Lecture 1 A Review of Goal-Oriented Error Estimation and Adaptive - PowerPoint PPT Presentation

Lecture 1 A Review of Goal-Oriented Error Estimation and Adaptive Methods Serge Prudhomme D´ epartement de math´ ematiques et de g´ enie industriel Polytechnique Montr´ eal DCSE Fall School 2019 TU Delft, The Netherlands, November 4-8, 2019 S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 1 / 38

Plan of Lectures Plan of Lectures Lecture 1: A Review of Goal-Oriented Error Estimation and Adaptive Methods Lecture 2: Adaptive Methods for Problems with Uncertain Coefficients and Bayesian Inference Lecture 3: Goal-oriented formulation of boundary-value problems Lecture 4: Adaptive Construction of PGD reduced-order models with respect to QoI’s S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 2 / 38

Outline Outline 1) Introduction 2) Estimation and control of discretization errors in quantities of interest Linear problems (adjoint and error representation). Extension to nonlinear problems. Extension to time-dependent problems. Multiphysics coupled problems: Micro-fluidics application. 3) Conclusions S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 3 / 38

Introduction Introduction Error estimation is useful for two purposes: 1. To provide a measure of the accuracy in approximations. 2. To control errors in those approximations (for example via mesh adaptation in the case of finite element solutions). Error estimates have been constructed with respect to: a) Norms associated with the solution function spaces. b) Quantities of interest ⇒ goal-oriented error estimation. S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 4 / 38

Introduction Flow around obstacle (Stokes flow) Top plots: 3 3 Residual-based Error Estimation 2 2 and adaptivity 1 1 0 0 3 4 5 6 3 4 5 6 Bottom plots: Goal-Oriented 3 3 Error Estimation and adaptivity 2 2 QoI = averaged 1 1 vorticity in lower left corner. 0 0 3 4 5 6 3 4 5 6 S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 5 / 38

Introduction Catenary (linearized) model: − Tu ′′ = f, in Ω = (0 , 1) u = u 0 , at x = 0 u = u 1 , at x = 1 Exact solution: u ( x ) = f 2 T x (1 − x ) + u 0 (1 − x ) + u 1 x Quantity of interest: � u 0 + u 1 � + f Deflection at center point: Q ( u ) = u (0 . 5) = 2 8 T � 1 � u 0 + u 1 � f Average deflection: Q ( u ) = u ( x ) dx = + 2 12 T 0 + f � � Slope at origin: Q ( u ) = u ′ (0) = u 1 − u 0 2 T S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 6 / 38

Introduction Green’s function Suppose here that u 0 = u 1 = 0 . Strong form of problem: Weak formulation: − Tu ′′ = f, Given f ∈ L 2 (Ω) , in Ω = (0 , 1) Find u ∈ V = H 1 0 (Ω) s.t. u = 0 , at x = 0 u = 0 , at x = 1 B ( u, v ) = F ( v ) ∀ v ∈ V where  � 1   B ( u, v ) = Tu ′ v ′ dx   0 � 1   F ( v ) = fv dx   0 S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 7 / 38

Introduction Green’s function Let Q ( u ) = u ( x 0 ) . Note that: � 1 Q ( u ) = u ( x 0 ) = u ( x ) δ ( x − x 0 ) dx 0 The Green function is the function G 0 = G 0 ( x ) ∈ V that yields: � 1 Q ( u ) = u ( x 0 ) = fG 0 dx = F ( G 0 ) 0 that is: Q ( u ) = F ( G 0 ) = B ( u, G 0 ) S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 8 / 38

Introduction Green’s function Q ( u ) = B ( u, G 0 ) = F ( G 0 ) Primal problem: Adjoint (dual) problem: − Tu ′′ = f in (0 , 1) Find G 0 ∈ V s.t. u = 0 at x = 0 , 1 B ( v, G 0 ) = Q ( v ) ∀ v ∈ V Weak form: Strong form: Given f ∈ L 2 (Ω) , find u ∈ V s.t. � 1 � 1 Tv ′ G ′ 0 dx = vδdx B ( u, v ) = F ( v ) ∀ v ∈ V 0 0 so that: Provide QoI: � 1 − TG ′′ 0 = δ in (0 , 1) Q ( u ) = u ( x ) δ ( x − x 0 ) dx G 0 = 0 at x = 0 , 1 0 S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 9 / 38

Introduction Green’s function Adjoint problem: − TG ′′ 0 = δ in (0 , 1) G 0 = 0 at x = 0 , 1 Analytical solution:  (1 − x 0 ) x , 0 ≤ x ≤ x 0   2 T G 0 ( x ) = x 0 (1 − x )   , x 0 ≤ x ≤ 1 2 T S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 10 / 38

Adjoints and QoIs Generalized Green Function Abstract linear BVP: Find u ∈ V, B ( u, v ) = F ( v ) , ∀ v ∈ V Quantity of interest: � Q ( u ) = u ( x ) k ( x ) dx Ω Adjoint (dual) problem: Find p ∈ V, B ( v, p ) = Q ( v ) , ∀ v ∈ V FE approximation: Let V h ⊂ V Find u h ∈ V h , ∀ v h ∈ V h B ( u h , v h ) = F ( v h ) , Error equation: Find e ∈ V, B ( e, v h ) = R ( u h ; v ) ≡ F ( v ) − B ( u h , v ) , ∀ v ∈ V S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 11 / 38

Error Representation Error Representation Goal is to estimate E = Q ( u ) − Q ( u h ) E = B ( u, p ) − B ( u h , p ) (From adjoint problem) = F ( p ) − B ( u h , p ) (From primal problem) = R ( u h ; p ) (From definition of residual) = R ( u h ; p − p h ) (From orthogonality property) E = Q ( u − u h ) = Q ( e ) (From linearity of Q ) = B ( e, p ) (From adjoint problem) = B ( e, p − p h ) (From orthogonality property) = R ( u h ; p − p h ) (From error equation) S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 12 / 38

Adaptive strategies Adaptive strategies Let ˜ p be a higher-order approximation of the adjoint solution on same mesh as u h , i.e. p ∈ V h,p +1 , ∀ v ∈ V h,p +1 Find ˜ B ( v, ˜ p ) = Q ( v ) , p h = Π h,p ˜ Approach 1: Using only ˜ p and ˜ p � E ≈ η = R ( u h ; ˜ p − ˜ p h ) = R K ( u h ; ˜ p − ˜ p h ) K u ∈ V h,p +1 , Approach 2: Using ˜ p , but also ˜ � E ≈ η = B (˜ u − u h , ˜ p − ˜ p h ) = B K (˜ u − u h , ˜ p − ˜ p h ) K S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 13 / 38

Adaptive strategies Adaptive Strategies Adaptive scheme (catenary exemple): η ≈ R ( u h ; ˜ p − ˜ p h ) = F (˜ p − ˜ p h ) − B ( u h , ˜ p − ˜ p h ) � 1 � 1 p h ) ′ dx Tu ′ = f (˜ p − ˜ p h ) dx − h (˜ p − ˜ 0 0 N e � � � � � p h ) + Tu ′′ Tu ′ = f (˜ p − ˜ h (˜ p − ˜ p h ) dx − h (˜ p − ˜ p h ) ds K ∂K K =1 N e � � 1 � � � ( f + Tu ′′ [ Tu ′ = h )(˜ p − ˜ p h ) dx − 2[ h ] ](˜ p − ˜ p h ) | x K i K K =1 i =1 , 2 � �� � η K Refinement criterion: | η K | If max K | η K | ≥ γ tol , then refine element K S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 14 / 38

Adaptive strategies Example: Elliptic problem 1.00 1.00 0.75 0.75 0.50 0.50 Y Y 0.25 0.25 0.00 0.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 X X Optimal? S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 15 / 38

Nonlinear Problems Nonlinear Problems Abstract nonlinear problem: Find u ∈ V, B ( u ; v ) = F ( v ) , ∀ v ∈ V where: V = Banach space B ( · ; · ) = differentiable semilinear form. Q ( u ) = a possibly nonlinear differentiable functional on V. S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 16 / 38

Nonlinear Problems Nonlinear Problems Linearization (Taylor with exact remainder): B ( u + w ; v ) = B ( u ; v ) + B ′ ( u ; w, v ) + ∆ B ( u, w, v ) Q ( u + w ) = Q ( u ) + Q ′ ( u ; w ) + ∆ Q ( u, w ) where 1 B ′ ( u ; v, z ) = lim θ [ B ( u + θv ; z ) − B ( u ; z )] θ → 0 1 Q ′ ( u ; v ) = lim θ [ Q ( u + θv ) − Q ( u )] θ → 0 � 1 B ′′ ( u + sw ; w, w, v )(1 − s ) ds ∆ B ( u, w, v ) = 0 � 1 Q ′′ ( u + sw ; w, w )(1 − s ) ds ∆ Q ( q, w ) = 0 S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 17 / 38

Nonlinear Problems Secant Form (exact) Representation � 1 B ′ ( su + (1 − s ) u h ; e, v ) ds B ( u ; v ) = B ( u h ; v ) + , ∀ v ∈ V 0 � �� � ≡ B s ( u,u h ; e,v )= secant form of B Then Q ( u ) − Q ( u h ) = Q ( e ) = B s ( u, u h ; e, p ) = B ( u ; p ) − B ( u h ; p ) = F ( p ) − B ( u h ; p ) = R ( u h ; p ) where p is the solution of the dual problem: B s ( u, u h ; v, p ) = Q ( v ) , Find p ∈ V such that ∀ v ∈ V S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 18 / 38

Nonlinear Problems Linearization approach: Error Equation, Adjoint problem Error Equation: Let e = u − u h B ( u ; v ) = B ( u h + e ; v ) = B ( u h ; v ) + B ′ ( u h ; e, v ) + ∆ B ( u h , e, v ) = F ( v ) So B ′ ( u h ; e, v ) + ∆ B ( u h , e, v ) = R ( u h ; v ) , Find e ∈ V, ∀ v ∈ V Dropping higher-order terms, error can be approximated by: B ′ ( u h ; ˆ Find ˆ e ∈ V, e, v ) = R ( u h ; v ) , ∀ v ∈ V Adjoint problem: B ′ ( u h ; v, p ) = Q ′ ( u h ; v ) , Find p ∈ V such that ∀ v ∈ V S. Prudhomme (Polytechnique Montr´ eal) Goal-oriented Error Estimation November 4-8, 2019 19 / 38