( A ) ∼ κ max h − 2 κ min Goal: iterative solvers robust in size and heterogeneities C. Pechstein et al abstract two-level Schwarz 2 / 24

Relation to other methods Graham, Lechner & Scheichl Scheichl & Vainikko Overlapping Schwarz MS coarse space Pechstein & Scheichl FETI (2 papers) boundary layers WPI for some patterns Galvis & Efendiev Overlapping Schwarz Dolean, Nataf, Spillane a) std. coarse space, WPI & Xiang / & Scheichl λ b) gen.EVP K = M κ κ Pechstein & Scheichl Overlapping Schwarz Weighted Poincare Ineq. Γ = λ M gen.EVP S κ κ Efendiev, Galvis, Lazarov & Willems Overlapping Schwarz, abstract SPD problems ξ λ THIS TALK gen.EVP K = K κ κ C. Pechstein et al abstract two-level Schwarz 3 / 24

Problem setting – I Given f ∈ ( V h ) ∗ find u ∈ V h ∀ v ∈ V h a ( u , v ) = � f , v � ⇐ ⇒ A u = f Assumption throughout: A symmetric positive definite (SPD) Examples: � Darcy a ( u , v ) = Ω κ ∇ u · ∇ v dx � Elasticity a ( u , v ) = Ω C ε ( u ) : ε ( v ) dx � a ( u , v ) = Ω ν curl u · curl v + σ u · v dx Eddy current Heterogeneities / high contrast / nonlinearities in parameters C. Pechstein et al abstract two-level Schwarz 4 / 24

Problem setting – II V h . . . FE space of functions in Ω based on mesh T h = { τ } 1 A given as set of element stiffness matrices 2 + connectivity (list of DOF per element) Assembling property: � a ( v , w ) = a τ ( v | τ , w | τ ) τ where a τ ( · , · ) symm. pos. semi-definite { φ k } n k = 1 (FE) basis of V h 3 on each element: unisolvence set of non-vanishing basis functions linearly independent fulfilled by standard FE continuous, N´ ed´ elec, Raviart-Thomas of low/high order Two more assumptions on a ( · , · ) later! 4 C. Pechstein et al abstract two-level Schwarz 5 / 24

Problem setting – II V h . . . FE space of functions in Ω based on mesh T h = { τ } 1 A given as set of element stiffness matrices 2 + connectivity (list of DOF per element) Assembling property: � a ( v , w ) = a τ ( v | τ , w | τ ) τ where a τ ( · , · ) symm. pos. semi-definite { φ k } n k = 1 (FE) basis of V h 3 on each element: unisolvence set of non-vanishing basis functions linearly independent fulfilled by standard FE continuous, N´ ed´ elec, Raviart-Thomas of low/high order Two more assumptions on a ( · , · ) later! 4 C. Pechstein et al abstract two-level Schwarz 5 / 24

Schwarz setting – I Overlapping partition: Ω = � N j = 1 Ω j ( Ω j union of elements) � � φ k : supp ( φ k ) ⊂ Ω j V j := span such that every φ k contained in one of those spaces, i.e. N � V h = V j j = 1 Example: adding “layers” to non-overlapping partition (partition and adding layers based on matrix information only!) C. Pechstein et al abstract two-level Schwarz 6 / 24

Schwarz setting – II Local subspaces: V j ⊂ V h j = 1 , . . . , N Coarse space (defined later): V 0 ⊂ V h Additive Schwarz preconditioner: N � M − 1 R ⊤ j A − 1 AS , 2 = R j j j = 0 where A j = R ⊤ j AR j : V j → V h natural embedding and R ⊤ j ↔ R ⊤ j C. Pechstein et al abstract two-level Schwarz 7 / 24

Partition of unity Definitions: � � k : supp ( φ k ) ∩ Ω j � = ∅ dof (Ω j ) := � � k : supp ( φ k ) ⊂ Ω j V j = span { φ k } k ∈ idof (Ω j ) idof (Ω j ) := � � imult ( k ) := # j : k ∈ idof (Ω j ) Partition of unity: (used for design of coarse space and for stable splitting) n 1 � � Ξ j v = imult ( k ) v k φ k for v = v k φ k k ∈ idof (Ω j ) k = 1 Properties: N � Ξ j v = v Ξ j v ∈ V j j = 1 C. Pechstein et al abstract two-level Schwarz 8 / 24

Overlapping zone / Choice of coarse space Overlapping zone: Ω ◦ = { x ∈ Ω j : ∃ i � = j : x ∈ Ω i } j Observation: Ξ j | Ω j \ Ω ◦ = id j Coarse space should be local : N � V 0 = V 0 , j where V 0 , j ⊂ V j j = 1 m j E.g. V 0 , j = span { Ξ j p j , k } k = 1 C. Pechstein et al abstract two-level Schwarz 9 / 24

Overlapping zone / Choice of coarse space Overlapping zone: Ω ◦ = { x ∈ Ω j : ∃ i � = j : x ∈ Ω i } j Observation: Ξ j | Ω j \ Ω ◦ = id j Coarse space should be local : N � V 0 = V 0 , j where V 0 , j ⊂ V j j = 1 m j E.g. V 0 , j = span { Ξ j p j , k } k = 1 C. Pechstein et al abstract two-level Schwarz 9 / 24

Choice of coarse space (continued) ASM theory needs stable splitting : N � v = v 0 + v j j = 1 Suppose v 0 = � N j = 1 Ξ j Π j v | Ω j where Π j . . . local projector | 2 a , Ω j = | Ξ j ( v − Π j v ) | 2 j + | Ξ j ( v − Π j v )) | 2 | Ξ j ( v − Π j v ) a , Ω ◦ a , Ω j \ Ω ◦ j � �� � v j HOW? C | v | 2 ≤ a , Ω j ( a , D . . . restriction of a to D ) “Minimal” requirements: Π j be a -orthogonal Stability estimate: | Ξ j ( v − Π j v ) | 2 ≤ c | v | 2 a , Ω ◦ a , Ω j j C. Pechstein et al abstract two-level Schwarz 10 / 24

Choice of coarse space (continued) ASM theory needs stable splitting : N � v = v 0 + v j j = 1 Suppose v 0 = � N j = 1 Ξ j Π j v | Ω j where Π j . . . local projector | 2 a , Ω j = | Ξ j ( v − Π j v ) | 2 j + | Ξ j ( v − Π j v )) | 2 | Ξ j ( v − Π j v ) a , Ω ◦ a , Ω j \ Ω ◦ j � �� � v j HOW? C | v | 2 ≤ a , Ω j ( a , D . . . restriction of a to D ) “Minimal” requirements: Π j be a -orthogonal Stability estimate: | Ξ j ( v − Π j v ) | 2 ≤ c | v | 2 a , Ω ◦ a , Ω j j C. Pechstein et al abstract two-level Schwarz 10 / 24

Abstract eigenvalue problem Gen.EVP per subdomain: Find p j , k ∈ V h | Ω j and λ j , k ≥ 0: ∀ v ∈ V h | Ω j a Ω j ( p j , k , v ) = λ j , k a Ω ◦ j (Ξ j p j , k , Ξ j v ) A j p j , k = λ j , k X j A ◦ j X j p j , k ( X j . . . diagonal) a D . . . restriction of a to D (properties of eigenfunctions discussed soon) In the two-level ASM: Choose first m j eigenvectors per subdomain: � � j = 1 ,..., N V 0 = span Ξ j p j , k k = 1 ,..., m j C. Pechstein et al abstract two-level Schwarz 11 / 24

Abstract eigenvalue problem Gen.EVP per subdomain: Find p j , k ∈ V h | Ω j and λ j , k ≥ 0: ∀ v ∈ V h | Ω j a Ω j ( p j , k , v ) = λ j , k a Ω ◦ j (Ξ j p j , k , Ξ j v ) A j p j , k = λ j , k X j A ◦ j X j p j , k ( X j . . . diagonal) a D . . . restriction of a to D (properties of eigenfunctions discussed soon) In the two-level ASM: Choose first m j eigenvectors per subdomain: � � j = 1 ,..., N V 0 = span Ξ j p j , k k = 1 ,..., m j C. Pechstein et al abstract two-level Schwarz 11 / 24

Comparison with existing works Galvis & Efendiev (SIAM 2010): � � κ ∇ p j , k · ∇ v dx = λ j , k κ p j , k v dx ∀ v ∈ V h | Ω j Ω j Ω j Efendiev, Galvis, Lazarov & Willems (submitted): � ∀ v ∈ V | Ω j a Ω j ( p j , k , v ) = λ j , k a Ω j ( ξ j ξ i p j , k , ξ j ξ i v ) i ∈ neighb ( j ) ξ j . . . partition of unity, calculated adaptively (MS) Our gen.EVP: a Ω j ( p j , k , v ) = λ j , k a Ω ◦ j (Ξ j p j , k , Ξ j v ) ∀ v ∈ V h | Ω j ⇒ λ j , k ∈ [ 0 , ∞ ] both matrices typically singular = C. Pechstein et al abstract two-level Schwarz 12 / 24

Convergence theory Definition: Each element appears in at most k 0 of the subdomains Ω j Assumption 1: a Ω j ( · , · ) SPD on span { φ k | Ω j } k ∈ dof (Ω j ) \ idof (Ω j ) Assumption 2: a Ω ◦ j ( · , · ) SPD on span { φ k | Ω j } k ∈ idof (Ω j ) \ idof (Ω j \ Ω ◦ j ) Theorem 0 < λ j , m j + 1 < ∞ : If for all j: � � 1 �� N κ ( M − 1 AS , 2 A ) ≤ ( 1 + k 0 ) 2 + k 0 ( 2 k 0 + 1 ) 1 + max λ j , m j + 1 j = 1 C. Pechstein et al abstract two-level Schwarz 13 / 24

Properties of eigenfunctions Gen.EVP per subdomain: Find p j , k ∈ V h | Ω j and λ j , k ∈ [ 0 , ∞ ] : a Ω j ( p j , k , v ) = λ j , k a Ω ◦ j (Ξ j p j , k , Ξ j v ) ∀ v ∈ V h | Ω j � �� � =: b j ( p j , k , v ) Eigenfunctions with λ < ∞ are Free DOFs: here for continuous a Ω j ( · , · ) -harmonic in Ω j \ Ω ◦ 1 j Q 1 -elements a Ω j ( · , · ) -harmonic w.r.t. ker (Ξ j | Ω j ) 2 Ass. 1 = ⇒ 2nd extension possible ⇒ b j ( · , · ) SPD on space Ass. 2 = of eigenfunctions with λ < ∞ C. Pechstein et al abstract two-level Schwarz 14 / 24

Sufficient conditions for Assumptions 1 and 2 Assumptions 1 and 2 hold if all a τ ( · , · ) definite curl ( ν curl u ) + u or if certain mixed “boundary” value problems solvable: Last conditions easy to check for concrete PDEs: kernel of Darcy controlled by one Dirichlet DOF kernel of 3D Elasticity controlled by three non-collinear DOFs C. Pechstein et al abstract two-level Schwarz 15 / 24