Axioms of Adaptivity Dirk Praetorius joint work with Carsten - PowerPoint PPT Presentation

Adaptive algorithms for computational PDEs January 5, 2016 Axioms of Adaptivity Dirk Praetorius joint work with Carsten Carstensen (Berlin), Michael Feischl (Sydney), Kris van der Zee (Nottingham) TU Wien Institute for Analysis and Scientific Computing

Optimal Convergence of Adaptive FEM Introduction Dirk Praetorius (TU Wien)

Optimal Convergence of Adaptive FEM Introduction Adaptive Algorithm initial mesh T 0 adaptivity parameter 0 < θ ≤ 1 For all ℓ = 0 , 1 , 2 , 3 , . . . iterate 1 SOLVE: compute discrete solution U ℓ for mesh T ℓ 2 ESTIMATE: compute indicators η ℓ ( T ) for all T ∈ T ℓ 3 MARK: find (minimal) set M ℓ ⊆ T ℓ s.t. � � η ℓ ( T ) 2 ≤ η ℓ ( T ) 2 θ T ∈T ℓ T ∈M ℓ 4 REFINE: refine (at least) all T ∈ M ℓ to obtain T ℓ +1 D¨ orfler: SINUM 33 (1996) Dirk Praetorius (TU Wien) – 1 –

Optimal Convergence of Adaptive FEM Introduction What is all about? 1 10 uniform O ( N − 1 / 2 ) 0 10 adaptive −1 10 error estimator −2 10 −3 10 O ( N − 3 / 2 ) −4 10 −5 10 −6 10 0 1 2 3 4 10 10 10 10 10 number of elements Feischl, Karkulik, Melenk, Praetorius: SINUM 51 (2013) Gantumur: Numer. Math. 123 (2013) Dirk Praetorius (TU Wien) – 2 –

Optimal Convergence of Adaptive FEM Introduction Mathematical Questions can we prove convergence of algorithm? can we guarantee optimal convergence rates? at least asymptotically what problem class can be covered? AFEM for 2nd order elliptic PDEs? ABEM for 2nd order elliptic PDEs? linear/nonlinear problems? goal-oriented adaptivity? D¨ orfler: SINUM 33 (1996) 324 citations Morin, Nochetto, Siebert: SINUM 38 (2000) 184 citations Binev, Dahmen, DeVore: Numer. Math. 97 (2004) 176 citations Stevenson: Found. Comput. Math. 7 (2007) 134 citations Cascon, Kreuzer, Nochetto, Siebert: SINUM 46 (2008) 140 citations Dirk Praetorius (TU Wien) – 3 –

Optimal Convergence of Adaptive FEM Introduction Axioms of Adaptivity? Carstensen, Feischl, Page, P. ’14 1 reproduces all results on rate optimality of adaptive algorithms independent of linear or nonlinear problem independent of discretization (e.g., FEM, BEM, FVM, coupled) equivalent estimators (not only residual estimators) inexact solvers 2 four properties (= axioms) of error estimator are sufficient two axioms are even necessary 3 problem + discretization enter only through proof of axioms Carstensen, Feischl, Page, Praetorius: CAMWA 67 (2014) Dirk Praetorius (TU Wien) – 4 –

Optimal Convergence of Adaptive FEM Outline Introduction 1 Axioms of Adaptivity 2 Optimal Standard Adaptivity 3 Optimal Goal-Oriented Adaptivity 4 Conclusions 5 Dirk Praetorius (TU Wien)

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Dirk Praetorius (TU Wien)

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Main Theorem on Adaptive Algorithms Theorem (Stevenson ’07, ..., Carstensen, Feischl, Page, P. ’14) validity of axioms (A1)–(A4) 0 < θ ≤ 1 η ℓ + n ≤ C q n η ℓ = ⇒ ∃ C > 0 ∃ 0 < q < 1 ∀ ℓ, n ≥ 0 � ∪ {T 0 } � T ∈ refine ( T 0 ) : # T ≤ N T N := s > 0 arbitrary 0 < θ ≪ 1 sufficiently small M ℓ has (essentially) minimal cardinality � =: � η � A s � N s (# T ℓ ) s η ℓ ≃ sup = ⇒ sup T opt ∈ T N η opt min ℓ ∈ N 0 N> 0 Carstensen, Feischl, Page, Praetorius: CAMWA 67 (2014) Dirk Praetorius (TU Wien) – 5 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Axioms of Adaptivity reduction (A2) estimator reduction stability (A1) quasi-orthogonality (A4) optimality of linear convergence D¨ orfler marking of ηℓ closure estimate discrete reliability (A3) optimal convergence of ηℓ overlay estimate optimal convergence efficiency of Uℓ Carstensen, Feischl, Page, Praetorius: CAMWA 67 (2014) Dirk Praetorius (TU Wien) – 6 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity The Axioms ∀T + ∀T ⋆ ∈ refine ( T + ) � η + ( T ) 2 � 1 / 2 � � η ⋆ ( T ) 2 � 1 / 2 � � � � � (A1) − � ≤ C stab | | | U ⋆ − U + | | | � T ∈T + ∩T ⋆ T ∈T + ∩T ⋆ � � η ⋆ ( T ) 2 ≤ q red η + ( T ) 2 + C red | | 2 (A2) | | U ⋆ − U + | | T ∈T ⋆ \T + T ∈T + \T ⋆ � | 2 ≤ C 2 η + ( T ) 2 (A3) | | | U ⋆ − U + | | rel T ∈R + where T + \T ⋆ ⊆ R + ⊆ T + , # R + ≤ C rel #( T + \T ⋆ ) ∀ ℓ, N ≥ 0 ∀ ε > 0 N � � ≤ C orth ( ε ) η 2 � | | 2 − εη 2 (A4) | | U k +1 − U k | | k ℓ k = ℓ Dirk Praetorius (TU Wien) – 7 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Poisson Model Problem Strong formulation in Ω ⊂ R d − ∆ u = f u = 0 on Γ = ∂ Ω Weak formulation find u ∈ H 1 0 (Ω) s.t. ˆ ˆ for all v ∈ H 1 ∇ u · ∇ v = fv 0 (Ω) Ω Ω Dirk Praetorius (TU Wien) – 8 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Residual Error Estimator for Poisson Model Problem Reliability and efficiency | | | u − U ⋆ | | | � η ⋆ � | | | u − U ⋆ | | | + osc ⋆ | | | · | | | = �∇ ( · ) � L 2 (Ω) � � η ⋆ ( T ) 2 � 1 / 2 η ⋆ = T ∈T ⋆ η ⋆ ( T ) 2 = h 2 T � f � 2 L 2 ( T ) + h T � [ ∂ n U ⋆ ] � 2 L 2 ( ∂T ∩ Ω) � � � 1 / 2 h 2 T � f − f T � 2 osc ⋆ := L 2 ( T ) T ∈T ⋆ Dirk Praetorius (TU Wien) – 9 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Axiom (A1): Stability on Non-Refined Elements (A1) Stability on non-refined elements, T ⋆ ∈ refine ( T + ) � η + ( T ) 2 � 1 / 2 � � η ⋆ ( T ) 2 � 1 / 2 � � � � � − � ≤ C stab | | | U ⋆ − U + | | | � T ∈T + ∩T ⋆ T ∈T + ∩T ⋆ verification for Poisson model problem: η ⋆ ( T ) 2 = h 2 T � f � 2 L 2 ( T ) + h T � [ ∂ n U ⋆ ] � 2 L 2 ( ∂T ∩ Ω) inverse triangle inequality + scaling arguments � � 1 / 2 � h T � [ ∂ n ( U ⋆ − U + )] � 2 LHS ≤ L 2 ( ∂T ∩ Ω) T ∈T + ∩T ⋆ � �∇ ( U ⋆ − U + ) � L 2 (Ω) Casc´ on, Kreuzer, Nochetto, Siebert: SINUM 46 (2008) Dirk Praetorius (TU Wien) – 10 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Axiom (A2): Reduction on Refined Elements (A2) Reduction on refined elements, T ⋆ ∈ refine ( T + ) � � η ⋆ ( T ) 2 ≤ q red η + ( T ) 2 + C red | | 2 | | U ⋆ − U + | | T ∈T ⋆ \T + T ∈T + \T ⋆ verification for Poisson model problem: η ⋆ ( T ) 2 = h 2 T � f � 2 L 2 ( T ) + h T � [ ∂ n U ⋆ ] � 2 L 2 ( ∂T ∩ Ω) � ( T ⋆ \T + ) = � ( T + \T ⋆ ) 2 h T for T ⋆ ∋ T ′ � T ∈ T + h T ′ ≤ 1 triangle inequality + Young inequality + scaling arguments q red ≈ 1 2 Casc´ on, Kreuzer, Nochetto, Siebert: SINUM 46 (2008) Dirk Praetorius (TU Wien) – 11 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Axiom (A3): Discrete Reliability (A3) Discrete reliability, T ⋆ ∈ refine ( T + ) exists R + ⊆ T + with T + \T ⋆ ⊆ R + # R + ≤ C rel #( T + \T ⋆ ) � | 2 ≤ C 2 η + ( T ) 2 | | | U ⋆ − U + | | rel T ∈R + discrete reliability = ⇒ reliability R + = T + \T ⋆ for FEM R + = patch( T + \T ⋆ ) for BEM / FVM Stevenson: Found. Comput. Math. 7 (2007) Dirk Praetorius (TU Wien) – 12 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Axiom (A4): Quasi-Orthogonality (A4) Quasi-orthogonality, for all ε > 0 and ℓ, N N � � | � ≤ C orth ( ε ) η 2 | 2 − εη 2 | | U k +1 − U k | | k ℓ k = ℓ verification for Poisson model problem Galerkin orthogonality + symmetry = ⇒ Pythagoras theorem | 2 + | | 2 = | | 2 | | | u − U k +1 | | | | U k +1 − U k | | | | u − U k | | quasi-orth. with C orth ( ε ) = C 2 telescoping series = ⇒ rel , ε = 0 N N � � | 2 � ≤ | � | | 2 = | 2 − | | 2 | | | U k +1 − U k | | | | u − U k | | | | u − U k +1 | | | | u − U ℓ | | k = ℓ k = ℓ Feischl, F¨ uhrer, Praetorius: SINUM 52 (2014) Dirk Praetorius (TU Wien) – 13 –

Optimal Convergence of Adaptive FEM Axioms of Adaptivity Axioms of Adaptivity reduction (A2) estimator reduction stability (A1) quasi-orthogonality (A4) optimality of linear convergence D¨ orfler marking of ηℓ closure estimate discrete reliability (A3) optimal convergence of ηℓ overlay estimate optimal convergence efficiency of Uℓ Carstensen, Feischl, Page, Praetorius: CAMWA 67 (2014) Dirk Praetorius (TU Wien) – 14 –

Optimal Convergence of Adaptive FEM Optimal Standard Adaptivity Dirk Praetorius (TU Wien)

Optimal Convergence of Adaptive FEM Optimal Standard Adaptivity Estimator Reduction Stability (A1) + Reduction (A2) = ⇒ Estimator Reduction ∀ 0 < θ ≤ 1 ∃ 0 < q est < 1 ∃ C est > 0 ∀ ℓ ∈ N 0 : η 2 ℓ +1 ≤ q est η 2 | 2 ℓ + C est | | | U ℓ +1 − U ℓ | | sketch: Young inequality + (A1) + (A2) + D¨ orfler marking q est = (1 + δ ) − θ (1 + δ − q red ) ≈ 1 − θ/ 2 C est = C 2 stab (1 + δ − 1 ) + C red Casc´ on, Kreuzer, Nochetto, Siebert: SINUM 46 (2008) Dirk Praetorius (TU Wien) – 15 –