A space-time Trefftz method for the second order wave equation Lehel Banjai The Maxwell Institute for Mathematical Sciences Heriot-Watt University, Edinburgh & Department of Mathematics, University of Novi Sad RICAM, 9th Nov 2016 Joint work with: Emmanuil Georgoulis (U of Leicester), Oluwaseun F Lijoka (HW) 1 / 34
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A space-time Trefftz method for the second order wave equation · Outline of the talk 1 Motivation 2 An interior-penalty space-time dG method 3 Damped wave equation 4 Numerical results
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A space-time Trefftz method for the second order waveequation
Lehel Banjai
The Maxwell Institute for Mathematical SciencesHeriot-Watt University, Edinburgh
&Department of Mathematics, University of Novi Sad
RICAM, 9th Nov 2016
Joint work with: Emmanuil Georgoulis (U of Leicester), Oluwaseun F Lijoka (HW)
1 / 34
Outline of the talk
1 Motivation
2 An interior-penalty space-time dG method
3 Damped wave equation
4 Numerical results
2 / 34
Outline
1 Motivation
2 An interior-penalty space-time dG method
3 Damped wave equation
4 Numerical results
3 / 34
Acoustic wave equation
u −∇ ⋅ a∇u = 0 in Ω × [0,T ],
u = 0 on ∂Ω × [0,T ],
u(x ,0) = u0(x), u(x ,0) = v0(x), in Ω.
Set-up
Initial data u0 ∈ H10(Ω), v0 ∈ L2(Ω).
a(x) piecewise constant 0 < ca < a(x) < Ca.
Unique solution exists with
u ∈ L2([0,T ]; H1
0(Ω)), u ∈ L2([0,T ]; L2
(Ω)), u ∈ L2([0,T ]; H−1
(Ω)).
Aim
Develop an efficient Trefftz type method:
Approximate u in terms of local solutions of the wave equation.
To do this develop and analyse a time-space dG method.
4 / 34
Acoustic wave equation
u −∇ ⋅ a∇u = 0 in Ω × [0,T ],
u = 0 on ∂Ω × [0,T ],
u(x ,0) = u0(x), u(x ,0) = v0(x), in Ω.
Set-up
Initial data u0 ∈ H10(Ω), v0 ∈ L2(Ω).
a(x) piecewise constant 0 < ca < a(x) < Ca.
Unique solution exists with
u ∈ L2([0,T ]; H1
0(Ω)), u ∈ L2([0,T ]; L2
(Ω)), u ∈ L2([0,T ]; H−1
(Ω)).
Aim
Develop an efficient Trefftz type method:
Approximate u in terms of local solutions of the wave equation.
To do this develop and analyse a time-space dG method.4 / 34
Frequency domain motivation
Frequency domain
Take cue from frequency domain
u(x) ≈k
∑j=1
fjeiωx⋅aj ,
where aj are directions, ∣aj ∣ = 1.
Motivation −∆u − ω2u = 0:
For large ω, minimize the number of degrees of freedom per wavelength.
Time-domain
Time-domain equivalent
u(x, t) ≈k
∑j=1
fj(t − x ⋅ aj)
≈k
∑j=1
p
∑`=0
αj ,`(t − x ⋅ aj)`.
5 / 34
Frequency domain motivation
Frequency domain
Take cue from frequency domain
u(x, ω) ≈k
∑j=1
fj(ω)e iωx⋅aj ,
where aj are directions, ∣aj ∣ = 1.
Time-domain
Time-domain equivalent
u(x, t) ≈k
∑j=1
fj(t − x ⋅ aj)
≈k
∑j=1
p
∑`=0
αj ,`(t − x ⋅ aj)`.
5 / 34
Frequency domain motivation
Frequency domain
Take cue from frequency domain
u(x, ω) ≈k
∑j=1
fj(ω)e iωx⋅aj ,
where aj are directions, ∣aj ∣ = 1.
Time-domain
Time-domain equivalent
u(x, t) ≈k
∑j=1
fj(t − x ⋅ aj)
≈k
∑j=1
p
∑`=0
αj ,`(t − x ⋅ aj)`.
5 / 34
Frequency domain motivation
Frequency domain
Take cue from frequency domain
u(x, ω) ≈k
∑j=1
fj(ω)e iωx⋅aj ,
where aj are directions, ∣aj ∣ = 1.
Time-domain
Time-domain equivalent
u(x, t) ≈k
∑j=1
fj(t − x ⋅ aj)
≈k
∑j=1
p
∑`=0
αj ,`(t − x ⋅ aj)`.
5 / 34
(A bit of) Literature on Trefftz methodsPlenty of literature in the frequency domain
O. Cessenat and B. Despres, Application of an ultra weak variationalformulation of elliptic PDEs to the two-dimensional Helmholtz equation,SIAM J. Numer. Anal., (1998)
R. Hiptmair, A. Moiola, and I. Perugia, Plane wave discontinuous Galerkinmethods for the 2D Helmholtz equation: analysis of the p-version, SIAM J.Numer. Anal., (2011).
Fewer in time-domain
S. Petersen, C. Farhat, and R. Tezaur, A space-time discontinuous Galerkinmethod for the solution of the wave equation in the time domain, Internat.J. Numer. Methods Engrg. (2009)
F. Kretzschmar, A. Moiola, I. Perugia, S. M. Schnepp, A priori error analysisof space-time Trefftz discontinuous Galerkin methods for wave problems,IMA JNA, (2015).
LB, E. Georgoulis, O Lijoka, A Trefftz polynomial space-time discontinuousGalerkin method for the second order wave equation, accepted in SINUM(2016).
6 / 34
(A bit of) Literature on Trefftz methodsPlenty of literature in the frequency domain
O. Cessenat and B. Despres, Application of an ultra weak variationalformulation of elliptic PDEs to the two-dimensional Helmholtz equation,SIAM J. Numer. Anal., (1998)
R. Hiptmair, A. Moiola, and I. Perugia, Plane wave discontinuous Galerkinmethods for the 2D Helmholtz equation: analysis of the p-version, SIAM J.Numer. Anal., (2011).
Fewer in time-domain
S. Petersen, C. Farhat, and R. Tezaur, A space-time discontinuous Galerkinmethod for the solution of the wave equation in the time domain, Internat.J. Numer. Methods Engrg. (2009)
F. Kretzschmar, A. Moiola, I. Perugia, S. M. Schnepp, A priori error analysisof space-time Trefftz discontinuous Galerkin methods for wave problems,IMA JNA, (2015).
LB, E. Georgoulis, O Lijoka, A Trefftz polynomial space-time discontinuousGalerkin method for the second order wave equation, accepted in SINUM(2016).
6 / 34
Outline
1 Motivation
2 An interior-penalty space-time dG method
3 Damped wave equation
4 Numerical results
7 / 34
DG setting
Time discretization 0 = t0 < t1 < ⋅ ⋅ ⋅ < tN = T , In = [tn, tn+1];τn = tn+1 − tn.
Spatial-mesh T n of Ω consisting of open simplices such thatΩ = ∪K∈TnK .
Space-time slabs Tn × In, h-space-time meshwidth.
The skeleton of the space mesh denoted Γn and Γn ∶= Γn−1 ∪ Γn.
Usual jump and average definitions (e = K+ ∩K− ∈ Γint)
The choice of stabilization parameters and convergence
Let τn = tn+1 − tn, h = diam(K), (x , t) ∈ K × (tn, tn+1),K ∈ Tn.
diam(K)/ρK ≤ cT , ∀K ∈ Tn, n = 0,1, . . . ,N − 1, where ρK is theradius of the inscribed circle of K .
Assume space-time elements star-shaped with respect to a ball.
Choice of parameters σ0 = p2cT C 2
a Cinv(cah)−1, σ1 = Cap3(hτn)
−1
σ2 = h(Caτn)−1.
Theorem
For sufficiently smooth solution and h ∼ τ
∣∣∣uh− uex∣∣∣ = O(hp−1/2
).
Proof uses truncated Taylor expansion.
20 / 34
Error estimate in mesh independent norm
Using a Gronwall argument we can show for v ∈ Sh,pn,Trefftz
∥v∥2Ω×In + ∥
√a∇v∥2
Ω×In ≤ τneC(tn+1−tn)/hn (∥v(t−n+1)∥2Ω + ∥
√a∇v(t−n+1)∥
2Ω) ,
hn ∶= minx∈Ω h(x , t), t ∈ In. Let τ = max τn and h = min hn. Then
∥V ∥2Ω×(0,T) + ∥
√a∇V ∥
2Ω×(0,T) ≤ CτeCτ/h∣∣∣V ∣∣∣
2⋆, ∀V ∈ V h,p
Trefftz.
Proposition
∥uh− uex∥
2Ω×[0,T ] + ∥∇uh
−∇uex∥2Ω×[0,T ]
≤ C infV ∈V h,p
Trefftz
(τeCτ/h∣∣∣V − uex∣∣∣2⋆
+ ∥V − uex∥2Ω×[0,T ] + ∥
√a∇(V − uex)∥
2Ω×[0,T ]).
21 / 34
Outline
1 Motivation
2 An interior-penalty space-time dG method
3 Damped wave equation
4 Numerical results
22 / 34
Wave equation with damping
Damped wave equation:u + αu −∆u = 0.
The extra term decreases the energy:
d
dtE(t) = −α∥u∥2,
hence only a minor modification to the DG formulation needed.
However: truncations of the Taylor expansion are no longer solutions!
Grysa, Maciag, Adamczyk-Krasa ’14 consider solutions of the form
e−αtp1(x , t) + p2(x , t)
with p1 and p2 polynomial in x and t.
23 / 34
Wave equation with damping
Damped wave equation:u + αu −∆u = 0.
The extra term decreases the energy:
d
dtE(t) = −α∥u∥2,
hence only a minor modification to the DG formulation needed.
However: truncations of the Taylor expansion are no longer solutions!
Grysa, Maciag, Adamczyk-Krasa ’14 consider solutions of the form
e−αtp1(x , t) + p2(x , t)
with p1 and p2 polynomial in x and t.
23 / 34
Wave equation with damping
Damped wave equation:u + αu −∆u = 0.
The extra term decreases the energy:
d
dtE(t) = −α∥u∥2,
hence only a minor modification to the DG formulation needed.
However: truncations of the Taylor expansion are no longer solutions!
Grysa, Maciag, Adamczyk-Krasa ’14 consider solutions of the form
e−αtp1(x , t) + p2(x , t)
with p1 and p2 polynomial in x and t.
23 / 34
Solution formula in 1D
Instead, use basis functions obtained by propagating polynomial initialdata:
u(x ,0) = u0(x) = xαj , u(x ,0) = v0(x) = 0,
andu0(x) = 0, v0(x) = xβj
with∣αj ∣ ≤ p ∣βj ∣ ≤ p − 1.
In 1D solution is then given by the d’Alambert-like formula
u(x , t) =1
2[u0(x − t) + u0(x + t)] e−αt/2
+α
4e−αt/2
∫
x+t
x−tu0(s)I0 (ρ(s)α2 ) +
t
ρ(s)I1 (ρ(s)α2 )ds
+1
2e−αt/2
∫
x+t
x−tv0(s)I0 (ρ(s)α2 )ds, ρ(s; x , t) =
√t2 − (x − s)2.
24 / 34
Solution formula in 1D
Instead, use basis functions obtained by propagating polynomial initialdata:
u(x ,0) = u0(x) = xαj , u(x ,0) = v0(x) = 0,
andu0(x) = 0, v0(x) = xβj
with∣αj ∣ ≤ p ∣βj ∣ ≤ p − 1.
In 1D solution is then given by the d’Alambert-like formula
u(x , t) =1
2[u0(x − t) + u0(x + t)] e−αt/2
+α
4e−αt/2
∫
x+t
x−tu0(s)I0 (ρ(s)α2 ) +
t
ρ(s)I1 (ρ(s)α2 )ds
+1
2e−αt/2
∫
x+t
x−tv0(s)I0 (ρ(s)α2 )ds, ρ(s; x , t) =
√t2 − (x − s)2.
24 / 34
Solution formula ctd.
Rearranging (the last term)
1
2te−αt/2
∫
1
0[v0(x + st) + v0(x − st)] I0 (αt2
√1 − s2)ds.
For example for v0(x) = x2
x2te−αt/2∫
1
0I0 (αt2
√1 − s2)ds + t3e−αt/2
∫
1
0s2I0 (αt2
√1 − s2)ds.
Corresponding term in 3D
te−αt/2
1
∫
0
−∫∂B(0,1)
[v0(x + tsz) +Dv0(x + tsz)] ⋅zdSz I0 (αt2
√1 − s2)ds
Hence need efficient representation of functions of the type
%j(t) = ∫1
0s j I0 (αt2
√1 − s2)ds
In Matlab: Chebfun is ideal.
25 / 34
Solution formula ctd.
Rearranging (the last term)
1
2te−αt/2
∫
1
0[v0(x + st) + v0(x − st)] I0 (αt2
√1 − s2)ds.
For example for v0(x) = x2
x2te−αt/2∫
1
0I0 (αt2
√1 − s2)ds + t3e−αt/2
∫
1
0s2I0 (αt2
√1 − s2)ds.
Corresponding term in 3D
te−αt/2
1
∫
0
−∫∂B(0,1)
[v0(x + tsz) +Dv0(x + tsz)] ⋅zdSz I0 (αt2
√1 − s2)ds
Hence need efficient representation of functions of the type
%j(t) = ∫1
0s j I0 (αt2
√1 − s2)ds
In Matlab: Chebfun is ideal.
25 / 34
Solution formula ctd.
Rearranging (the last term)
1
2te−αt/2
∫
1
0[v0(x + st) + v0(x − st)] I0 (αt2
√1 − s2)ds.
For example for v0(x) = x2
x2te−αt/2∫
1
0I0 (αt2
√1 − s2)ds + t3e−αt/2
∫
1
0s2I0 (αt2
√1 − s2)ds.
Corresponding term in 3D
te−αt/2
1
∫
0
−∫∂B(0,1)
[v0(x + tsz) +Dv0(x + tsz)] ⋅zdSz I0 (αt2
√1 − s2)ds
Hence need efficient representation of functions of the type
%j(t) = ∫1
0s j I0 (αt2
√1 − s2)ds
In Matlab: Chebfun is ideal.25 / 34
Changes to the analysis
Polynomial in space Ô⇒ the same discrete inverse inequalities used+ the extra term decreases energy Ô⇒ choice of parameters,stability and quasi-optimality proof identical.
Approximation properties of the discrete space (in 1D): Away from the boundary, in each space-time element K × (t−, t+)
project solution to K and neighbouring elements at time t− topolynomials and propagate.
At boundary, extend exact solution anti-symmetrically and againproject and propagate.
26 / 34
Outline
1 Motivation
2 An interior-penalty space-time dG method
3 Damped wave equation
4 Numerical results
27 / 34
One dimensional settingSimple 1D setting:
Ω = (0,1), a ≡ 1.
Initial data
u0 = e−( x−5/8
δ)2
, v0 = 0, δ ≤ δ0 = 7.5 × 10−2.
Interested in having few degrees of freedom for decreasing δ ≤ δ0.
0 0.2 0.4 0.6 0.8 1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
t = 1.5, δ = δ0
0 0.2 0.4 0.6 0.8 1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
t = 1.5, δ = δ0/4
Energy of exact solution
exact energy = 12∥ux(x ,0)∥2
Ω ≈ 2δ−1∫
∞
−∞y 2e−2y2
dy = δ−1
√π
2√
2.
We compare with full polynomial space.Note for polynomial order p