A spacetime DPG method for acoustic waves

Transcription

1 A spacetime DPG method for acoustic waves Rainer Schneckenleitner JKU Linz January 30, 2018 ( JKU Linz ) Spacetime DPG method January 30, / 41

2 Overview 1 The transient wave problem 2 The broken weak formulation 3 Verification of the density condition 4 The ideal DPG method and a priori & a posteriori error estimates 5 Implementation of practical DPG and numerical results ( JKU Linz ) Spacetime DPG method January 30, / 41

3 Outline 1 The transient wave problem 2 The broken weak formulation 3 Verification of the density condition 4 The ideal DPG method and a priori & a posteriori error estimates 5 Implementation of practical DPG and numerical results ( JKU Linz ) Spacetime DPG method January 30, / 41

4 Problem formulation The wave equation Let Ω 0 R d be the spatial domain with boundary Ω 0 and let Ω = Ω 0 (0, T ) be the spacetime cylinder, with T > 0. Furthermore, let f L 2 (Ω) and g L 2 (Ω) d. Then the considered first order system for the wave equation is given by t q grad x µ = g, t µ div x q = f. Additionally, the wave equation is equipped with homogeneous initial and boundary conditions, i.e., µ t=0 = 0, q t=0 = 0, µ Ω0 (0,T ) = 0. ( JKU Linz ) Spacetime DPG method January 30, / 41

5 The formal wave operator Wave operator can be seen as first order distributional derivative operator A : L 2 (Ω) d+1 D (Ω) d+1 [ ] t u Au = q grad x u µ, t u µ div x u q where u L 2 (Ω) d+1 is split into u = [ uq u µ ] with u q L 2 (Ω) d and u µ L 2 (Ω). ( JKU Linz ) Spacetime DPG method January 30, / 41

6 The formal wave operator Theorem Let Ω be open. The space endowed with the norm is a Hilbert space. Proof. Blackboard. W (Ω) := {u L 2 (Ω) d+1 : Au L 2 (Ω) d+1 } = W u W = ( u 2 + Au 2 ) 1/2 ( JKU Linz ) Spacetime DPG method January 30, / 41

7 The formal wave operator The formal adjoint of A is the operator A and satisfies (Au, v) = (u, Av) for all u, v D(Ω) d+1. Furthermore, we introduce D : W W by Du, v W = (Au, v) + (u, Av) for all u, v W, where W denotes the dual space of W, and, W denotes the duality pairing in W. Assume u, v D(Ω) d+1, then Du, v W = u q (n t v q n x v µ ) + u µ (n t v µ n x v q )ds, Ω where n = (n x, n t ) is the outward unit normal to Ω R d+1 ( JKU Linz ) Spacetime DPG method January 30, / 41

8 The unbounded wave operator Definition of an unbounded operator, again denoted by A Domain dom(a) of A takes initial and boundary conditions into account We partition the spacetime boundary Ω into Moreover, we define Γ 0 = Ω 0 {0}, Γ T = Ω 0 {T }, Γ b = Ω 0 [0, T ], V = {v D(Ω) d+1 : v Γ0 = 0, v µ Γb = 0}, V = {v D(Ω) d+1 : v ΓT = 0, v µ Γb = 0}. ( JKU Linz ) Spacetime DPG method January 30, / 41

9 The unbounded wave operator Now we can define the unbounded operator with A : dom(a) L 2 (Ω) d+1 L 2 (Ω) d+1, [ ] t u Au = q grad x u µ, t u µ div x u q dom(a) = {u W : Du, v W = 0 for all v V }. D(Ω) d+1 dom(a) W = A is densly defined and has an adjoint A A equals A when applied to dom(a ) = {v L 2 (Ω) d+1 : l L 2 (Ω) d+1 s. t. (Au, v) = (u, l) for all u dom(a)} ( JKU Linz ) Spacetime DPG method January 30, / 41

10 The unbounded wave operator Throughout we denote dom(a) and dom(a ) endowed with the topology of W as V and V, respectively V and V are closed subsets of W = A, A are closed operators We have V = D(V ) := {w W : f, w W = 0 for all f D(V )} as well as the inclusions V V, V V. ( JKU Linz ) Spacetime DPG method January 30, / 41

11 Outline 1 The transient wave problem 2 The broken weak formulation 3 Verification of the density condition 4 The ideal DPG method and a priori & a posteriori error estimates 5 Implementation of practical DPG and numerical results ( JKU Linz ) Spacetime DPG method January 30, / 41

12 The broken weak formulation Partition of Ω into a mesh Ω h of open elements K with Ω = K Ω h K "Broken" analogue W h of W given by W h = {v L 2 (Ω) d+1 : A h v L 2 (Ω) d+1 }. Let A h be the wave operator applied element by element, i.e., (A h v) K = A(v K ), v W (K), K Ω h The operator D h : W h W h is defined by D h u, v h := D h u, v Wh = (A h u, v) + (u, A h v) for all u, v W h and, h denotes the duality pairing in W h ( JKU Linz ) Spacetime DPG method January 30, / 41

13 The broken weak formulation Let D h,v denote the restriction to V, i.e., and we define Q with is a complete space. q Q = D h,v = D h V Q := R(D h,v ) inf v D 1 h,v ({q}) v W ( JKU Linz ) Spacetime DPG method January 30, / 41

14 The broken weak formulation The bilinear form b((v, ρ), w) = (v, A h w) + ρ, w h on (L 2 (Ω) d+1 Q) W h leads to Broken weak formulation Let F W h. Find: u L2 (Ω) d+1 and λ Q such that b((u, λ), w) = F (w) for all w W h. (1) ( JKU Linz ) Spacetime DPG method January 30, / 41

15 The broken weak formulation Well-posedness of (1) is guaranteed, if V = D(V ), (2) A : V L 2 (Ω) d+1 is a bijection. (3) Theorem Suppose V is dense in V and V is dense in V. Then the conditions (2) and (3) are satisfied. Proof. Blackboard. ( JKU Linz ) Spacetime DPG method January 30, / 41

16 Outline 1 The transient wave problem 2 The broken weak formulation 3 Verification of the density condition 4 The ideal DPG method and a priori & a posteriori error estimates 5 Implementation of practical DPG and numerical results ( JKU Linz ) Spacetime DPG method January 30, / 41

17 Verification of the density condition Application of the density result for a hyperrectangle, i.e., for some a i > 0. Ω = Ω 0 (0, T ), Ω 0 = d (0, a i ), Theorem On the previously defined Ω, V is dense in V and V is dense in V. i=1 ( JKU Linz ) Spacetime DPG method January 30, / 41

18 Verification of the density condition Proof. The proof is divided into three steps. 1 Extension: Here, we extend a function in V using spatial reflections to a domain which has larger spatial extent than Ω. The operations R i, x = x 2x i e i, R i,+ x = x + 2(a i x i )e i perform reflections of x about x i = 0 and x i = a i for i = 1,..., d. The extended domains Q i are obtained recursively by Q 0 = Ω, Q i, = R 1 i, Q i 1, Q i,+ = R 1 i,+ Q i 1, Q i = Q i, Q i 1 Q i,+. The final extended domain is Q = Q d. ( JKU Linz ) Spacetime DPG method January 30, / 41

19 Verification of the density condition Then we need even and odd extensions of functions. Let G i,e, G i,o : L 2 (Q i 1 ) L 2 (Q i ) be defined by f (R i, x, t) if (x, t) Q i,, G i,e f (x, t) = f (R i,+ x, t) if (x, t) Q i,+, f (x, t) if (x, t) Q i 1. f (R i, x, t) if (x, t) Q i,, G i,o f (x, t) = f (R i,+ x, t) if (x, t) Q i,+, f (x, t) if (x, t) Q i 1. For vector valued functions v L 2 (Ω) d+1, we define G i v(x, t) = (G i,e v i )e i + j i (G i,o v j )e j. ( JKU Linz ) Spacetime DPG method January 30, / 41

20 Verification of the density condition Next, we define where E k = G k G k 1 G 1, E k = G k G k+1 G d, G i w(x, t) = (G i,ew i )e i + (G i,ow j )e j. j i with G G i,ow(x, t) = w(x, t) w(r 1 i, i,ew(x, t) = w(x, t) + w(r 1 i, x) w(r 1 i,+ x), x) + w(r 1 i,+ x). ( JKU Linz ) Spacetime DPG method January 30, / 41

21 Verification of the density condition It holds that (Ev, w) Q = (v, E w) for all v L 2 (Ω) d+1, w L 2 (Q) d+1. It can be proven that for any v V, AEv L 2 (Q) d+1, AEv coincides with EAv and Ev W (Q) ( JKU Linz ) Spacetime DPG method January 30, / 41

22 Verification of the density condition 2 Translation: In this step we translate up the previously obtained extension in time coordinate. Let v V and Ẽv be the extension of Ev by 0 to R d+1. For τ δ, δ > 0, the translation operator in time direction, i.e., (τ δ w)(x, t) = w(x, t δ) it holds that lim τ δg g L δ 0 2 (R d+1 ) = 0 for all g L2 (R d+1 ). With the restriction H δ from R d+1 to Q δ = d i=1 ( a i, 2a i ) ( δ, T + δ) it must be verified that AH δ τ δ Ẽv = H δ τ δ Ẽv. In particular we have H δ τ δ Ẽv W (Q δ ) whenever v V. ( JKU Linz ) Spacetime DPG method January 30, / 41

23 Verification of the density condition 3 Mollification: In this step we consider a v V and mollify the time-translated extension τ δ Ẽv. The used mollifier is given by where ρ 1 (x, t) = ρ ε (x, t) = ε (d+1) ρ 1 (ε 1 x, ε 1 t), { k exp( 1 1 x 2 t 2 ) if x 2 + t 2 < 1, 0 if x 2 + t 2 1 with k such that R d+1 ρ 1 = 1. To end this proof it suffices to show that is in V and v v ε Ω W ε 0 0. v ε = ρ ε τ δ Ẽv ( JKU Linz ) Spacetime DPG method January 30, / 41

24 Outline 1 The transient wave problem 2 The broken weak formulation 3 Verification of the density condition 4 The ideal DPG method and a priori & a posteriori error estimates 5 Implementation of practical DPG and numerical results ( JKU Linz ) Spacetime DPG method January 30, / 41

25 The ideal DPG method and a priori & a posteriori error estimates Approximation of the broken weak formulation by ideal DPG method Find: u h U h L 2 (Ω) d+1 and λ h Q h Q such that b((u h, λ h ), w h ) = F (w h ) for all w h T (U h Q h ), (4) where T : L 2 (Ω) d+1 Q W h is such that (T (v, ρ), w) h = b((v, ρ), w) for all w W h and any (v, ρ) L 2 (Ω) d+1 Q. The mixed formulation Find: ε h W h and (u h, λ h ) (U h Q h ) such that (ε h, w) h + b((u h, λ h ), w) = F (w) for all w W h, b((v, ρ), ε h ) = 0 is equivalent to formulation (4), see Seminar 08. for all (v, ρ) U h Q h ( JKU Linz ) Spacetime DPG method January 30, / 41

26 The ideal DPG method and a posteriori error estimates The expression η = ε h Wh = ε h 2 W (K) K Ω h 1/2 is an efficient and reliable a posteriori error estimator, see Seminar 08. ( JKU Linz ) Spacetime DPG method January 30, / 41

27 The ideal DPG method and a priori error estimates For a priori estimates we distinguish between Case A:: Ω h is a geometrically conforming mesh of (d+1)-simplices V h = {u V C(Ω) d+1 : u K P p+1 (K) d+1 for all K Ω h }, U h = {u L 2 (Ω) d+1 : u K P p (K) d+1 for all K Ω h }, Case B:: Ω h is a geometrically conforming mesh of hyperrectangles V h = {u V C(Ω) d+1 : u K Q p+1 (K) d+1 for all K Ω h }, U h = {u L 2 (Ω) d+1 : u K Q p (K) d+1 for all K Ω h }, where P p (K) and Q p (K) denote spaces of polynomials of total degree p, and of degree at most p in each variable, respectively. Q h = D h V h ( JKU Linz ) Spacetime DPG method January 30, / 41

28 The ideal DPG method and a priori error estimates Theorem Let u V H s+1 (Ω) d+1 and λ = D h u solve (1). Let U h and V h be as in one of the previously introduced cases depending on the mesh type, and Q h = D h V h. Then for (d 1)/2 < s p + 1. Proof. Blackboard. u u h + λ λ h Q ch s u H s+1 (Ω) d+1 ( JKU Linz ) Spacetime DPG method January 30, / 41

29 Outline 1 The transient wave problem 2 The broken weak formulation 3 Verification of the density condition 4 The ideal DPG method and a priori & a posteriori error estimates 5 Implementation of practical DPG and numerical results ( JKU Linz ) Spacetime DPG method January 30, / 41

30 Implementation of practical DPG and numerical results W h replaced by Y r h Case A:: Y r h = {w W h(ω) : w K P r (K) d+1 }, Case B:: Y r h = {w W h(ω) : w K Q r (K) d+1 }. Considered the mixed formulation Find: ε h Y r h and (u h, λ h ) (U h Q h ) such that (ε h, w) h + b((u h, λ h ), w) = F (w) for all w Y r h, b((v, ρ), ε h ) = 0 for all (v, ρ) U h Q h (5) r = p + d + 1 ( JKU Linz ) Spacetime DPG method January 30, / 41

31 Implementation of practical DPG and numerical results For the implementation of (5), λ h = D h z h for some z h V h and (ε h, w) h + b((u h, D h z h ), w) = F (w) for all w Y r h, b((v, D h r), ε h ) = 0 for all (v, r) U h V h (6) is considered. Decomposition of V h into V 0 h = {z V h : z K = 0 for all K Ω h } and remainder V 1 h = V h\v 0 h. b((v, D h V 0 h ), w) = 0, thus replace V h by V 1 h in (6). ( JKU Linz ) Spacetime DPG method January 30, / 41

32 Implementation of practical DPG and numerical results Yields the matrix equation [ A B B 0 ] [ e x ] = [ f 0 ], (7) where e and x are the vectors of coefficients in the basis expansion of ε h Y r h and (u h, z h ) U h V h, respectively, [A] kl = (y l, y k ) h, [B 0 ] ki = b((u i, 0), y k ), [B 1 ] = b((0, D h z j ), y k ) and B = [B 0, B 1 ]. r = p + d + 1 N (A) = N (B 0 ) = {0} B 1 may have a nontrivial kernel ( JKU Linz ) Spacetime DPG method January 30, / 41

33 Techniques to solve despite the null space Technique 1: Remaining orthogonal to null space in conjugate gradients Instead of (7) solve Schur complement system } B A {{ 1 B } x = B A 1 f =:C by means of CG ker C = ker B = ker B 1 Convergence if K n (C, r 0 ) remains l 2 orthogonal to ker C for all n x 0 = 0 = r 0 = B A 1 f R(B ) = (ker C) C n r 0 is orthogonal to ker C for all n 1 ( JKU Linz ) Spacetime DPG method January 30, / 41

34 Techniques to solve despite the null space Technique 2: Regularization of the linear system Rewriting B A 1 Bx = B A 1 f in block form yields [ ] B 0 A 1 B 0 B0 A 1 B 1 B1 A 1 B 0 B1 x = B A 1 f A 1 B 1 Solving the invertible system [ B 0 A 1 B 0 B0 A 1 B 1 B1 A 1 B 0 B1 A 1 B 1 + αm ] x = B A 1 f with the mass matrix M jl = (z l, z j ) and α > 0, e.g., α = ( JKU Linz ) Spacetime DPG method January 30, / 41

35 Convergence rates in two-dimensional spacetime Ω = (0, 1) 2, homogeneous boundary and initial conditions, the exact solution of the second order wave equation is given by φ(x, t) = sin(πx)sin 2 (πt) which results in a solution u = [ πcos(πx)sin 2 (πt) πsin(πx)sin(2πt) ] of the first order system g = 0, f = π 2 sin(πx)(2cos(2πt) + sin 2 (πt)) ( JKU Linz ) Spacetime DPG method January 30, / 41

36 Convergence rates in two-dimensional spacetime Figure: L 2 convergence rates for u h, taken from [J. Gopalakrishnan and P. Sepúlveda, (2017)] ( JKU Linz ) Spacetime DPG method January 30, / 41

37 Adaptivity Ω = (0, 1) 2, f = g = 0, homogeneous boundary conditions, i.e., µ = 0 and non-zero initial conditions with µ t=0 = φ 0, q t=0 = φ 0 φ 0 (x) = exp( 1000((x 0.5) 2 )) ( JKU Linz ) Spacetime DPG method January 30, / 41

38 Adaptivity Figure: Numerical pressure µ, adaptive refinement, setting p = 3, taken from [J. Gopalakrishnan and P. Sepúlveda, (2017)] ( JKU Linz ) Spacetime DPG method January 30, / 41

39 Convergence rates in three-dimensional spacetime Ω = (0, 1) 3, homogeneous boundary and initial conditions, the exact solution of the second order wave equation is given by which results in a solution φ(x, t) = sin(πx)sin(πy)t 2 u = of the first order system πcos(πx)sin(πy)t 2 πcos(πy)sin(2πx)t 2 2sin(πx)sin(πy)t g = 0, f = sin(πx)sin(πy)(2 + 2π 2 t 2 ) ( JKU Linz ) Spacetime DPG method January 30, / 41

40 Convergence rates in three-dimensional spacetime Figure: L 2 convergence rates for u h, taken from [J. Gopalakrishnan and P. Sepúlveda, (2017)] ( JKU Linz ) Spacetime DPG method January 30, / 41

41 References [1] L. Demkowicz, J. Gopalakrishnan, S. Nagaraj, and P. Sepúlveda. A spacetime DPG method for the Schrödinger equation. SIAM Journal on Numerical Analysis, 55(4): , [2] A. Ern, J.-L. Guermond, and G. Caplain. An intrinsic criterion for the bijectivity of Hilbert operators related to Friedrichs systems. Communications in Partial Differential Equations, 32: , [3] J. Gopalakrishnan and P. Sepúlveda. A spacetime DPG method for acoustic waves. arxiv: [math. NA], Sept ( JKU Linz ) Spacetime DPG method January 30, / 41