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

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