Aposteriori error analysis of timestepping schemes for the wave - PowerPoint PPT Presentation

Aposteriori error analysis of timestepping schemes for the wave equation using elliptic reconstruction techniques Omar Lakkis Mathematics — University of Sussex — Brighton, England UK a talk based on joint work with E.H. Georgoulis (Athens GR & Leicester GB), C. Makridakis (Sussex GB & Crete GR), J.M. Virtanen (Leicester GB) 6 January 2016 Adaptive algorithms for computational partial differential equations University of Birmingham England UK Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 1 / 36

Outline 1 The wave equation and backward Euler 2 The wave equation, backward Euler and energy A user’s guide to the elliptic reconstruction Aposteriori estimates for backward Euler 3 A Baker’s trick Main result for backward Euler Aposteriori estimates for the Leapfrog method 4 Verlet, Newmark, Leapfrog, Cosine, etc. Numerical results Closing 5 Remarks Credits Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 2 / 36

The model linear wave equation a second order hyperbolic problem Initial–boundary value problem ∂ tt u = ∆ u + f , u | spatial boundary = 0 and u (0) = u 0 , ∂ t u (0) = v 0 . Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 3 / 36

The model linear wave equation a second order hyperbolic problem Initial–boundary value problem ∂ tt u = ∆ u + f , u | spatial boundary = 0 and u (0) = u 0 , ∂ t u (0) = v 0 . As a first order system � 0 � u � � � u � � 0 � 1 ∂ t = + . v ∆ 0 v f Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 3 / 36

The model linear wave equation a second order hyperbolic problem Initial–boundary value problem ∂ tt u = ∆ u + f , u | spatial boundary = 0 and u (0) = u 0 , ∂ t u (0) = v 0 . As a first order system � 0 � u � � � u � � 0 � 1 ∂ t = + . v ∆ 0 v f Simplest timestepping scheme: Backward–Euler for the system (Bernardi and S¨ uli, 2005): � � u n � u n − 1 � � � � � 1 − k n 0 = + . v n v n − 1 + k n ∆ 1 f ( t n ) Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 3 / 36

The model linear wave equation a second order hyperbolic problem Initial–boundary value problem ∂ tt u = ∆ u + f , u | spatial boundary = 0 and u (0) = u 0 , ∂ t u (0) = v 0 . As a first order system � 0 � u � � � u � � 0 � 1 ∂ t = + . v ∆ 0 v f Simplest timestepping scheme: Backward–Euler for the system (Bernardi and S¨ uli, 2005): � � u n � u n − 1 � � � � � 1 − k n 0 = + . v n v n − 1 + k n ∆ 1 f ( t n ) Resulting equation form is a 2-step (timestep = k n ) method u n − u n − 1 − u n − 1 − u n − 2 − k n ∆ u n = k n f ( t n ) . k n k n − 1 Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 3 / 36

Previous work Goal-oriented duality approach: W. Bangerth and R. Rannacher (1999) and Wolfgang Bangerth and Rolf Rannacher (2001) Direct Galerkin orthogonality with energy approach: Bernardi and S¨ uli (2005) Semidiscrete analysis: Picasso (2010) Heuristic-based adaptive methods: Wiberg & Li (1998), Schemann & Bornemann (1998), Romero & Lacoma (2006). Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 4 / 36

Spatially semidiscrete schemes Suppose the exact elliptic operator A : Dom A → Ran A , e.g, − ∆ : H 1 0 (Ω) → H − 1 (Ω) is discretized as A : → V V (8.1) V �→ AV : � AV , Φ � = � A V | Φ � ∀ Φ ∈ V then a spatially semidiscrete method for the wave equation takes the system form � U ( t ) � � 0 � � U ( t ) � � � 1 0 d t + = Π V f ( t ) V ( t ) A 0 V ( t ) where Π V is the L 2 (Ω) -orthogonal projection. Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 5 / 36

Spatially semidiscrete schemes Suppose the exact elliptic operator A : Dom A → Ran A , e.g, − ∆ : H 1 0 (Ω) → H − 1 (Ω) is discretized as A : → V V (9.1) V �→ AV : � AV , Φ � = � A V | Φ � ∀ Φ ∈ V then a spatially semidiscrete method for the wave equation takes the system form � U ( t ) � � 0 � � U ( t ) � � � 1 0 d t + = Π V f ( t ) V ( t ) A 0 V ( t ) where Π V is the L 2 (Ω) -orthogonal projection. Equation form looks very nice (it is) to analyze d tt U ( t ) + AU ( t ) = Π V f ( t ) . Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 5 / 36

Energy methods for the wave equation Combining the elliptic reconstruction to energy estimates for the system, one recovers the estimates of Bernardi and S¨ uli (2005). Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 6 / 36

Energy methods for the wave equation Combining the elliptic reconstruction to energy estimates for the system, one recovers the estimates of Bernardi and S¨ uli (2005). Combining the elliptic reconstruction to energy estimates for the equation form, one recovers more general estimates, but only in the spatially discrete scheme. Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 6 / 36

Energy methods for the wave equation Combining the elliptic reconstruction to energy estimates for the system, one recovers the estimates of Bernardi and S¨ uli (2005). Combining the elliptic reconstruction to energy estimates for the equation form, one recovers more general estimates, but only in the spatially discrete scheme. Remark (Failure of fully discrete analysis in energy for equation) is due to “bad” time reconstruction, using polynomials, and there seems to be no way out. . . Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 6 / 36

Energy methods for the wave equation Combining the elliptic reconstruction to energy estimates for the system, one recovers the estimates of Bernardi and S¨ uli (2005). Combining the elliptic reconstruction to energy estimates for the equation form, one recovers more general estimates, but only in the spatially discrete scheme. Remark (Failure of fully discrete analysis in energy for equation) is due to “bad” time reconstruction, using polynomials, and there seems to be no way out. . . Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 6 / 36

Fully discrete scheme We consider the fully discrete scheme for the initial value wave problem for each n = 1 , . . . , N , find U n ∈ V n h such that � ∂ 2 U n , V � + a ( U n , V ) = � f n , V � ∀ V ∈ V n h , where f n := f ( t n , · ) , the backward second and first finite differences ∂ 2 U n := ∂U n − ∂U n − 1 , k n with U n − U n − 1  , for n = 1 , 2 , . . . , N, ∂U n :=  k n V 0 := π 0 u 1 for n = 0 ,  Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 7 / 36

A User’s Guide to the Elliptic Reconstruction u ( · , t ) ∈ H 1 0 (Ω) exact solution at fixed time t H 1 0 (Ω) u

A User’s Guide to the Elliptic Reconstruction u ( · , t ) ∈ H 1 0 (Ω) exact solution at fixed time t H 1 0 (Ω) U ( · , t ) ∈ V time-dependent V -FE approximation of u ( · , t ) . u V (conforming) U

A User’s Guide to the Elliptic Reconstruction u ( · , t ) ∈ H 1 0 (Ω) exact solution at fixed time t H 1 0 (Ω) U ( · , t ) ∈ V time-dependent V -FE approximation of u ( · , t ) . u want: w intermediate object ? w between u and U V (conforming) U

A User’s Guide to the Elliptic Reconstruction u ( · , t ) ∈ H 1 0 (Ω) exact solution at fixed time t H 1 0 (Ω) U ( · , t ) ∈ V time-dependent V -FE approximation of u ( · , t ) . u want: w intermediate object ? w (= R V U ) between u and U ǫ want: U ∈ V an elliptic V -FE V (conforming) approximation of w , error Galerkin ⊥ ǫ = U − w , U

A User’s Guide to the Elliptic Reconstruction u ( · , t ) ∈ H 1 0 (Ω) exact solution at fixed time t H 1 0 (Ω) U ( · , t ) ∈ V time-dependent V -FE approximation of u ( · , t ) . u ρ want: w intermediate object ? w (= R V U ) between u and U ǫ want: U ∈ V an elliptic V -FE V (conforming) approximation of w , error Galerkin ⊥ ǫ = U − w , U Galerkin orthogonality ⇒ aposteriori bounds on � ǫ � are available “off the shelf”, Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 8 / 36

A User’s Guide to the Elliptic Reconstruction u ( · , t ) ∈ H 1 0 (Ω) exact solution at fixed time t H 1 0 (Ω) U ( · , t ) ∈ V time-dependent V -FE approximation of u ( · , t ) . u ρ want: w intermediate object ? w (= R V U ) between u and U ǫ want: U ∈ V an elliptic V -FE V (conforming) approximation of w , error Galerkin ⊥ ǫ = U − w , U Galerkin orthogonality ⇒ aposteriori bounds on � ǫ � are available “off the shelf”, w not computable, but time dependent error ρ := w − u satifies original PDE with computable data. Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 8 / 36

Combining the original wave equation (21.1) ∂ tt u + A u = f Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 9 / 36

Combining the original wave equation (22.1) ∂ tt u + A u = f semidiscrete scheme ∂ tt U + AU = Π V f (22.2) Omar Lakkis (Sussex, GB) Aposteriori wave Birmingham, 2016-01-06 9 / 36