An Adaptive Space-Time Boundary Element Method for the Wave Equation

Transcription

1 W I S S E N T E C H N I K L E I D E N S C H A F T An Adaptive Space-Time Boundary Element Method for the Wave Equation Marco Zank and Olaf Steinbach Institut für Numerische Mathematik AANMPDE(JS)-9-16, Strobl 5 July

2 Table of contents 1. Laplace transform method 2. Energy approach 3. One-dimensional wave equation 4. Numerical examples 2

3 Laplace transform method [Bamberger, Ha Duong 1986] u + u tt = 0 in Ω R +, u Γ R + = g on Γ R +, i.c. 1. Apply Laplace transform to the wave equation and end up with the Helmholtz equation in the frequency domain for ω C 0 : U(ω) ω 2 U(ω) = 0 in Ω, γ int 0 U(ω) = G(ω) on Γ 2. Reformulate the Helmholtz equation via potentials for ω C 0 : Ṽ (ω): H 1/2 (Γ) H 1 (Ω), U(ω) = Ṽ (ω)λ(ω) 3. Analyse the bie explicitly in terms of ω C σ0 for fixed σ 0 > 0 : V (ω) C(σ 0 ) ω, V (ω) 1 C(σ0 ) ω 2 4. Apply Paley-Wiener theorem and convolution theorem for the inverse transform to the time domain: u = Ṽ λ := L 1 (ω Ṽ (ω)λ(ω)) 3

4 Bilinear form and loss of time regularity Bilinear form in the frequency domain for ω {Im ω σ 0 > 0} : ψ, ιωv (ω)ϕ ϕ, ψ H 1/2 (Γ) Define bilinear form for σ σ 0 > 0 a σ (λ, τ) := 0 e 2σt τ(t), Loss of regularity in time for σ σ 0 > 0 ( V dλ ) (t) dt. dt Γ V 1 V : H 3 σ(r, H 1/2 (Γ)) H 0 σ(r, H 1/2 (Γ)), V 1 V : H 2; 1/2 σ,γ H 0; 1/2 σ,γ. 4

5 An energy approach [Aimi et al. 2008] For any solution u of the wave equation tt u(x, t) u(x, t) = 0 for x R d \ Γ, t (0, T ), t u(x, 0) = u(x, 0) = 0 for x R d \ Γ, u(x, t) = g(x, t) for (x, t) Σ := Γ [0, T ] it holds 0 = ( tt u u) t u ( 1 = t 2 ( tu) ) 2 u 2 ( t u u). 5

6 With Green s formula it follows 0 = 1 { ( t u(x, T )) 2 + u(x, T ) 2} dx 2 R d \Γ T Define the energy E(T ) := t γ int 0 u(, t), [[γ 1u]](, t) Γ dt. R d \Γ { ( t u(x, T )) 2 + u(x, T ) 2} dx and represent the solution by u = Ṽ λ. Hence, E(T ) = T Define the bilinear form 0 t (V λ)(t), λ(t) Γ dt = t (V λ), λ Σ. a E (λ, ϕ) := t (V λ), ϕ Σ. 6

7 One-dimensional wave equation Q := (0, L) (0, T ) and Σ := {0, L} [0, T ]. Single layer potential: Ṽ w(x, t) = 1 2 t x w 0 (s)ds for t R, x R \ {0, L} and a density w = t x L ( w0 w L ). w L (s)ds 7

8 Single layer operator V : L 2 (Σ) H 1 {0} (Σ) ( V w(t) = 1 t 0 w 0(s)ds + ) t L 0 w L (s)ds 2 t L 0 w 0 (s)ds + t 0 w L(s)ds for t [0, T ] with L 2 (Σ) := H 1 {0} (Σ) := {v = ( v0 { v = ( v0 v L ) } : v 0, v L L 2 (0, T ), ) } : v v 0, v L H 1 (0, T ), γv 0 (0) = γv L (0) = 0. L 8

9 Theorem (Aimi et al. 2008) The bilinear form a E : L 2 (Σ) L 2 (Σ) R, a E (w, v) := t (V w), v L2 (Σ), is bounded and coercive: a E (ϕ, ϕ) C(T ) ϕ 2 L 2 (Σ) ϕ L 2 (Σ) Find w L 2 (Σ) for given g H{0} 1 (Σ) such that a E (w, v) = 1 2 tg, v L 2 (Σ) + t(k g), v L 2 (Σ) where the double layer operator is given by ( ) g0 K g(t) = K (t) = 1 ( ) gl (t L) g L 2 g 0 (t L) v L 2 (Σ), for t [0, T ]. 9

10 Decomposition Σ = N 0 +N L i=1 τ i. A conforming ansatz space for L 2 (Σ) is S t 0 (Σ) :=S0 t 0 (0, T ) S t 0 L (0, T ) {( )} ϕ 0 N0 {( =span 0,i 0 span 0 i=1 { } =span ˆϕ 0 N0 +N L i. The resulting linear system is i=1 V t w = g ϕ 0 L,j )} NL with V t R (N 0+N L ) (N 0 +N L ), g R N 0+N L and w R N 0+N L. j=1 10

11 A posteriori error estimator [Steinbach, Schulz 2000] η i := γ int 1 u w t Approximate solution in Q : L 2 (τ i ) ũ t := Ṽ w t W g Neumann trace of the approximate solution: Local error estimator: for i = 1,..., N 0 + N L w t := γ int 1 ũ t = 1 2 w t + K w t + D g η i := w t w t L 2 (τ i ) for i = 1,..., N 0 + N L For a parameter θ [0, 1] refine all elements τ i where η i θ max j η j. 11

12 Numerical examples with L = 3 and T = 6 12

13 Numerical examples with L = 3 and T = 6 12

14 Numerical examples with L = 3, T = 6 L γ int L N 0 + N L 1 u w t 2 (Σ) eoc u ũ t L 2 (Q) eoc e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

15 Numerical examples with L = 3, T = 6 1 γ 1 int u w t L 2 (Σ) uniform θ = 0.05 N ,000 10,000 degrees of freedom N : =N 0 + N L 14

16 Numerical examples with L = 3 and T = 6 15

17 Numerical examples with L = 3 and T = 6 15

18 Numerical examples with L = 3, T = 6 L γ int L N 0 + N L 1 u w t 2 (Σ) eoc u ũ t L 2 (Q) eoc e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e

19 Numerical examples with L = 3, T = 6 10 γ 1 int u w t L 2 (Σ) uniform θ = 0.05 θ = 0.5 N 1 N 1/ ,000 10,000 degrees of freedom N : =N 0 + N L 17

20 Numerical examples with L = 3, T = 6, θ = t 2 1 x

21 Numerical examples with L = 3, T = 6, θ = t 2 1 x

22 Numerical examples with L = 3, T = 6, θ = t 2 1 x

23 Numerical examples with L = 3, T = 6, θ = t 2 1 x

24 Numerical examples with L = 3, T = 6, θ = t 2 1 x

25 Summary and outlook Laplace transform method Loss of regularity in time Energy approach One-dimensional wave equation A posteriori error estimator Error analysis for 1D? Bilinear form for 2D, 3D? 19

26 [1] AIMI, A., DILIGENTI, M., GUARDASONI, C., AND PANIZZI, S. A space-time energetic formulation for wave propagation analysis by BEMs. Riv. Mat. Univ. Parma (7) 8 (2008), [2] BAMBERGER, A., AND HA DUONG, T. Formulation variationnelle pour le calcul de la diffraction d une onde acoustique par une surface rigide. Math. Methods Appl. Sci. 8, 4 (1986), [3] GLÄFKE, M. Adaptive methods for time domain boundary integral equations. PhD thesis, [4] HA-DUONG, T. On retarded potential boundary integral equations and their discretisation. In Topics in computational wave propagation, vol. 31 of Lect. Notes Comput. Sci. Eng. Springer, Berlin, 2003, pp [5] SAYAS, F.-J. Retarded Potentials and Time Domain Boundary Integral Equations: A Road Map. Springer, [6] SCHULZ, H., AND STEINBACH, O. A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem. Calcolo 37, 2 (2000),