Generalised FEs: Domain Decomposition, Optimal Local Approximation - PowerPoint PPT Presentation

Initial Remarks Complicated variation of α ( x ) on many scales ( h ≪ diam(Ω)) Hard to resolve by “geometric” coarse mesh! Goal A: Efficient & scalable multilevel parallel solver robust w.r.t. mesh size h ( ⇔ w.r.t. problem size n : O ( n ) cost) robust w.r.t. coefficients α ( x ) ! Goal B: Simulate on coarse mesh where α is not resolved ! Discretisation in “special” coarse space V H → Upscaling Approximation depends on (subgrid) variation & contrast in α ! Robust multiscale space is expensive for general coefficients Unless we have periodicity, scale separation, multiple RHSs, parameter dependence, not clear why Goal B over Goal A Coefficient-robust theory for Goal B much less well developed ! Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 5 / 38

Domain Decomposition / Multigrid Theory for Varying Coefficients Coarse grids resolve coefficient Bramble, Pasciak & Schatz, 88 & 89 ; Mandel, 93 ; Dryja, Smith & Widlund, 94 ; Wang & Xie, 94 ; Chan & Mathew, 94 ; Dryja, Sarkis & Widlund, 96 ; Sarkis, 97 ; Klawonn & Widlund, 01 , Mandel & Dohrmann, 03 ; Toselli & Widlund, 05 ; Xu & Zhu, 08 ; etc Coarse grids do not resolve coefficient Graham & Hagger, 99 ; Graham, Lechner & RS, 07 ; Pechstein & RS, 08 ; Van lent, RS & Graham 09 ; Galvis & Efendiev 10 ; Dolean, Nataf, RS & Spillane, 11 ; RS, Vassilevski & Zikatanov, 11 ; Efendiev, Galvis, Lazarov & Willems, 12 ; Spillane, Dolean, Hauret, Nataf et al, 14 ; Heinlein, Klawonn & Rheinbach, 16 ; Gander & Loneland, 17 ; etc Ideas from Algebraic Multigrid literature Alcouffe, Brandt, Dendy et al, 81 ; Ruge, St¨ uben, 87 ; Vassilevski, 92 ; Vanek, Mandel & Brezina, 96 ; Chartier, Falgout, Henson et al, 03 ; Falgout, Vassilevski & Zikatanov 05 ; Vassilevski, 08 ; etc Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 6 / 38

Types of Multiscale Methods & Theory (incomplete list) Adaptive FEs ..., [Babuska, Rheinboldt, 1978] Generalised FEs [Babuska, Osborn, 1983] Numerical Upscaling ..., [Durlofsky, 1991] Multiscale Finite Elements [Hou, Wu, 1997], ... Variational Multiscale Method [Hughes et al, 1998] Multigrid Based Upscaling [Moulton, Dendy, Hyman, 1998] Multiscale Finite Volume Methods [Jenny, Lee, Tchelepi, 2003] Heterogeneous Multiscale Method [E, Engquist, 2003] Multiscale Mortar Spaces [Arbogast, Wheeler et al, 2007] (& other DD based methods) Adaptive Multiscale FVMs/FEs [Durlovsky, Efendiev, Ginting, 2007] Energy minimising bases [Dubois, Mishev, Zikatanov, 2009] Locally spectral (Generalised MsFEM) [Efendiev, Galvis, Wu, 2010] ... etc ... Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 7 / 38

Simplifying Assumptions & Theory (incomplete list of refs) 1 Periodicity ⇒ Homogenisation theory ..., [Hou, Wu, 1997], ... 2 Scale Separation ..., [Abdulle, 2005], ... 3 Inclusions and simple interfaces [Chu, Graham, Hou, 2010] (high contrast, no periodicity, no scale separation) 4 Bound in special flux norm [Berlyand, Owhadi, 2010] (high contrast, no periodicity, no scale separation) 5 Low contrast [Larson, Malqvist, ’07], [Owhadi, Zhang, ’11], [Grasedyck et al, ’11], [Babuska, Lipton, ’11], [Malqvist, Peterseim, ’14] (no periodicity, no scale separation) 6 Exploit links to DD theory [RS, Vassilevski, Zikatanov, 2011] (weighted Poincar´ e, stable quasi-interpolant, weighted Bramble-Hilbert) Combine 5 and 6 to cover more general high contrast coeffs. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 8 / 38

Simplifying Assumptions & Theory (incomplete list of refs) 1 Periodicity ⇒ Homogenisation theory ..., [Hou, Wu, 1997], ... 2 Scale Separation ..., [Abdulle, 2005], ... 3 Inclusions and simple interfaces [Chu, Graham, Hou, 2010] (high contrast, no periodicity, no scale separation) 4 Bound in special flux norm [Berlyand, Owhadi, 2010] (high contrast, no periodicity, no scale separation) 5 Low contrast [Larson, Malqvist, ’07], [Owhadi, Zhang, ’11], [Grasedyck et al, ’11], [Babuska, Lipton, ’11], [Malqvist, Peterseim, ’14] (no periodicity, no scale separation) 6 Exploiting links to DD [RS, Vassilevski, Zikatanov, 2011] (weighted Poincar´ e, stable quasi-interpolant, weighted Bramble-Hilbert) Combine 5 and 6 to cover more general high contrast coeffs. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 8 / 38

Classical theory and more recent ideas Classical theory in H 1 and H 1 / 2 -norm based on standard Poincar´ e inequalities and robustness of weighted L 2 -projections [Bramble, Xu, Math Comp 91] , . . . (for resolving coarse grids!) More recent ideas directly in the energy norm based on weighted Poincar´ e type inequalities [Galvis, Efendiev, 2010], [Pechstein, RS, 2011 & 2012] and an abstract Bramble-Hilbert Lemma ← − This Talk! (for energy minimising coarse spaces) [RS, Vassilevski, Zikatanov, MMS 2011] Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 9 / 38

Subspace Correction Methods (e.g. two-level Schwarz or multigrid ) Problem (in variational form) : Find u h ∈ V h s.t. � a ( u h , v h ) ≡ α ∇ u h · ∇ v h = ( f , v h ) for all v h ∈ V h . Ω Precondition by solving (exactly or approximately) in subspaces V 0 , V 1 , . . . V L ⊂ V h in parallel (additive) or successively (multiplicative) χ 1 χ χ 3 Two-level overlapping Schwarz 2 V ℓ = { v h ∈ V h : supp( v h ) ⊂ Ω ℓ } with Ω 2 Ω Ω 3 1 overlapping partitioning { Ω ℓ } L ℓ =1 of Ω and V 0 = span { Φ j ∈ V h : j = 1 , . . . , N } (abstract) � L M − 1 R T ℓ A − 1 add A = ℓ R ℓ A A ℓ = restriction of A to subspace Ω ℓ � �� ℓ =0 (assume overlap δ � H ) = P ℓ Geometric Multigrid & BPX similar with V ℓ = p.w. lin. FE space on nested triangulations {T h ℓ } L ℓ =0 Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 10 / 38

Two-level Overlapping Schwarz – Abstract Theory � Let α v 2 d x � v � 2 0 , α = (weighted L 2 -norm) Ω Analogously to classical theory (in H 1 -seminorm and L 2 -norm) we have: Theorem (Two-level Schwarz) [RS, Vassilevski, Zikatanov, MMS 2011] If there exists an operator Π : V h → V 0 such that, for all v ∈ V h , � Π v � 2 a ≤ C 1 � v � 2 � v − Π v � 2 0 , α ≤ C 2 � v � 2 and (1) a a (stability) (weak approximation) then κ ( M − 1 add A ) � C 1 + C 2 . The hidden constant is independent of α , L , h . Similar result for geometric multigrid (different norm � · � ∗ induced by smoother) Main question: How to choose Π and how to prove (1) ? Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 11 / 38

“Nodal” Coarse Spaces, e.g. Piecewise Linears V 0 = V H cts. p.w. linears on a shape-regular grid T H No assumption that coefficient is resolved on T H ! Theorem [RS, Vassilevski, Zikatanov, SINUM 2012] For all K ∈ T H , let C P K > 0 be the best constant s.t. for all v ∈ V h K H 2 � v � 2 inf ξ ∈ R � v − ξ � 2 0 , α ,ω K ≤ C P ( WPI ) a (with a slight variation near Dirichlet boundaries). Then κ ( M − 1 add A ) � max K ∈T H C P K � supp(Φ j ) α v d x with Π v = � j v α v α � j Φ j and = supp(Φ j ) α d x . j (WPI) linked directly to local quasi-monotonicity [Pechstein, RS, 2012] Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 12 / 38

When is Poincar´ e constant independent of contrast in α ? Careful theory in [Pechstein, RS, 2012] linking robustness to quasi-monotonicity . e constant C P Bounds for the effective Poincar´ T : Darker colour means higher permeability. * X X* η min η η min min O (1 + log( H O ( H O ( η O (1) η )) η ) h ) (a) (b) 4 6 5 7 3 η 5 8 1 2 3 1 7 * X 4 9 6 C P T � α 2 * 2 8 X 9 η α 1 H η Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 13 / 38

Numerical Example ( Geometric Multigrid ) Ω = (0 , 1) 2 , uniform grids {T ℓ } L ℓ =0 with L = 4 and h L = 1 / 384. Two “islands” not alligned with T 0 and T 1 where α ( x ) = � α ( α ( x ) = 1 elsewhere) C P C P K bounded for all ω K K not bdd. on some ω K λ − 1 # MG Its (tol = 10 − 8 ) λ − 1 # MG Its (tol = 10 − 8 ) � α 1 1 10 1 1.69 10 1.72 10 10 2 2.75 14 3.87 19 10 3 3.32 12 14.5 23 10 4 3.42 10 115.5 70 10 5 3.42 10 1125 76 In right table islands closer to each other! Guiding principle for choice of “nodal” coarse spaces T H sufficiently fine ( locally ) s.t. α ( x ) quasi-monotone on all ω K When it is difficult to ensure quasi-monotonicity on all ω K − → Coefficient-dependent Coarse Spaces ! Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 14 / 38

Energy minimising coarse spaces = Generalised FEs Suppose { Ω ℓ } L ℓ =1 is an overlapping partition of Ω and { χ ℓ } L ℓ =1 an associate partition of unity w. � χ ℓ � ∞ � 1 & �∇ χ ℓ � ∞ � δ − 1 � H − 1 ℓ ℓ (This could be a set of FE basis functions and their supports.) Local Energy Minimization subject to Functional Constraints For each subdomain Ω ℓ , assume that we have a collection of linear j =1 ⊂ V h (Ω ℓ ) ′ and let functionals { f ℓ, j } m ℓ v ∈ V h (Ω ℓ ) � v � 2 Ψ ℓ, j = arg min subject to f ℓ, k (Ψ ℓ, j ) = δ jk . (2) a , Ω ℓ Now, with I h the standard nodal interpolant onto V h , let V H = span { Φ ℓ, j } with Φ ℓ, j = I h ( χ ℓ Ψ ℓ, j ) , ℓ = 1 , L , j = 1 , m ℓ . (“glueing” together the locally energy minimising bases via a partition of unity) Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 15 / 38

Energy minimising coarse spaces = Generalised FEs Importance of energy minimization noted in AMG literature : Explicitly: [Mandel, Brezina & Vanek, 99] ; [Wan, Chan & Smith, 99] ; [Xu & Zikatanov, 04] ; [Brannick, Brezina et al, 05] (implicitly in all AMG methods) Theorem [RS, Vassilevski, Zikatanov, MMS 2011] If ∀ v ∈ V h (Ω ℓ ) the local quasi-interpolant Π ℓ v = � j f ℓ, j ( v )Ψ ℓ, j satisfies � Π ℓ v � a , Ω ℓ � � v � a , Ω ℓ � Ω ℓ α | v − Π ℓ v | 2 d x H 2 ℓ � u � 2 � a , Ω ℓ add A ) � 1 with Π v = � L � m ℓ then κ ( M − 1 j =1 f ℓ, j ( v )Φ ℓ, j . ℓ =1 The assumptions of this theorem follow from a ’novel’ abstract approximation result related to the Bramble-Hilbert Lemma . Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 16 / 38

An abstract Bramble–Hilbert Lemma – Tool 1 Suppose V ⊂ H and H Hilbert with norm � · � , a is an abstract k =1 ⊂ V ′ symmetric continuous bilinear form on V × V and { f k } m and define (as in the specific case above) , for all v ∈ V , v ∈ V | v | 2 ψ k = arg min a , subject to f j ( ψ k ) = δ jk j , k = 1 , . . . , m . Make the following assumptions: A1. a is positive semi-definite and defines a semi-norm | · | a on V � � v � 2 + | v | 2 and a defines a norm on V . A2. For all q ∈ R m there exists a v q ∈ V with f k ( v q ) = q k , and � v q � � c q � q � l 2 ( R m ) . � m A3. � v � 2 ≤ c a | v | 2 k =1 | f k ( v ) | 2 , a + c f for all v ∈ V . Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 17 / 38

An abstract Bramble–Hilbert Lemma – Tool 1 Theorem (RS, Vassilevski, Zikatanov, MMS 2011) Let Assumptions A1-3 hold. Then π u = � k f k ( u ) ψ k satisfies � u − π u � ≤ √ c a | u | a | π u | a ≤ | u | a and for all u ∈ V . (Note that this is independent of the constants c q and c f in A2 and A3 .) Proof. First one notes that given u ∈ V , π u minimizes energy subject to f k ( v ) = f k ( u ). Thus | π u | a ≤ | u | a by construction. Secondly, from A3 , the fact that f k ( v − Π v ) = 0 ∀ k and the stability estimate, we get � m � v − Π v � 2 c a | v − Π v | 2 | f ( v − Π v ) | 2 ≤ a + c f l =1 c a | v − Π v | 2 a ≤ 2 c a ( | v | 2 a + | Π v | 2 a ) ≤ 4 c a | v | 2 = a . (can lose factor 2 by more careful bound) Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 18 / 38

In our specific model problem considered above Assumption A1 is naturally satisfied on any subdomain Ω ℓ � Ω ℓ α v 2 d x ( weighted L 2 -norm ! ) with H = L 2 (Ω ℓ ) and � v � = Assumption A2 simply means the functionals { f k } should be linearly independent. Coarse space robustness reduced to verifying Assumption A3 For one functional reduces to (WPI) and quasi-monotonicity. For more then one functional opens possibility of coefficient robustness even for non-quasi-monotone coefficients. More importantly: can be applied also to other problems, e.g. elasticity, Stokes, ... Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 19 / 38

Choice of functionals – ‘Knob’ 1 � [Galvis, Efendiev ’10] : f ℓ, j ( v ) = Ω ℓ α Ψ ℓ, j v d x where Ψ ℓ, j is the 1 j th eigenfunction of matrix pencil of local stiffness & mass matrix � � [RS, Vassilevski, Zikatanov ’11] : f ℓ, j ( v ) = Ω ℓ, j α v d x / Ω ℓ, j α d x 2 where { Ω ℓ, j } m ℓ j =1 is partitioning of Ω ℓ s.t. (WPI) holds on each Ω ℓ, j (Construction of Ψ ℓ, j requires solution of m ℓ local saddle point systems .) � [Dolean, Nataf, RS, Spillane ’12] : f ℓ, j ( v ) = ∂ Ω ℓ α Ψ ℓ, j v d s where 3 Ψ ℓ, j is j th eigenfunction of Dirichlet-to-Neumann operator on ∂ Ω ℓ � � � � � [Spillane et al ’14] : f ℓ, j ( v ) = Ω ℓ α ∇ χ ℓ Ψ ℓ, j · ∇ χ ℓ v d x where 4 Ψ ℓ, j is j th eigenfct. of matrix pencil stiffness matrix & a ( χ ℓ · , χ ℓ · ) ← − GenEO – see below! But also in multiscale literature: � [Babuska, Lipton ’11] : f ℓ, j ( v ) = ω ℓ α ∇ Ψ ℓ, j · ∇ v d x with ω ℓ ⊂ Ω ℓ 1 � � [Peterseim, RS ’16] (LOD): f ℓ, j ( v ) = Ω ℓ αχ j v d x / Ω ℓ αχ j d x 2 [Owhadi ’17] (Gamblets): Hierarchy of functionals similar to Case 2 3 Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 20 / 38

Verification of Assumption A3 for Cases 1 and 2 � Ω ℓ α v 2 d x Recall on subdomain Ω ℓ chose H = L 2 (Ω ℓ ) and � v � = Case 2: Applying (WPI) on each subsubdomain Ω ℓ, j : � m ℓ � � v � 2 ≤ α |∇ v | 2 d x + α max H d ℓ, j H 2 | f ℓ, j ( v ) | 2 C P ℓ ℓ � �� Ω ℓ, j j =1 =: c f � m ℓ ≤ c a � v � 2 | f ℓ, j ( v ) | 2 C P ℓ, j ) H 2 a + c f with c a := (max ℓ j j =1 � Case 1: Recall f ℓ, j ( v ) = Ω ℓ α Ψ ℓ, j v d x with Ψ ℓ, j the eigenfunction related to pair of energy and weighted L 2 -inner product. Thus: m ℓ � � v � 2 ≤ c a � v � 2 | f ℓ, j ( v ) | 2 c a := λ − 1 a + c f with ℓ, m ℓ +1 j =1 Also random energy minimisation methods possible [Buhr, Smetana ’18] Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 21 / 38

Contrast-robust approximation theory for LOD & GFEM How can we use this abstract Bramble-Hilbert Lemma to obtain a contrast-robust approximation theory for LOD or GFEM? Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 22 / 38

Localizable Orthogonal Decomposition (LOD) FE space V H := span { Φ j } associated with (coarse) FE mesh T H Quasi-interpolation operator Π : V h → V H with � ( v , Φ j ) L 2 (Ω) Π v := Φ j (1 , Φ j ) L 2 (Ω) (Π invertible on V H !) j Decomposition V h = V H ⊕ V f V f with h := kernel Π = { v ∈ V h | Π v = 0 } h For each v ∈ V h define the fine scale projection P f v ∈ V f h by a ( P f v , w ) = a ( v , w ) for all w ∈ V f (global!) h a –Orthogonal Decomposition [Malqvist, Peterseim, ’11] V h = V ms H ⊕ V f a ( V ms H , V f with V ms := (1 − P f ) V H and h ) = 0 h H Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 23 / 38

Modified (multiscale) nodal basis { Φ j | j = 1 , . . . , N } ⊂ V H denotes classical nodal basis j := P f Φ j ∈ V f ϕ f h denotes the fine scale correction of Φ j Ideal multiscale FE space � � V ms Φ j − ϕ f = span j | j = 1 , . . . , N H Example ϕ f Φ j − ϕ f = Φ j - j j �� ∈ V ms ∈ V f ∈ V H H h Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 24 / 38

Exponential decay and localisation Define nodal patches ω j , k of k -th order around vertex x j of T H x x x x ω j , 1 ω j , 2 ω j , 3 ω j , 4 Can show that | ϕ f j | H 1 (Ω \ ω j , k ) � γ k | ϕ f j | H 1 (Ω) (with γ < 1). Define ϕ f j , k ∈ V f h ( ω j , k ) := { v ∈ V f h | supp v ⊂ ω j , k } (the localised correction) s.t. w ∈ V f a ( ϕ f j , k , w ) = a (Φ j , w ) for all h ( ω j , k ) Localized multiscale FE spaces V ms H , k := span { Φ H j − ϕ f j , k | j = 1 , . . . , N } Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 25 / 38

Multiscale Coarse Problem & Approximation Result Multiscale approximation Seek u ms H , k ∈ V ms H , k such that a ( u ms for all v ∈ V ms H , k , v ) = ( f , v ) H , k dim V ms H , k = dim V H = N & basis functions have local support Overlap of the supports is proportional to the parameter k Theorem (Malqvist & Peterseim, 2011) | u − u ms H , k | H 1 (Ω) � k d γ k � f � H − 1 (Ω) + H � f � L 2 (Ω) + | u − u h | H 1 (Ω) Thus, provided k � log γ ( 1 H ) and h is suff’ly small we have optimal O ( H ) convergence without any assumptions on scales or regularity. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 26 / 38

But unfortunately γ → 1 as the contrast α max → ∞ and the α min hidden constant depends also on α max α min ⇓ Thus for high contrast (in theory) no localization ! Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 27 / 38

A contrast-robust theory (work again in weighted norms) Theorem (Peterseim & RS, 2016) If ∃ linear, cont. quasi-interpolation operator Π : V h → V H s.t. (Π | V H ) − 1 Π v H = v H , for all v H ∈ V H (QI1) H − 2 T � v − Π v � 2 0 ,α, T + � v − Π v � 2 a , T ≤ C 2 � v � 2 (QI2) a ,ω T , for all v ∈ V h and T ∈ T H (QI3) for all v H ∈ V H there exists a v ∈ V h , s.t. Π v = v H , supp v ⊂ supp v H and � v � a ≤ C 3 � v H � a . then (with some universal constant m � 1) � � m e − k H � u − u ms α max H , k � a � H � f � H − 1 (Ω) + � f � L 2 (Ω) + � u − u h � a α min α − 1 / 2 min Thus, provided k � ln( α max 1 H ) and h suff’ly small we have optimal α min O ( H ) convergence without assumptions on regularity or contrast. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 28 / 38

A suitable quasi-interpolation operator For simplicity assume α p.w. constant w.r.t. some grid T η , with h < η < H , but not by T H ( T H ⊂ T η ⊂ T H nested) N � ( α v , Φ j ) L 2 (Ω) Choose Π v := Φ j ( again weighted! ) ( α, Φ j ) L 2 (Ω) j =1 Theorem [Peterseim, RS ’16] For all T ∈ T H , let C P T > 0 be the best constant s.t. inf ξ ∈ R � v − ξ � 2 0 , α ,ω T ≤ C P T H 2 T � v � 2 ∀ v ∈ V h . ( WPI ) a ,ω T Then H − 2 T � v − Π v � 2 0 ,α, T + � v − Π v � 2 a , T � C 2 � v � 2 a where C 2 � H η max T ∈T H C P T , i.e. Assumption (QI2). � � 2 H Moreover, (QI1) and (QI3) hold with C 3 � . η Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 29 / 38

Discussion & Tool 2 In summary, we do get contrast independent convergence rates, but so far only under fairly stringent assumptions on the type of coefficient variation (i.e. locally quasi-monotone & p.w. constant w.r.t. T η for moderate H /η ) Key tool: Weighted Caccioppoli-type Inequality Let ω ⊂ Ω s.t. dist( ∂ω, ∂ Ω) > δ > 0. Then � u � a ,ω ≤ 2 δ − 1 � u � 0 ,α, Ω for all a -harmonic u on Ω. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 30 / 38

Ideas for non-quasi-monotone coefficients – Work in Progress! Adapt grid or enrich local space or change functionals ! LOD : Refine base grid T H locally where C P T depends on contrast (similar to multiresolution idea in gamblets) GFEM : Use eigenproblem with Ω ℓ ⊂ Ω ∗ ℓ and combine (abstract) Bramble-Hilbert (Tool 1) with (weighted) Caccioppoli (Tool 2) [Babuska, Lipton ’11], [Smetana, Patera ’16], [Buhr, Smetana ’18] Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 31 / 38

Beyond scalar elliptic problems Linear elasticity equations: � � � a ( u , v ) := C ( x ) ε ( u ) : ε ( v ) d x = f · v d x + ( σ · n ) · v d x ∀ v ∈ V Ω Ω Γ small length scales ( < mm), high contrast and strongly anisotropic CERTEST (EPSRC Project) STEAM (Turing/Royce Project) Bristol, Bath, Exeter, Heidelberg,... Exeter, Heidelberg, Imperial,... Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 32 / 38

Change Eigenproblem: GenEO [Spillane et al ’14] – Tool 3 Key lemma in subspace correction theory to bound κ ( M − 1 add A ): Lions’ Lemma – Stable splitting � L � L � v ℓ � 2 a ≤ C 2 0 � v � 2 ∃ C 0 > 0 : ∀ v ∈ V h : ∃ v ℓ ∈ V ℓ : v = v ℓ and a ℓ =0 ℓ =0 Key observation in [Spillane, Dolean, Hauret, Nataf, Pechstein, RS ’14]: Lemma (Local sufficient condition) – Tool 3 � v ℓ � 2 a , Ω ℓ ≤ C 2 1 � v � 2 Suppose that ∃ C 1 > 0 : ∀ ℓ = 1 , . . . , L : a , Ω ℓ . Then the splitting above is stable with C 2 0 = 2 + k 0 C 2 1 + 2 k 2 0 C 2 1 (where k 0 is the maximal #subdomains any degree of freedom belongs to) Choose v ℓ := χ ℓ ( v − v 0 ). Motivates following (variational) eigenproblem : � � � � a Ω ℓ ψ ℓ, j , v = λ j a Ω ℓ χ ℓ ψ ℓ, j , χ ℓ v ∀ v ∈ V h (Ω ℓ ) (full overlap case) � � a , Ω ℓ ≤ λ − 1 get � v ℓ � 2 ℓ, m ℓ +1 � v � 2 and w. V 0 := span I h ( χ ℓ ψ ℓ, j ) : ℓ ≤ L , j ≤ m ℓ a , Ω ℓ Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 33 / 38

Toy Composite Example for Demonstration - Cantilever Flat composite plate [0 , 100mm] × [0 , 20mm] Cantilever under uniform pressure (top surface) 12 Layers - 11 weak interfaces [ ∓ 45 ◦ / 0 ◦ / 90 ◦ / ± 45 ◦ / ∓ 45 ◦ / 90 ◦ / 0 ◦ / ± 45 ◦ ] . 20-node serendipity elements Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 34 / 38

GenEO Modes & Numerical Results (Benchmarking) [Butler, Dodwell, Reinarz, Sandhu, RS, Seelinger ’19] Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 35 / 38

Industrially motivated problem (with over 2 × 10 8 DOFs) Wingbox section with defect under internal fuel pressure (ply-scale stress resolution!!) Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 36 / 38

Parallel Efficiency of HPC Implementation up to 15,360 cores Dune HPC implementation of GenEO within Parallel performance on UK National HPC Cluster [Butler, Dodwell, Reinarz, Sandhu, RS, Seelinger ’19] This scale of computations brings composites problems that would otherwise be unthinkable into the feasible range. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 37 / 38

Current Work & Final Remarks Extend to nonlinear elasticity & composite failure More complicated geometries & bigger overlap GenEO as a GFEM : first results in [Dodwell, Sandhu, RS ’17] (different functional as [Babuska, Lipton], [Buhr, Smetana]: ‘Knob’ 1 ) Bayesian inference: surrogate in multilevel MCMC ‘Knob’ 2 : Choice of partition of unity (seems to have big effect) ‘Knob’ 3 : ARPACK eigensolver vs. randomised eigensolver Some initial experiments below! Theoretical Aim: Prove contrast-independent approximation results for (versions of) LOD and GFEM ! THANK YOU! [If anybody is up for rock climbing on Wed afternoon, please let me know!] Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

References RS , PS Vassilevski & LT Zikatanov, Weak Approximation 1 Properties of Elliptic Projections with Functional Constraints, Multiscale Model Sim (SIAM) 9 , 2011. N Spillane, V Dolean, P Hauret, F Nataf, C Pechstein & RS , 2 Abstract Robust Coarse Spaces for Systems of PDEs via Generalized Eigenproblems in the Overlaps, Numer Math 126 , 2014. D Peterseim & RS , Robust Numerical Upscaling of Elliptic Multiscale 3 Problems at High Contrast, Comput Meth Appl Math 16 , 2016. TJ Dodwell, A Sandhu & RS , Customized Coarse Models for Highly 4 Heterogeneous Materials, in ”Bifurcation and Degradation of Geomaterials with Engineering Applications” (Papamichos et al Eds.), Springer Series in Geomechanics and Geoengineering, 2017. R Butler, TJ Dodwell, A Reinarz, A Sandhu, RS & L Seelinger, High- 5 performance dune modules for solving large-scale, strongly anisotropic elliptic problems with applications to aerospace composites arXiv:1901.05188, 2019. Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

Eigenfunctions for Different Partitions of Unity (scalar elliptic) Coefficient function Harmonic POU - 1st Mode First 6 eigenmodes in each domain: [Arne Strehlow] P.O.U. λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 p.w. const. Ω 1 0 . 00272 0 . 00536 0 . 19743 0 . 22077 0 . 28599 0 . 34094 Ω 2 0 . 00315 0 . 00749 0 . 20680 0 . 22085 0 . 30189 0 . 34094 Sarkis Ω 1 0 . 01963 0 . 05366 0 . 09788 1 . 05319 1 . 05517 1 . 05974 Ω 2 0 . 01599 0 . 04153 0 . 09416 0 . 99473 1 . 05318 1 . 05327 Harmonic Ω 1 0 . 03357 0 . 21091 0 . 78878 1 . 05086 1 . 05326 1 . 05974 Ω 2 0 . 03444 0 . 28577 0 . 83536 1 . 00547 1 . 00852 1 . 00915 Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

Eigenfunctions for Different Partitions of Unity (scalar elliptic) ’Disconnected’ cross [Arne Strehlow] (4 subdomains) Piecewise constant POU Harmonic POU Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

GenEO as GFEM for scalar elliptic problem [Tim Dodwell] First 5 eigenfunctions on Ω 6 (16 subdomains; a ( x ) is log-normal sample): Coarse Model Fine model err = � u h − R T H U H � 2 dim V h = 4 × 10 4 dim V H = 320 � u h � 2 ( m = 20 , O = 5) Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

Coarse Approximation Error (channels & islands) Figure: Parameter Distribution Rob Scheichl (Heidelberg) CIRM – Luminy, Sep 2019 Generalised Finite Elements 38 / 38

Generalised FEs: Domain Decomposition, Optimal Local Approximation - PowerPoint PPT Presentation

Generalised FEs: Domain Decomposition, Optimal Local Approximation & Model Order Reduction Robert Scheichl Institute of Applied Mathematics & Interdisciplinary Centre for Scientific Computing Universit at Heidelberg Collaborators:

Image Processing A case study for a domain decomposed MPI code Domain Decomposition 1

A generalised multi-receiver radio network and its decomposition into independent

On a two-level domain decomposition preconditioner for 3D flows in anisotropic highly

Domains and Games Glynn Winskel, Cambridge Generalised domain theories: stable domain theory,

A Domain Decomposition Method for Large Scale Simulations of Two-phase Flows with Moving Contact

Efficient Realization of Geometric Constraint Introduction Systems via Optimal Recursive

From optimal cubature formulae to Chebyshev lattices: a way towards generalised Clenshaw-Curtis

Optimal investment under multiple defaults: a BSDE-decomposition approach en PHAM Huy

BDDC Domain Decomposition Algorithms Olof B. Widlund Courant Institute, New York University 75th

Implementing Generalised Alt Gavin Lowe Implementing Generalised Alt 02 CSO for dummies

Time Domain Decomposition Methods Martin J. Gander martin.gander@math.unige.ch University of

Some Domain Decomposition Methods for Discontinuous Coefficients Marcus Sarkis WPI RICAM-Linz,

CFD exercise Regular domain decomposition Reusing this material This work is licensed under a

CFD exercise Regular domain decomposition Reusing this material This work is licensed under a

M A PHYS or the development of a parallel algebraic domain decomposition solver in the course of

CFD example Regular domain decomposition Fluid Dynamics The study of the mechanics of fluid

Decomposition of Database Preferences on the Power Set of the Domain Patrick Roocks Institute

Hp-spectral FEMs in fast domain decomposition algorithms V . KORNEEV and A . RYTOV

Spacetime domain decomposition methods for linear and nonlinear diffusion problems Michel

Efficient Domain Generalization via Common-Specific Low-Rank Decomposition * Sunita Sarawagi 1

From Surface Equivalence Principle to Modular Domain Decomposition Florian Muth* Hermann

A Domain Decomposition Method for Quasilinear Elliptic PDEs Using Mortar Finite Elements

Lagrangian Decomposition for Optimal Cost Partitioning Florian Pommerening 1 oger 1 Malte Helmert

Adaptive Coarse Spaces and Multiple Search Directions: Tools for Robust Domain Decomposition