Local convergence of adaptive finite element methods for nonlinear - PowerPoint PPT Presentation

Finite element methods Adaptive FEM Convergence for nonlinear PDE CRM/McGill Applied Mathematics Seminar Local convergence of adaptive finite element methods for nonlinear problems Gantumur Tsogtgerel McGill University Joint with Michael Holst and Yunrong Zhu November 2, 2009

Finite element methods Adaptive FEM Convergence for nonlinear PDE Model problem − u ′′ ( x ) = f ( x ) in I = ( 0, 1 ) , and u ( 0 ) = u ( 1 ) = 0. Define A : C 2 ( I ) → C 0 ( I ) by u ∈ C 2 ( I ) . Au = − u ′′ for Let D = C ∞ 0 ( I ) ∗ . Then C 0 ( I ) ⊂ L 1 ( I ) ⊂ D by � 1 w ∈ L 1 ( I ) , v ∈ C ∞ � w , v � = wv � � w � L 1 � v � C 0 for 0 ( I ) . 0 We can extend A to A : D → D by � Au , v � = − � u , v ′′ � for any v ∈ C ∞ 0 ( I ) . Note that if u ∈ C 2 ( I ) we have � 1 � 1 � 1 1 1 � u ′ v ′ = u ′ v ′ � − u ′′ v = − u ′′ v uv ′′ . 0 + 0 − � � � � 0 0 0

Finite element methods Adaptive FEM Convergence for nonlinear PDE Weak formulation � 1 � 1 1 � u ′ v ′ � � u ′ � L 2 � v ′ � L 2 � Au , v � = − u ′′ v = uv 0 + � � 0 0 is an inner product on C ∞ 0 ( I ) . The induced norm is � 1 � u � 2 ( u ′ ) 2 H 1 = � Au , u � = 0 0 ( I ) wrt to this norm is denoted by H 1 0 ( I ) . and the completion of C ∞ 0 ( I ) ∗ is bounded, linear, and invertible. 0 ( I ) → H − 1 ( I ) ≡ H 1 Thus A : H 1 In particular, for any f ∈ H − 1 ( I ) the following equation has a unique solution Au = f .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Galerkin method Let X be a Hilbert space, and A : X → X ∗ be a bounded linear operator satisfying � Av , v � � α 2 � v � 2 for any v ∈ X ( α > 0 ) . X Then � A · , ·� is an inner product on X , inducing a norm � · � A equivalent to � · � X . So A is invertible. Au = f ⇔ � Au , v � = � f , v � v ∈ X . for all Let X h ⊂ X be a linear subspace. Consider u h ∈ X h such that � Au h , v � = � f , v � v ∈ X h . for all This gives the Galerkin orthogonality � A ( u − u h ) , v � = 0 v ∈ X h for all or u − u h ⊥ A X h . � u − u h � A = inf v ∈ X h � u − v � A .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Finite element method X = H 1 0 ( I ) . Let T h = { [ 0, h ] , [ h , 2 h ] , . . . , [ 1 − h , 1 ] } , and X h = { v ∈ C 0 0 ( I ) : v is linear on each of e ∈ T h } . Let { φ i } be a basis of X h , and put u h = � i U i φ i � i U i � Aφ i , φ k � = � f , φ k � k = 1, . . . , m . v ∈ X h � u − v � A � h s − 1 � u � H s � u − u h � A = inf for any s � 2. In general, and piecewise polynomial elements of order d , we have � u − u h � H 1 � h s − 1 � u � H s for any s � d . In n -dimension, the number of degrees of freedom N ∼ h − n , so � u − u h � H 1 � N − s − 1 n � u � H s for any s � d .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Saddle point problems � − ∆ u + grad p = f � − ∆ � grad Au = f A = or with . div 0 div u = 0 0 ) n × L 2 . Let X h ⊂ X , and consider u h ∈ X h such that Set X = ( H 1 � Au h , v � = � f , v � v ∈ X h . for all The following theorem is due to Jinchao Xu and Ludmil Zikatonov. Theorem (Babuˇ ska-Brezzi-Ladyzhenskaya condition) The above problem is uniquely solvable if and only if � Aw , v � = α h > 0. w ∈ X h sup inf � w � X � v � X v ∈ X h Moreover, the latter implies � u − u h � � � A � v ∈ X h � u − v � X . inf α h

Finite element methods Adaptive FEM Convergence for nonlinear PDE Semilinear problems − ∆u + u q = f Ω ⊂ R 3 , in with u = 0 on ∂Ω . s = 5 − q u ∈ H 1 ⇒ u ∈ L 6 ⇒ u q ∈ L 6 /q ⇒ u q ∈ H − 1 + s with . 2 If f ∈ L 2 , then (− ∆ ) − 1 ( f − u q ) ∈ H 1 + s . Hence φ : H 1 0 → H 1 0 : u �→ (− ∆ ) − 1 ( f − u q ) is compact if q < 5 . By Schauder, there exists u ∈ H 1 0 such that φ ( u ) = u . Galerkin approximation of a locally unique solution is locally quasi-optimal.

Finite element methods Adaptive FEM Convergence for nonlinear PDE A posteriori error estimates X = H 1 0 ( I ) . Let T h = { [ 0, x 1 ] , [ x 1 , x 2 ] , . . . , [ x m , 1 ] } , and X h = { v ∈ C 0 0 ( I ) : v is linear on each of e ∈ T h } . For any v ∈ X and its linear interpolant v h ∈ X h , we have � 1 � f − Au h , v � = � f − Au h , v − v h � = f ( v − v h ) − u ′ h ( v − v h ) ′ 0 � x i + 1 � x i + 1 � f ( v − v h ) − u ′ = h ( v − v h ) � � x i x i i � � C � f � L 2 ( x i , x i + 1 ) � v − v h � L 2 ( x i , x i + 1 ) i � � C � f � L 2 ( x i , x i + 1 ) h i � v � H 1 ( x i , x i + 1 ) i implying that � � � f − Au h � 2 h 2 i � f � 2 η 2 H − 1 � C L 2 ( x i , x i + 1 ) = i . i i

Finite element methods Adaptive FEM Convergence for nonlinear PDE Lower bound Let f be piecewise constant wrt T h , and ϕ be a “bubble” function supported in ( x i , x i + 1 ) . � x i + 1 � x i + 1 � f � 2 L 2 ( x i , x i + 1 ) � f · fϕ = ( u − u h ) ′ ( fϕ ) ′ x i x i � � u − u h � H 1 ( x i , x i + 1 ) � fϕ � H 1 � � u − u h � H 1 ( x i , x i + 1 ) � f � L 2 ( x i , x i + 1 ) h − 1 i implying that η i = h i � f � L 2 ( x i , x i + 1 ) � � u − u h � H 1 ( x i , x i + 1 ) .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Interior node property Let f be piecewise constant wrt T h , and ϕ ∈ X ℓ ( ℓ < h ) be a “bubble” function supported in ( x i , x i + 1 ) . � x i + 1 � x i + 1 � f � 2 L 2 ( x i , x i + 1 ) � f · fϕ = ( u ℓ − u h ) ′ ( fϕ ) ′ x i x i � � u ℓ − u h � H 1 ( x i , x i + 1 ) � fϕ � H 1 � � u ℓ − u h � H 1 ( x i , x i + 1 ) � f � L 2 ( x i , x i + 1 ) h − 1 i implying that η i = h i � f � L 2 ( x i , x i + 1 ) � � u ℓ − u h � H 1 ( x i , x i + 1 ) .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Adaptive finite element method Start with some initial mesh T 0 . Set k = 0 , and repeat • Solve for the Galerkin solution u k on the mesh T k • Estimate the error indicators { η i } over the elements of T k • Refine the elements of T k with largest error, to get T k + 1 • k + + Questions: • u k → u ? • � u k − u � H 1 � ρ k with ρ < 1 ? • dim X k ∼ ?

Finite element methods Adaptive FEM Convergence for nonlinear PDE Linear convergence From the Galerkin orthogonality � A ( u − u i + 1 ) , v � = 0 for all v ∈ X i + 1 , taking v = u i + 1 − u i , we have � u − u i � 2 A = � u − u i + 1 � 2 A + � u i + 1 − u i � 2 A . So if � u i + 1 − u i � A � c � u − u i � A , with constant c ∈ ( 0, 1 ) , we have � u − u i + 1 � 2 A = � u − u i � 2 A − � u i + 1 − u i � 2 A � ( 1 − c 2 ) � u − u i � 2 A .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Quasi-orthogonality for semilinear problems Let us consider a ( u , v ) + ( f ( u ) , v ) = 0, ∀ v ∈ H We have � u − u i � 2 a = � u − u i + 1 � 2 a + � u i + 1 − u i � 2 a + 2 a ( u − u i + 1 , u i + 1 − u i ) a ( u − u i + 1 , u i + 1 − u i ) = ( f ( u ) − f ( u i + 1 ) , u i + 1 − u i ) � C � f ( u ) − f ( u i + 1 ) � L 2 � u i + 1 − u i � L 2 � C � u − u i + 1 � L 2 � u i + 1 − u i � L 2 � Ch i + 1 � u − u i + 1 � H 1 � u i + 1 − u i � H 1

Finite element methods Adaptive FEM Convergence for nonlinear PDE Abstract argument Let F : X → X ∗ be continuous, and consider � F ( u ) , v � = 0 v ∈ X . for all Suppose that an AFEM generated the following • subspaces X 0 , X 1 , . . . ⊂ X • approximations u 0 , u 1 , . . . , solutions to � F ( u k ) , v k � = 0 ∀ v k ∈ X k Let X ∞ = ∪ i X i , and let u ∞ ∈ X ∞ be such that � F ( u ∞ ) , v � = 0 ∀ v ∈ X ∞ . If u k are locally quasi-optimal, then � u ∞ − u k � X � inf v ∈ X k � u ∞ − v � X → 0 as k → ∞ . For any v ∈ X , we have � F ( u ∞ ) , v � = � F ( u ∞ ) − F ( u k ) , v � + � F ( u k ) , v � � � F ( u ∞ ) − F ( u k ) � X ∗ � v � X + | � F ( u k ) , v � | → 0, provided lim k → ∞ � F ( u k ) , v � = 0 .

Finite element methods Adaptive FEM Convergence for nonlinear PDE Error estimate and marking X = H 1 0 ( Ω ) . η k : T k → R error estimator corresponding to T k and u k . If D ⊂ Ω is a union of some elements in T k , then � F ( u k ) , v � � η k ( D ) � v � H 1 ( D ) + η k ( Ω \ D ) � v � H 1 ( Ω \ D ) , and for some f ∈ L 2 ( Ω ) η k ( D ) � � u k � H 1 ( D ) + � f � L 2 ( D ) . Let M k ⊂ T k be the marked elements to refine. Then η k ( τ ) � C max σ ∈ M k η k ( σ ) τ ∈ T k \ M k