optimally blended finite/spectral element scheme for wave … · 2011-09-23 · optimally blended...
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Optimally Blended Finite/Spectral Element Scheme
for Wave Propagation
Mark Ainsworth(joint work with Hafiz Abdul Wajid, COMSATS, Pakistan)
Mathematics and Statistics, Strathclyde University,Glasgow, Scotland.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 1/54
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Outline
(Very) Quick Overview of Spectral Element Method
Typical Practical Application
Analysis and Properties
Optimal Blending
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 2/54
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Model Problem
Seek u : (−1, 1)× (0, T ) 7→ C:
utt − uxx = f(x, t) in (−1, 1)× (0, T ),
subject to u(x, 0) = ut(x, 0) = 0, x ∈ (−1, 1) and
u(−1, t) = g(t), ux(1, t) + ut(1, t) = 0 for t ∈ (0, T )
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 3/54
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Variational Formulation
Seek u ∈ H1(−1, 1) : u(−1, t) = g(t)
d2
dt2(u, v) +
d
dtuv|x=1 + (ux, vx) = (f, v),
for all v ∈ H1(−1, 1) : v(−1) = 0, where (u, v) =∫ 1
−1uv dx.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 4/54
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Variational Formulation
Seek u ∈ H1(−1, 1) : u(−1, t) = g(t)
d2
dt2(u, v) +
d
dtuv|x=1 + (ux, vx) = (f, v),
for all v ∈ H1(−1, 1) : v(−1) = 0, where (u, v) =∫ 1
−1uv dx.
Discretise by introducing a finite dimensional subspace X ⊂ H1(−1, 1).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 4/54
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Semi-Discretisation in Space
Seek U ∈ X : U(−1, t) = g(t)
d2
dt2(U, v) +
d
dtUv|x=1 + (Ux, vx) = (f, v),
for all v ∈ X : v(−1) = 0.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 5/54
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Semi-Discretisation in Space
Seek U ∈ X : U(−1, t) = g(t)
d2
dt2(U, v) +
d
dtUv|x=1 + (Ux, vx) = (f, v),
for all v ∈ X : v(−1) = 0.
Introduce basis φiNi=0 for X , and write
U(x, t) =N∑
i=0
αi(t)φi(x).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 5/54
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Semi-Discrete Scheme
Find αiNi=0 such that
N∑
i=0
((φi, φj)
d2αi
dt2+ φi(1)φj(1)
dαi
dt+ (φi,x, φj,x)αi
)= (f, φj)
for j = 1, . . . , N , with
αi(0) =dαi
dt(0) = 0
with α0(t) = g(t), t ∈ (0, T ).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 6/54
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Semi-Discrete Scheme
Find ~α such that
Md2~α
dt2+C
d~α
dt+K~α = ~r
with
~α(0) =d~α
dt(0) = ~0
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 7/54
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Semi-Discrete Scheme
Find ~α such that
Md2~α
dt2+C
d~α
dt+K~α = ~r
with
~α(0) =d~α
dt(0) = ~0
and
M ij =
∫ 1
−1
φi(x)φj(x) dx ‘Mass Matrix’
Kij =
∫ 1
−1
φ′
i(x)φ′
j(x) dx ‘Stiffness’
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 7/54
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Fully Discrete Scheme
Simple leapfrog scheme in time
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
where ~α0 = ~α1 = ~0, and
M ij =
∫ 1
−1
φi(x)φj(x) dx Mass
Kij =
∫ 1
−1
φ′
i(x)φ′
j(x) dx Stiffness
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 8/54
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Fully Discrete Scheme
Simple leapfrog scheme in time
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
where ~α0 = ~α1 = ~0, and
M ij =
∫ 1
−1
φi(x)φj(x) dx Mass
Kij =
∫ 1
−1
φ′
i(x)φ′
j(x) dx Stiffness
... requires inversion of mass matrix M at each time step.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 8/54
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Spectral Element Method: General Idea
Spectral element spatial discretisation gives
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+ K~αn = ~rn
where ~α0 = ~α1 = ~0, and
M ij ≈
∫ 1
−1
φi(x)φj(x) dx Mass
Kij ≈
∫ 1
−1
φ′
i(x)φ′
j(x) dx Stiffness
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 9/54
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Spectral Element Method: General Idea
Spectral element spatial discretisation gives
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+ K~αn = ~rn
where ~α0 = ~α1 = ~0, and
M ij ≈
∫ 1
−1
φi(x)φj(x) dx Mass
Kij ≈
∫ 1
−1
φ′
i(x)φ′
j(x) dx Stiffness
Spectral element method characterised by particular combinations of
quadrature rule for integrals
nature and form of basis functions.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 9/54
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Spectral Element Method: Quadrature Rule
Gauss-Lobatto quadrature rule:
∫ 1
−1
f(ζ) dζ ≈ Qp(f) =
p∑
ℓ=2
wℓf(ζℓ) +2
p(p+ 1)[f(−1) + f(1)],
where
Nodes: ζℓpℓ=2 zeros of L′
p
Weights: wℓ = 2/p(p− 1)[Lp−1(ζℓ)]2 > 0
with Lp Legendre polynomial of degree p. Rule exact for f ∈ P2p−1.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 10/54
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Spectral Element Method: Basis Functions
Lagrange basis functions φℓp+1ℓ=1 :
φℓ ∈ Pp : φℓ(ζm) =
1, ℓ = m
0, ℓ 6= m
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 11/54
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Spectral Element Method: Basis Functions
Lagrange basis functions φℓp+1ℓ=1 :
−1 −0.5 0 0.5 1−0.5
0
0.5
1
1.5Quadratic GL−Shape Functions
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Spectral Element Method: Basis Functions
Lagrange basis functions φℓp+1ℓ=1 :
−1 −0.5 0 0.5 1−0.5
0
0.5
1
1.5Cubic GL−Shape Functions
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Spectral Element Method: Basis Functions
Lagrange basis functions φℓp+1ℓ=1 :
−1 −0.5 0 0.5 1−0.5
0
0.5
1
1.5Order six GL−Shape Functions
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 11/54
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... so what’s the big deal?
Stiffness matrix K:
Kℓm = Qp(φ′
ℓφ′
m),
recall φ′
ℓ ∈ Pp−1 and quadrature rule exact for P2p−1, so
Kℓm = (φ′
ℓ, φ′
m) = Kℓm
Hence, stiffness matrix unchanged.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 12/54
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... so what’s the big deal?
Stiffness matrix K:
Kℓm = Qp(φ′
ℓφ′
m),
recall φ′
ℓ ∈ Pp−1 and quadrature rule exact for P2p−1, so
Kℓm = (φ′
ℓ, φ′
m) = Kℓm
Hence, stiffness matrix unchanged.... so what is the big deal?
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... so what’s the big deal?
Mass matrix M :
M ℓm = Qp(φℓφm) =
p+1∑
k=1
wkφℓ(ζk)φm(ζk).
(recall φℓ(ζk) = δℓk)
M ℓm =
p+1∑
k=1
wkδℓkδmk = wℓδℓm.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 13/54
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... so what’s the big deal?
Mass matrix M is diagonal
M =
w1 0 0 · · · 0
0 w2 0 · · · 0
0 0 w3 · · · 0...
......
. . . 0
0 0 0 0 wp+1
... finite elements with ‘Gauss-point mass lumping’Ref: G. Cohen, Higher-Order Numerical Methods for Transient WaveEquations, Springer 2002.
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Spectral Element Method
Spectral Element/Gauss-point Mass Lumped Finite Element Scheme
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+ K~αn = ~rn
where ~α0 = ~α1 = ~0, with
M = Diag(w1, . . . , wp+1)
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 14/54
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Spectral Element Method
Spectral Element/Gauss-point Mass Lumped Finite Element Scheme
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+ K~αn = ~rn
where ~α0 = ~α1 = ~0, with
M = Diag(w1, . . . , wp+1)
Same structure and approach applicable to
multiple spatial dimensions
acoustics, electromagnetics, structures, ...
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 14/54
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Spectral Element Method
Spectral Element/Gauss-point Mass Lumped Finite Element Scheme
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+ K~αn = ~rn
where ~α0 = ~α1 = ~0, with
M = Diag(w1, . . . , wp+1)
Same structure and approach applicable to
multiple spatial dimensions
acoustics, electromagnetics, structures, ...
Advantages:
retains geometric flexibility of finite elements
obtain extremely efficient time-stepping.
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Application: Seismic Simulation
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Application: Seismic Simulation
Earthquake: 9/9/2001 Hollywood CA MW = 4.3.
Surface topography of Southern California viewed from Southeast.
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Seismic Simulation: Hollywood 9/9/2001 MW = 4.3
• Hollywood (Full)• Hollywood (Zoom)
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Seismic Simulation: Computational Details
average distance between grid points at the surface roughly 335m
mesh contains 672,768 elements
time step size of 9 msec and 20,000 timesteps in total.
polynomial degree p = 4
each element contains 3(p+ 1)3 = 375 degrees of freedom
136 million degrees of freedom in total
144 processor Beowulf PC cluster using MPI
6.5 hours needed to compute seismograms with a duration of 3 min
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Seismic Simulation: Computational Details
average distance between grid points at the surface roughly 335m
mesh contains 672,768 elements
time step size of 9 msec and 20,000 timesteps in total.
polynomial degree p = 4
each element contains 3(p+ 1)3 = 375 degrees of freedom
136 million degrees of freedom in total
144 processor Beowulf PC cluster using MPI
6.5 hours needed to compute seismograms with a duration of 3 min
... state of the art back in 2004. Same technology SPECFEM3D beingused for global seismic simulation–see cover of Science, May 2005.
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Seismic Simulation: Cross-Section of Portion of Mesh
Note: Highly structured translation invariant mesh in bulk of the domain.
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Basic Issues in Computational Wave Propagation
A numerical scheme should, as far as possible:
propagate waves at correct speed i.e. control numerical dispersion;
preserve conserved quantities (energy, momentum, ...) i.e. controlnumerical dissipation;
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 20/54
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Basic Issues in Computational Wave Propagation
A numerical scheme should, as far as possible:
propagate waves at correct speed i.e. control numerical dispersion;
preserve conserved quantities (energy, momentum, ...) i.e. controlnumerical dissipation;
... but there is no silver bullet.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 20/54
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Basic Issues in Computational Wave Propagation
A numerical scheme should, as far as possible:
propagate waves at correct speed i.e. control numerical dispersion;
preserve conserved quantities (energy, momentum, ...) i.e. controlnumerical dissipation;
Every scheme needs at least two degrees of freedom per wavelength justto resolve the wave, unless we know something about structure of thesolution and take it into account.
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Basic Issues in Computational Wave Propagation
A numerical scheme should, as far as possible:
propagate waves at correct speed i.e. control numerical dispersion;
preserve conserved quantities (energy, momentum, ...) i.e. controlnumerical dissipation;
Every scheme needs at least two degrees of freedom per wavelength justto resolve the wave, unless we know something about structure of thesolution and take it into account.Fairly poor understanding of properties and behaviour of high orderspectral element schemes for approximation of waves
how high should the order be?
how much numerical dispersion? ... numerical dissipation?
how does behaviour compare with (consistent) finite elementscheme?
how compare to the ‘two degrees of freedom per wavelength’?Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 20/54
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Numerical Observations
Consider the model problem with time-harmonic excitation:
utt − uxx = 0
subject to
u(−1, t) = exp(iωt), ux(1, t) + ut(1, t) = 0, t ∈ (0, T ).
Solution has form u(x, t) = U(x) exp(−iωt) where U satisfies U(−1) = 1,U ′(1)− iωU(1) = 0 and
−U ′′(x)− ω2U(x) = 0, x ∈ (−1, 1).
True solution U(x) = exp(iω(x+ 1)).
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Piecewise Linear Approximation
Finite element method (exact quadrature) on uniform mesh of size h:
(K − κ2
M + iκC)~U = ~r
where κ = ωh/2, C = ~eN~etN ,
K =1
h
1 −1 0 · · · 0
−1 2 −1 · · · 0...
......
. . . −1
0 0 0 −1 1
; M =
h
6
2 1 0 · · · 0
1 4 1 · · · 0...
......
. . . 1
0 0 0 1 2
.
Easy to invert M of course, but becomes more problematic onunstructured grids in higher dimensions.
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Piecewise Linear Approximation
Spectral element method (GLL quadrature) on uniform mesh of size h:
(K − κ2
M − iκC)~U = ~r
where κ = kh/2, C = ~eN~etN ,
K =1
h
1 −1 0 · · · 0
−1 2 −1 · · · 0...
......
. . . −1
0 0 0 −1 1
; M =
h
2
1 0 0 · · · 0
0 2 0 · · · 0...
......
. . . 0
0 0 0 0 1
.
Mass matrix M diagonal.
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Piecewise Linear Approximation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
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Piecewise Linear Approximation
Finite elements exhibit phase lead. Spectral elements phase lag.
Magnitude is same for both schemes.
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Piecewise Quadratic Approximation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
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Piecewise Quadratic Approximation
Finite elements exhibit phase lead. Spectral elements phase lag.
Magnitude is smaller for spectral elements.
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Piecewise Cubic Approximation
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
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Piecewise Cubic Approximation
Finite elements exhibit phase lead. Spectral elements phase lag.
Magnitude is even smaller for spectral elements.
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How to measure this behaviour?
Dispersion relation for wave equation
utt − uxx = 0
obtained by seeking non-trivial solution of form u(x, t) = U(x)e−iωt, whereU has Bloch wave property
U(x+ h) = eikhU(x).
Gives k = ±ω.
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How to measure this behaviour?
Dispersion relation for wave equation
utt − uxx = 0
obtained by seeking non-trivial solution of form u(x, t) = U(x)e−iωt, whereU has Bloch wave property
U(x+ h) = eikhU(x).
Gives k = ±ω.Seek non-trivial discrete Bloch wave uh(x, t) = Uh(x)e
−iωt such that
Uh(x+ h) = eikhUh(x)
with k to be determined.
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How to measure this behaviour?
Dispersion relation for wave equation
utt − uxx = 0
obtained by seeking non-trivial solution of form u(x, t) = U(x)e−iωt, whereU has Bloch wave property
U(x+ h) = eikhU(x).
Gives k = ±ω.Seek non-trivial discrete Bloch wave uh(x, t) = Uh(x)e
−iωt such that
Uh(x+ h) = eikhUh(x)
with k to be determined.
k − k measures phase error
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Example: First Order SEM
Spectral elements (p = 1):
1
h(Uj−1 + 2Uj − Uj+1)− ω2hUj = 0
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Example: First Order SEM
Spectral elements (p = 1):
1
h(Uj−1 + 2Uj − Uj+1)− ω2hUj = 0
Seek non-trivial discrete U of form Uj = Ceikjh, such that
1
h
(e−ikh − 2 + eikh
)− ω2h = 0.
Hence,
cos kh = 1−Ω2
2, Ω = ωh
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Example: First Order SEM
Spectral elements (p = 1):
1
h(Uj−1 + 2Uj − Uj+1)− ω2hUj = 0
Seek non-trivial discrete U of form Uj = Ceikjh, such that
1
h
(e−ikh − 2 + eikh
)− ω2h = 0.
Hence,
cos kh = 1−Ω2
2, Ω = ωh
For Ω ≪ 1, obtain
kh− ωh =Ω3
24+O(Ω)5.
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Discrete Dispersion Relation (Ω = ωh)
Spectral Element Scheme
Order p Rp(Ω) cos−1Rp(Ω)− Ω
1 1− Ω2
2Ω3
24
2 Ω4−22Ω2+482(Ω2+24)
Ω5
2880
3 −Ω6+92Ω4−1680Ω2+3600
2(Ω4+60Ω2+1800)Ω7
604800
......
...
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Discrete Dispersion Relation (Ω = ωh)
Spectral Element Scheme
Order p Rp(Ω) cos−1Rp(Ω)− Ω
1 1− Ω2
2Ω3
24
2 Ω4−22Ω2+482(Ω2+24)
Ω5
2880
3 −Ω6+92Ω4−1680Ω2+3600
2(Ω4+60Ω2+1800)Ω7
604800
......
...
Finite Element Scheme (Ainsworth, SINUM 2004)
Order p Rp(Ω) cos−1Rp(Ω)− Ω
1 −2Ω2+6Ω2+6 −Ω3
24
2 3Ω4−104Ω2+240
Ω4+16Ω2+240 − Ω5
1440
3 −4Ω6+540Ω4−11520Ω2+25200
Ω6+30Ω4+1080Ω2+25200 − Ω7
201600...
......
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General Form of Dispersion Relation for Finite Elements
Discrete dispersion relation for order p elements:
cos(kh) = Rp(hω)
where Rp is the rational function
Rp(2κ) =[2No/2No − 2]κ cot κ − [2Ne + 2/2Ne]κ tanκ
[2No/2No − 2]κ cot κ + [2Ne + 2/2Ne]κ tanκ.
and Ne = ⌊p/2⌋ and No = ⌊(p+ 1)/2⌋.
Simple form in terms of Padé approximants ...
(Ainsworth, SINUM 2004)
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General Form of Dispersion Relation Spectral Elements
Define sequences ap∞
p=0 and bp∞
p=0 recursively by the rule
ap+1 = − 2p+1κ bp + ap−1,
bp+1 = 2p+1κ ap + bp−1,
for p ∈ N with a0 = 1, a1 = 1, b0 = 0 and b1 = 1/κ. Then, the discretedispersion relation for spectral element method is given by
Rp(2κ) = cos(kh) = (−1)p[ap (κbp−1 + pap) + bp (κap−1 − pbp)
ap (κbp−1 + pap)− bp (κap−1 − pbp)
].
(Ainsworth and Wajid, SINUM 2008)
Not related to Padé approximants! (c.f. finite element case)
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Phase Error for ωh ≪ 1
Phase error for finite element method satisfies
kh− ωh = −1
2
[p!
(2p)!
]2(ωh)
2p+1
2p+ 1+O (ωh)2p+3
(Ainsworth, SINUM 2004)
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Phase Error for ωh ≪ 1
Phase error for finite element method satisfies
kh− ωh = −1
2
[p!
(2p)!
]2(ωh)
2p+1
2p+ 1+O (ωh)2p+3
(Ainsworth, SINUM 2004)
Phase error for spectral element method satisfies
kh− ωh =1
2p
[p!
(2p)!
]2(ωh)2p+1
2p+ 1+O (ωh)
2p+3.
(Ainsworth and Wajid, SINUM 2008)
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Phase Error for ωh ≪ 1
Phase error for finite element method satisfies
kh− ωh = −1
2
[p!
(2p)!
]2(ωh)
2p+1
2p+ 1+O (ωh)2p+3
(Ainsworth, SINUM 2004)
Phase error for spectral element method satisfies
kh− ωh =1
2p
[p!
(2p)!
]2(ωh)2p+1
2p+ 1+O (ωh)
2p+3.
(Ainsworth and Wajid, SINUM 2008)
Finite elements exhibit phase lead and spectral elements exhibitphase lag.
Spectral elements a factor 1/p times more accurate than finiteelements.
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Numerical Dispersion for Large Order and Large Wavenumber
0 10 20 30 40 50 6010
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
102
104
p
|Rp−
cos(
ω h
)|
Case: ω h = 80 ω h >> 1
Error in the discrete dispersion relation with full integrationError in the discrete dispersion relation with reduced integration
Note: Very sharp transition between garbage and essentially exactModern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 33/54
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Numerical Dispersion for Large Order and Large Wavenumber
Suppose that ωh ≫ 1. Then the error Ep = cos kh− cosωh in the discretedispersion relation passes through distinct phases as the order p ∈ N isincreased:
Unstable Phase: For p = O(1), Ep ≈ (−1)p (ωh)2
2 .
Oscillatory Phase: For 1 ≪ 2p+ 1 < ωh− o(ωh)1/3, Ep oscillates anddecays to O(1) as p is increased.
Transition Zone: For ωh− o(ωh)1/3 < 2p+ 1 < ωh+ o(ωh)1/3, theerror Ep oscillates without further decrease.
Super-Exponential Decay: For 2p+ 1 > ωh+ o(ωh)1/3, Ep decreasesat a super-exponential rate.
(Ainsworth and Wajid, SINUM 2008)
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
Degree p approximation corresponds to p+ 1 modes per element (0, h).
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
Degree p approximation corresponds to p+ 1 modes per element (0, h).
=⇒ π(2p+ 2)/ωh modes per wavelength λ = 2π/ω
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
Degree p approximation corresponds to p+ 1 modes per element (0, h).
=⇒ π(2p+ 2)/ωh modes per wavelength λ = 2π/ω
Wave essentially resolved when 2p+ 1 = ωh+ o(ωh)1/3
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
Degree p approximation corresponds to p+ 1 modes per element (0, h).
=⇒ π(2p+ 2)/ωh modes per wavelength λ = 2π/ω
Wave essentially resolved when 2p+ 1 = ωh+ o(ωh)1/3
Hence,
π modes per wavelength are required to resolve wave.
Provides rigorous justification of ‘well-known’ rule of thumb governing useof spectral element methods.
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
Degree p approximation corresponds to p+ 1 modes per element (0, h).
=⇒ π(2p+ 2)/ωh modes per wavelength λ = 2π/ω
Wave essentially resolved when 2p+ 1 = ωh+ o(ωh)1/3
Hence,
π modes per wavelength are required to resolve wave.
Provides rigorous justification of ‘well-known’ rule of thumb governing useof spectral element methods.
... spectral element method not bad at all, in view of fact that no generalscheme can suffice with fewer than two degrees of freedom perwavelength.
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One Consequence of the Analysis
What is significance of condition 2p+ 1 ≈ ωh which appears in analysis?
Degree p approximation corresponds to p+ 1 modes per element (0, h).
=⇒ π(2p+ 2)/ωh modes per wavelength λ = 2π/ω
Wave essentially resolved when 2p+ 1 = ωh+ o(ωh)1/3
Hence,
π modes per wavelength are required to resolve wave.
Provides rigorous justification of ‘well-known’ rule of thumb governing useof spectral element methods.
... spectral element method not bad at all, in view of fact that no generalscheme can suffice with fewer than two degrees of freedom perwavelength.
Ref: Ainsworth and Wajid, SINUM 2008
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 35/54
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Blended Finite Element/Spectral Element Schemes
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 36/54
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Blended FEM/SEM Schemes
Idea: Can we obtain better scheme by averaging (blending) finiteelements and spectral elements?
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 37/54
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Blended FEM/SEM Schemes
Finite Element Scheme:
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 37/54
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Blended FEM/SEM Schemes
Finite Element Scheme:
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
Spectral Element Scheme:
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 37/54
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Blended FEM/SEM Schemes
Finite Element Scheme:
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
Spectral Element Scheme:
1
(∆t)2M
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
Blended FEM/SEM Scheme:
1
(∆t)2M τ
(~αn+1 − 2~αn + ~αn−1
)+
1
∆tC
(~αn − ~αn−1
)+K~αn = ~rn
where, for τ ∈ [0, 1] (to be determined)
M τ = (1− τ)M + τM .
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 37/54
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Blended FEM/SEM Schemes
Finite elements exhibit phase lead. Spectral elements phase lag.
Magnitude is same for both schemes. Suggests a symmetricweighting τ = 1/2.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 38/54
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Blended FEM/SEM Schemes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 38/54
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Blended FEM/SEM Schemes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 38/54
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Blended FEM/SEM Schemes
Dispersion analysis reveals for blended scheme (τ ∈ [0, 1]):
kh− ωh =1
24(ωh)3(2τ − 1) +O(ωh)5.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 39/54
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Blended FEM/SEM Schemes
Dispersion analysis reveals for blended scheme (τ ∈ [0, 1]):
kh− ωh =1
24(ωh)3(2τ − 1) +O(ωh)5.
... so choice τ = 1/2 is optimal and gives
kh− ωh =1
480(ωh)5 +O(ωh)7.
two additional orders of accuracy;
improved coefficient of leading order term.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 39/54
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Blended FEM/SEM Schemes
A quote taken from the conclusion of the highly influential article of Marfurt(Geophysics, 49, 1984):
Frequency domain finite element solutions employing weighted averageof consistent and lumped masses yield the most accurate results,and they promise to be the most cost-effective method for [severalapplications in wave propagation]. Marfurt (1984).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 40/54
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Blended FEM/SEM Schemes
A quote taken from the conclusion of the highly influential article of Marfurt(Geophysics, 49, 1984):
Frequency domain finite element solutions employing weighted averageof consistent and lumped masses yield the most accurate results,and they promise to be the most cost-effective method for [severalapplications in wave propagation]. Marfurt (1984).
Despite this bold statement, none of the citing articles or any other overthe past 20 years presents an optimal blended scheme for elements ofhigher than first order.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 40/54
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Blended FEM/SEM Schemes
A quote taken from the conclusion of the highly influential article of Marfurt(Geophysics, 49, 1984):
Frequency domain finite element solutions employing weighted averageof consistent and lumped masses yield the most accurate results,and they promise to be the most cost-effective method for [severalapplications in wave propagation]. Marfurt (1984).
Despite this bold statement, none of the citing articles or any other overthe past 20 years presents an optimal blended scheme for elements ofhigher than first order.Is it possible to obtain a general result?
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 40/54
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Blended FEM/SEM Schemes
Let p ≥ 2 and τ ∈ [0, 1]. Then, error in discrete wavenumber for blendedscheme is given by
kh− ωh =1
2
[p!
(2p)!
]2(ωh)2p+1
2p+ 1(1 + 1/p)τ − 1+O(ωh)2p+3.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 41/54
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Blended FEM/SEM Schemes
Let p ≥ 2 and τ ∈ [0, 1]. Then, error in discrete wavenumber for blendedscheme is given by
kh− ωh =1
2
[p!
(2p)!
]2(ωh)2p+1
2p+ 1(1 + 1/p)τ − 1+O(ωh)2p+3.
Optimal blending parameter given by τ = p/(p+ 1).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 41/54
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Blended FEM/SEM Schemes
Let p ≥ 2 and τ ∈ [0, 1]. Then, error in discrete wavenumber for blendedscheme is given by
kh− ωh =1
2
[p!
(2p)!
]2(ωh)2p+1
2p+ 1(1 + 1/p)τ − 1+O(ωh)2p+3.
Optimal blending parameter given by τ = p/(p+ 1).
With this choice, we find that
kh− ωh =4
2p− 1
[(p+ 1)!
(2p+ 2)!
]2(ωh)2p+3
2p+ 3+O(ωh)2p+5.
two additional orders of accuracy;
coefficient of leading order term much smaller.
Ref: Ainsworth and Wajid, SINUM 2010.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 41/54
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Blended FEM/SEM Schemes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 42/54
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Blended FEM/SEM Schemes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 42/54
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Blended FEM/SEM Schemes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 42/54
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Blended FEM/SEM Schemes
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1.5
−1
−0.5
0
0.5
1
1.5
2
U
Numerical real wave obtained using the spectral element schemeNumerical real wave obtained using the finite element schemeNumerical real wave obtained using the optimal schemeExact real wave
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 42/54
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What about d > 1 dimensions?
Can we find τ such that blending mass matrices for FEM/SEM scheme in2D has higher accuracy?
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 43/54
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What about d > 1 dimensions?
Can we find τ such that blending mass matrices for FEM/SEM scheme in2D has higher accuracy?Bad news: there is no choice giving higher accuracy.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 43/54
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What about d > 1 dimensions?
Can we find τ such that blending mass matrices for FEM/SEM scheme in2D has higher accuracy?Bad news: there is no choice giving higher accuracy.Don’t give up yet. Look in detail at structure of 2D equations on tensorproduct grid:
K ⊗M +M ⊗K − ω2M ⊗M (FEM)
and
K ⊗ M + M ⊗K − ω2M ⊗ M (SEM)
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 43/54
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What about d > 1 dimensions?
Can we find τ such that blending mass matrices for FEM/SEM scheme in2D has higher accuracy?Bad news: there is no choice giving higher accuracy.Don’t give up yet. Look in detail at structure of 2D equations on tensorproduct grid:
K ⊗M +M ⊗K − ω2M ⊗M (FEM)
and
K ⊗ M + M ⊗K − ω2M ⊗ M (SEM)
Suggests that in multi-dimensions should look for blending in the form
K ⊗Mτ +Mτ ⊗K − ω2Mτ ⊗Mτ (Blended)
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 43/54
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What about d > 1 dimensions?
Can we choose τ so that
K ⊗Mτ +Mτ ⊗K − ω2Mτ ⊗Mτ (Blended)
has higher order accuracy?
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 44/54
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What about d > 1 dimensions?
Can we choose τ so that
K ⊗Mτ +Mτ ⊗K − ω2Mτ ⊗Mτ (Blended)
has higher order accuracy?
Yes ... and for all dimensions d;
Optimal τ = p/(p+ 1) ... same as in 1D case;
Obtain two additional orders of accuracy and improved constant (asin 1D case).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 44/54
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What about d > 1 dimensions?
Can we choose τ so that
K ⊗Mτ +Mτ ⊗K − ω2Mτ ⊗Mτ (Blended)
has higher order accuracy?
Yes ... and for all dimensions d;
Optimal τ = p/(p+ 1) ... same as in 1D case;
Obtain two additional orders of accuracy and improved constant (asin 1D case).
but ...
nobody wants to assemble mass matrices for both SEM and FEM orto form Kronecker products.
what to do with non-affine elements?
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 44/54
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New ‘Non-Standard’ Quadrature Rules
For τ ∈ [0, 1), define
∫ 1
−1
f(x)dx ≈ Q(p)τ (f) =
p∑
j=0
wjf(ξj)
where ξj zeros of Pp+1 − τPp−1 and
wj =2[p(1 + τ) + τ ]
p(p+ 1)Pp(ξj)[P ′
p+1(ξj)− τP ′
p−1(ξj)].
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 45/54
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New ‘Non-Standard’ Quadrature Rules
For τ ∈ [0, 1), define
∫ 1
−1
f(x)dx ≈ Q(p)τ (f) =
p∑
j=0
wjf(ξj)
where ξj zeros of Pp+1 − τPp−1 and
wj =2[p(1 + τ) + τ ]
p(p+ 1)Pp(ξj)[P ′
p+1(ξj)− τP ′
p−1(ξj)].
Then
nodes are distinct and contained in (−1, 1);
weights well-defined and positive;
rule has precision 2p− 1 (under-integration).
... non-standard but perfectly reasonable quadrature rule.(Ainsworth-Wajid, SINUM 2010)
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‘AW-Quad’ Quadrature Rules
ξ w
p = 1 ±0.816496580927726 1.000000000000000
p = 2 0.000000000000000 1.230769230769231
±0.930949336251263 0.384615384615385
p = 3 ±0.96433527587956 0.199826014447922
±0.429352058315787 0.800173985552078
p = 4 0.000000000000000 0.693766937669377
±0.978315678013417 0.121787277062268
±0.638731398345590 0.531329254103044
... non-standard but perfectly reasonable quadrature rules.
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‘AW-Quad’ Quadrature Rules
If you have FEM code on quads/hexahedra: Replace Gauss-Legendrequadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1), and solve.
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‘AW-Quad’ Quadrature Rules
If you have FEM code on quads/hexahedra: Replace Gauss-Legendrequadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1), and solve.
OR
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 47/54
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‘AW-Quad’ Quadrature Rules
If you have FEM code on quads/hexahedra: Replace Gauss-Legendrequadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1), and solve.
OR
If you have SEM code on quads/hexahedra: Replace Gauss-Lobattoquadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1) and solve (don’tchange basis functions).
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 47/54
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‘AW-Quad’ Quadrature Rules
If you have FEM code on quads/hexahedra: Replace Gauss-Legendrequadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1), and solve.
OR
If you have SEM code on quads/hexahedra: Replace Gauss-Lobattoquadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1) and solve (don’tchange basis functions).
Resulting approximation is precisely the optimally blended FEM/SEMapproximation.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 47/54
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‘AW-Quad’ Quadrature Rules
If you have FEM code on quads/hexahedra: Replace Gauss-Legendrequadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1), and solve.
OR
If you have SEM code on quads/hexahedra: Replace Gauss-Lobattoquadrature rule with ‘AW-Quad’ rule with τ = p/(p+ 1) and solve (don’tchange basis functions).
Resulting approximation is precisely the optimally blended FEM/SEMapproximation.
... trivial implement the method at virtually no additional overhead.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 47/54
![Page 103: Optimally Blended Finite/Spectral Element Scheme for Wave … · 2011-09-23 · Optimally Blended Finite/Spectral Element Scheme for Wave Propagation Mark Ainsworth (joint work with](https://reader036.vdocument.in/reader036/viewer/2022062918/5edc9580ad6a402d66674f54/html5/thumbnails/103.jpg)
Pekeris Waveguide (thanks to Jeff Zitelli, Univ. of Texas)
Planar case: harmonic excitation on line zS .Axisymmetric case: harmonic excitation at point zS .
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 48/54
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Pekeris Waveguide - True Solution
Real part of pressure. Frequency 200Hz. Line source.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 49/54
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Pekeris Waveguide - Mesh
PML used to truncate domain.Polynomial order of elements (yellow = quartic, green = cubic).Mesh size: hk/(2p+ 1) ≈ 0.4 at 200Hz.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 50/54
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Pekeris Waveguide - FEM vs Optimal Blending
Magnitude of error for Standard FEM (top) and Optimal Blending (bottom).In this example, we take a 2D waveguide of 1km and frequency of 200Hz.Relative error in L2 norm:Standard FEM 18.1%; Optimally Blended 4.44%.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 51/54
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Pekeris Waveguide - FEM vs Optimal Blending
Comparison of errors in standard FEM and Optimally Blended Methodover a range of frequencies.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 52/54
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Pekeris Waveguide - Axisymmetric Case
Same behaviour observed in axi-symmetric case also!
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 53/54
![Page 109: Optimally Blended Finite/Spectral Element Scheme for Wave … · 2011-09-23 · Optimally Blended Finite/Spectral Element Scheme for Wave Propagation Mark Ainsworth (joint work with](https://reader036.vdocument.in/reader036/viewer/2022062918/5edc9580ad6a402d66674f54/html5/thumbnails/109.jpg)
Blended FEM/SEM Schemes: The story so far ...
general result for optimal blending and error analysis;
numerical performance is promising;
novel non-standard quadrature rules means easily implemented in anexisting code.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 54/54
![Page 110: Optimally Blended Finite/Spectral Element Scheme for Wave … · 2011-09-23 · Optimally Blended Finite/Spectral Element Scheme for Wave Propagation Mark Ainsworth (joint work with](https://reader036.vdocument.in/reader036/viewer/2022062918/5edc9580ad6a402d66674f54/html5/thumbnails/110.jpg)
Blended FEM/SEM Schemes: The story so far ...
general result for optimal blending and error analysis;
numerical performance is promising;
novel non-standard quadrature rules means easily implemented in anexisting code.
References:
M. Ainsworth, “Discrete dispersion relation for hp-version finiteelement approximation at high wave number,” SINUM, vol. 42, no. 2,2004.
M. Ainsworth and H. A. Wajid, “Dispersive and dissipative behavior ofthe spectral element method,” SINUM, vol. 47, no. 5, 2009.
M. Ainsworth and H. A. Wajid, “Optimally blended spectral-finiteelement scheme for wave propagation and nonstandard reducedintegration,” SINUM, vol. 48, no. 1, 2010.
Modern Techniques for Numerical Solution of PDEs, ACMAC, Crete, September 2011 – p. 54/54