Error estimates for the Galerkin finite element approximation for a - PowerPoint PPT Presentation

General linear vibration model Modal damping Galerkin approximation Error estimates Error estimates for the Galerkin finite element approximation for a linear second order hyperbolic equation with modal damping Alna van der Merwe Department of Mathematical Sciences Auckland University of Technology New Zealand SANUM 2016 22 March 2016 Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Outline 1 General linear vibration model 2 Modal damping 3 Galerkin approximation 4 Error estimates Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates General linear vibration model Let X , W and V denote Hilbert spaces such that V ⊂ W ⊂ X Space Inner product Norm ( · , · ) X � · � X X Inertia space W c ( · , · ) � · � W Energy space V b ( · , · ) � · � V J is an open interval containing zero, or of the form [0 , τ ) or [0 , ∞ ). Problem G such that u ′ is continuous at � � Given a function f : J → X, find a function u ∈ C J ; V 0 , and for each t ∈ J, u ( t ) ∈ V , u ′ ( t ) ∈ V , u ′′ ( t ) ∈ W , and � � � � � � � � c u ′′ ( t ) , v + a u ′ ( t ) , v + b u ( t ) , v = f ( t ) , v for each v ∈ V , (1) X while u (0) = u 0 and u ′ (0) = u 1 Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates General linear vibration model Assumptions We assume that the following additional properties hold E1 V is dense in W and W is dense in X E2 There exists a constant C b such that � v � W ≤ C b � v � V for each v ∈ V E3 There exists a constant C c such that � v � X ≤ C c � v � W for each v ∈ W E4 The bilinear form a is nonnegative, symmetric and bounded on V Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Viscous type damping | a ( u , v ) | ≤ K a � u � W � v � W M. Basson and N. F. J. van Rensburg (2013) Galerkin finite element approximation of general linear second order hyperbolic equations, Numerical Functional Analysis and Optimization, 34:9, 976 - 1000 DOI: 10.1080/01630563.2013.807286 Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Modal damping a ( u , v ) = µ b ( u , v ) + η c ( u , v ); µ ≥ 0 , η ≥ 0 Viscous damping (air damping, external damping) Material damping (strain rate damping, Kelvin-Voigt damping, internal damping) Example: Euler-Bernoulli beam model with viscous damping and internal damping Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Hyperbolic heat conduction Bounded domain Ω ⊂ R 3 Conservation of heat energy: ρ c p ∂ t T = −∇ · q + f Density ρ , the specific heat c p , temperature T Heat source term f , heat flux q Fourier’s law: Cattaneo-Vernotte law: Generalised dual-phase-lag: q = − k ∇ T q ( r , t + τ q ) = − k ∇ T ( r , t ) q ( r , t + τ q ) = − A ∇ T ( r , t + τ T ) q + τ q ∂ t q = − k ∇ T q + τ q ∂ t q = − A ∇ T − τ T A ∂ t ∇ T Heat equation: Hyperbolic heat equation: Generalised dual-phase-lag model: ∂ t T = c 2 ∇ 2 T τ q ∂ 2 t T + ∂ t T = c 2 ∇ 2 T γ 2 ∂ 2 t T + γ 1 ∂ t T − τ T ∇ · ( A ∇ ( ∂ t T )) = ∇ · ( A ∇ T ) Thermal conductivity k , and c 2 = k ρ c p Time delay τ q in heat flux, and time delay τ T for temperature gradient γ 1 = ρ c p , and γ 2 = τ q ρ c p Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Galerkin approximation S h is a finite dimensional subspace of V Problem G h Given a function f : J → X, find a function u h ∈ C 2 ( J ) such that u ′ h is continuous at 0 and for each t ∈ J, u h ( t ) ∈ S h and X for each v ∈ S h , � u ′′ � � u ′ � � � � � c h ( t ) , v + a h ( t ) , v + b u h ( t ) , v = f ( t ) , v (2) while u h (0) = u h h (0) = u h 0 and u ′ 1 1 are elements of S h as close as possible to u 0 and u 1 The initial values u h 0 and u h Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Semi-discrete approximation A projection is used to find an estimate for the discretization error e h ( t ) = u ( t ) − u h ( t ) The projection operator P is defined by b ( u − Pu , v ) = 0 for each v ∈ S h The idea of the projection method is to split the error e h ( t ) into two parts e h ( t ) = e p ( t ) + e ( t ) = ( u ( t ) − Pu ( t )) + ( Pu ( t ) − u h ( t )) Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Fundamental estimate Lemma If the solution u of Problem G satisfies Pu ∈ C 2 ( J ) , then for any t ∈ J, � t √ � �� � e ( t ) � V + � e ′ ( t ) � W ≤ 2 � e (0) � V + � e ′ (0) � W + � � e ′′ p � W + η � e ′ p � W 0 The proof is based on the brief outline given in Strange and Fix, An Analysis of the Finite Element Method (1973) for the undamped wave equation. The energy expression E ( t ) for e ( t ) forms the central concept in the proof E ( t ) = 1 2 c ( e ′ ( t ) , e ′ ( t )) + 1 2 b ( e ( t ) , e ( t )) = 1 W + 1 2 � e ′ ( t ) � 2 2 � e ( t ) � 2 V Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Projection error There exists a subspace H ( V , k ) of V , and positive constants C and α (depending on V and k ) such that for u ∈ H ( V , k ), � u − Pu � V ≤ Ch α � u � H ( V , k ) , where � · � H ( V , k ) is a norm or semi-norm associated with H ( V , k ) k is a positive integer determined by the regularity of the solution u Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Galerkin approximation: semi-discrete approximation u h ( t ) = � n j =1 u j ( t ) φ j if { φ j } forms a basis for S h . Semi-discrete approximation u ( t ) = ¯ u ′′ ( t ) + C ¯ u ′ ( t ) + K ¯ M ¯ F ( t ) u ( t ) = [ u 1 ( t ) u 2 ( t ) . . . u n ( t )] t ¯ Approximation of ¯ u ( t k ) is ¯ u k Fully discrete approximation ( δ t ) − 2 M [¯ u k − 1 ] + (2 δ t ) − 1 C [¯ u k = ¯ u k +1 − 2¯ u k + ¯ u k +1 − ¯ u k − 1 ] + K ¯ F ( t k ) Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Galerkin approximation: fully discrete approximation j =1 u j u k = ( u 1 k , u 2 k , . . . , u n k ), then u h k = � n If ¯ k φ j is the approximation for u h ( t k ). Problem G h D Assume that ρ 0 and ρ 1 are positive numbers such that ρ 0 + 2 ρ 1 = 1. Find a sequence k } ⊂ S h such that for each k = 1 , 2 , . . . , N − 1, { u h c ( δ t − 2 [ u h k − 1 ] , v ) + a ((2 δ t ) − 1 [ u h k +1 − 2 u h k + u h k +1 − u h k − 1 ] , v ) (3) b ( ρ 1 u h k +1 + ρ 0 u h k + ρ 1 u h + k − 1 , v ) = ( ρ 1 f ( t k +1 ) + ρ 0 f ( t k ) + ρ 1 f ( t k − 1 ) , v ) X , for each v ∈ S h while u h 0 = d h and u h 1 − u h − 1 = (2 δ t ) v h . Alna van der Merwe Error estimates – modal damping

General linear vibration model Modal damping Galerkin approximation Error estimates Galerkin approximation: fully discrete approximation Error estimate � u ( t k ) − u h � u ( t k ) − u h ( t k ) � W + � u h ( t k ) − u h k � W ≤ k � W K 2 h α + K 1 δ t 2 . ≤ Alna van der Merwe Error estimates – modal damping