Discontinuous Galerkin methods Lecture 3 - Brown University

RMMC 2008
Discontinuous Galerkin methods
Lecture 3
Jan S Hesthaven
Brown University
-0.75
Jan.Hesthaven@Brown.edu x
y
y
1
0.75
0.5
0.25
0
-0.25
-0.5
-1
-1 -0.5 0 0.5 1
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
1.203
1.142
1.081
1.019
0.958
0.896
0.835
0.774
0.712
0.651
0.590
0.528
0.467
0.405
0.344
0.283
0.221
0.160
0.099
0.037
y
1
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
8.4 Scattering about a vertical cylinder in a finite-width channel -0.0881 147
10
0.75
9
test the consistency of the imposed bound-
8
7
ary conditions and investigate 0.5
6how
the geo-
0
cal model based on the linear Padé (2,2) ro-
0.02
tational velocity version we set up a test-0.25 for
0.01
open-channel flow. We seek to model the scat-
0
-0.5
tering of an incident wave field which is prop-
-0.01
y
-0.02
der positioned -0.03 in the middle of a finite-width
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
x
-1
-1 -0.5 0 0.5 1
x
y
-1
-1 -0.5 0 0.5 1
x
8.4 Scattering about a vertical cylinder in a finitewidth
channel
A numerical study is carried out to both
metric representation of the the domain may 0.25
affect the computed solution. In a numeri-
agating toward a bottom-mounted rigid cylin-
channel.
1
-0.75
Due to the symmetry, the solution is mathe-
y
0.02
0.01
0
-0.01
-0.02
-0.03
-0.0028
-0.0072
-0.0117
-0.0162
-0.0207
-0.0252
-0.0297
-0.0342
-0.0387
-0.0432
-0.0477
-0.0522
-0.0567
-0.0612
-0.0657
-0.0702
-0.0747
-0.0791
-0.0836
1.95E-06
1.70E-06
1.45E-06
1.20E-06
9.45E-07
6.94E-07
4.43E-07
1.92E-07
-5.90E-08
-3.10E-07
-5.61E-07
-8.12E-07
-1.06E-06
-1.31E-06
-1.57E-06
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-1
x
-1 -0.5 0 0.5 1
Figure 8.12: Scattering x of waves about a
Darmstadt International Workshop, October 2004 – p.79
cylinder in a finite-width channel.

A brief overview of what’s to come
• Lecture 1: Introduction, Motivation, History
• Lecture 2: Basic elements of DG-FEM
• Lecture 3: Linear systems and some theory
• Lecture 4: A bit more theory and discrete stability
• Lecture 5: Attention to implementations
• Lecture 6: Nonlinear problems and properties
• Lecture 7: Problems with discontinuities and shocks
• Lecture 8: Higher order/Global problems

Lecture 3
• Let’s briefly recall what we know
• Constructing fluxes for linear systems
• The local basis and Legendre polynomials
• Approximation theory on the interval

Let us recall
We already know a lot about the basic DG-FEM
• Stability is provided by carefully choosing the
numerical flux.
• Accuracy appear to be given by the local solution
representation.
• We can utilize major advances on monotone
schemes to design fluxes.
• The scheme generalizes with very few changes to
very general problems -- multidimensional systems
of conservation laws.

Let us recall
We already know a lot about the basic DG-FEM
• Stability is provided by carefully choosing the
numerical flux.
• Accuracy appear to be given by the local solution
representation.
• We can utilize major advances on monotone
schemes to design fluxes.
• The scheme generalizes with very few changes to
very general problems -- multidimensional systems
of conservation laws.
At least in principle -- but what can we actually prove ?

2.4 Interlude on linear hyperbolic problems
For But linear systems, first a the bit construction more of on thefluxes upwind numerical flux is particularly
simple and we will discuss this in a bit more detail. Important application
areas include Maxwell’s equations and the equations of acoustics and
elasticity. Let us briefly look a little more carefully at linear
To systems illustrate the basic approach, let us consider the two-dimensional
system
Q(x) ∂u
+ ∇ ·F = Q(x)∂u
∂t ∂t + ∂F 1
∂x + ∂F linear systems, the construction of the upwind numerical flux is particrly
simple and we will discuss this in a bit more detail. Important appliion
areas include Maxwell’s equations and the equations of acoustics and
sticity.
To illustrate the basic approach, let us consider the two-dimensional
tem
Q(x)
2
=0, (2.19)
∂y
where the flux is assumed to be given as
F =[F1, F 2] =[A1(x)u, A2(x)u] .
Furthermore, we will make the natural assumption that Q(x) is invertible and
symmetric for all x ∈ Ω. To formulate the numerical flux, we will need an
∂u
+ ∇ ·F = Q(x)∂u
∂t ∂t + ∂F 1
∂x + ∂F 2
=0, (2.19)
∂y
ere the flux is assumed to be given as
F =[F1, F 2] =[A1(x)u, A2(x)u] .
thermore, Prominent we will make examples the natural areassumption
that Q(x) is invertible and
metric • Acoustics
for all x ∈ Ω. To formulate the numerical flux, we will need an
• Electromagnetics
• Elasticity
In such cases we can derive exact upwind fluxes

∂t ∂x ∂y
assume approximation that Q(x) of ˆn ·F andutilizing Ai(x) information vary smoothly from both throughout sides of the Ω. interface. In
anExactly rewrite Linear how Eq. this systems (2.19) information as and is combined fluxes should follow the dynamics of the
equations.
Let us first assume that Q(x) and Ai(x) vary smoothly throughout Ω. In
Q(x) thisAssume case, we can first rewrite that Eq. all (2.19) coefficients as vary smoothly
∂u
rested in the formulation Q Q(x) of a flux+ along A1(x) the normal, + A2(x) ˆn, we+
B(x)u
operator
∂t ∂x ∂y
ll low-order terms; for example, it vanishes if Ai is constant.
rested in Π
where the = (ˆnxA1(x) formulation B collects all low-order terms; for example, it vanishe
∂u + ˆnyA2(x)) of a flux
∂u . along the normal, ˆn, we
operator+
Since A1(x) we are + interested A2(x) + inB(x)u the formulation =0, of a flux along
∂t ∂x ∂y
r that will consider the operator
−1 Π = SΛS −1 ,
diagonal matrix, Λ, has purely real entries; that is, Eq. (2.19) is a
yperbolic system [142]. We express this as
Q(x) ∂u ∂u ∂u
s all low-order Π = (ˆnxA1(x) terms; ˆn ·F = for + A1(x) Πu. + example, ˆnyA2(x)) + A2(x) it vanishes . + B(x)u if Ai is =0, constant.
∂t ∂x ∂y
terested in the formulation of a flux Π along = (ˆnxA1(x) the normal, + ˆnyA2(x)) ˆn, we .
the linear system can be understood by considering Q−1 ing to the elements of Λ that have positive and negative signs, re-
Π.
r that
Λ = Λ + + Λ − ,
The flux along a normal ˆn is then
ewhere operator B collects all low-order terms; for example, it vanishes if Ai is constant.
Since we Note are interested inˆn particular ·F in= the Πu. formulation that of a flux along the normal, ˆn, we
will consider Π = the(ˆnxA1(x) operator + ˆnyA2(x)) . ˆn ·F = Πu.
the linear system can be understood by considering Q
Π = (ˆnxA1(x) + ˆnyA2(x)) .
−1 at Q
Π.
−1Π can be diagonalized as
Q −1 Π = SΛS −1 Thus, the nonzero elements of Λ
,
− correspond to those elements
racteristic vector S−1u where the direction of propagation is ophe
normal (i.e., they are incoming components). In contrast, those
orresponding to Λ + 36 2 The key ideas
reflect components propagating along ˆn (i.e.,
Note in particular that
ˆn ·F = Πu.
The dynamics of the linear system can be understood by considering Q−1Π. Let us assume that Q−1Π can be diagonalized as
Q −1 Π = SΛS −1 lar that
ˆn ·F = Πu.
of the linear system can be understood by considering Q
,
where the diagonal matrix, Λ, has purely real entries; that is, Eq. (2.19) is a
−1Π. that Q−1Π can be diagonalized as
Q −1 Π = SΛS −1 The dynamics of the linear system can be understood by c
Let us assume that Q
,
onal matrix, Λ, has purely real entries; that is, Eq. (2.19) is a
−1Π can be diagonalized as
Q −1 Π = SΛS −1 at Q
,
where the diagonal matrix, Λ, has purely real entries; that
strongly hyperbolic system [142]. We express this as
−1Π can be diagonalized as
Q −1 Π = SΛS −1 al matrix, Λ, has purely real entries; that is, Eq. (2.19) is a
lic system [142]. We express this , as
al matrix, Λ, Λ has = purely Λ real entries; that is, Eq. (2.19) is a
ic system [142]. We express this as
+ + Λ − ,
the elements of Λ that have positive and negative signs, rethe
nonzero elements of Λ− eaving through the boundary). The basic picture is illustrated in
here we, for completeness, also illustrate a λ2 = 0 eigenvalue (i.e.,
agating mode).
his basic understanding, it is clear that a numerical upwind flux can
d as
(ˆn ·F) correspond to those elements
∗ = QS Λ + S −1 u − + Λ − S −1 u + Now diagonalize this as
u
,
–
u * u **
u +
λ2 λ1 λ3 and we obtain
Fig. 2.3. Sketch of the characteristic wave speeds of a t
Λ = Λ + −
+ + Λ − −1 boundary between , two states, u − and u + . The two intermed

know that
with λ > 0; that is, the wave corresponding to λ1 is entering the domain, the
λ1 = −λ, λ2 =0, λ3 = λ,
Linear systems and fluxes
∀i : −λiQ[u − − u + ] + [(Πu) − − (Πu) + ]=0
wave corresponding to λ3 is leaving, and λ2 corresponds to a stationary wave
must hold across each wave. This is also known as the Ra
condition and is a simple consequence of conservation of u ac
discontinuity. To appreciate this, consider the scalar wave eq
∂u
Consider the problem + λ∂u =0, x ∈ [a, b].
∂t ∂x
Integrating u over the interval, we have
−
with λ > 0; that is, the wave corresponding to λ1 is entering the domain, the
wave corresponding to λ3 is leaving, and λ2 corresponds to a stationary wave
as illustrated in Fig. 2.3.
Following the well-developed theory or Riemann solvers [218, 303], we
know that
∀i : −λiQ[u
λ
− − u + ] + [(Πu) − − (Πu) + as illustrated in Fig. 2.3.
Following the well-developed theory or Riemann solvers [218, 303], we
know that
∀i : −λiQ[u
]=0, (2.20)
must hold across each wave. This is also known as the Rankine-Hugoniot
condition and is a simple consequence of conservation of u across the point of
discontinuity. To appreciate this, consider the scalar wave equation
− − u + ] + [(Πu) − − (Πu) + ]=0, (2.20)
must hold across each wave. This is also known as the Rankine-Hugoniot
condition and is a simple consequence of conservation of u across the point of
discontinuity. To appreciate this, consider the scalar wave equation
∂u
+ λ∂u =0, x ∈ [a, b].
∂t ∂x
For non-smooth coefficients, it is a little more complex
d
dt
Then we clearly have
b
Integrating over the interval, ∂u we have
∂t a
∂x
b
d
Integrating over the interval, we have
u dx = −λ (u(b, t) − u(a, t)) = f(a, t) − f(b
a + λ∂u =0, x ∈ [a, b]. b
since f u= dx λu. = −λ On(u(b, thet) other − u(a, hand, t)) = f(a, since t) −the f(b, wave t), is propagati
dt a
speed, λ, we also have
b
d
dt
u dx = d
dt
u +
since f = λu. d On the u dx other = −λ hand,
dt (u(b, t) since − u(a, thet)) wave = f(a, is propagating t) − f(b, t), at a constant
speed, λ, we also b
a have
b
(λt − a)u − +(b − λt)u + = λ(u −
since f = λu. d On the other ahand,
since the wave is propagating at a constant
speed, λ, we alsou dt
have dx = d − +
(λt − a)u +(b− λt)u
dt
= λ(u − − u + ).
Taking a → x− and b → x + a
, we recover the jump conditions
d b
d − + Taking a → x − +
− and b → x + , we recover the jump conditions

d
dt
b
u dx = d
(λt − a)u − +(b − λt)u + = λ(u − − u + ).
Linear systems and fluxes
a dt
a →Hence, x by simple mass conservation, we achieve
− and b → x + erive the proper numerical upwind flux for such cases, we need to
careful. One should keep in mind that the much simpler local Laxhs
flux also works in this case, albeit most likely leading to more
ion than if the upwind flux , weis recover used. the jump conditions
simplicity, we assume that we have only three entries in Λ, given as
−λ(u − − u + )+(f − − f + )=0.
for a → x− ,b→ x +
λ1 = −λ, λ2 =0, λ3 = λ,
0; that is, the wave corresponding to λ1 is entering the domain, the
eralization to Eq. (2.20) is now straightforward.
These are the Rankine-Hugoniot conditions
rresponding to λ3 is leaving, and λ2 corresponds to a stationary wave
rated in Fig. 2.3.
wingFor the the well-developed general system, theory orthese Riemann are solvers [218, 303], we
at
36 2 The key ideas
∀i : −λiQ[u − − u + ] + [(Πu) − − (Πu) + ]=0, (2.20)
u * u **
ld across each wave. This is also known as the Rankine-Hugoniot
λ1 n and is a simple consequence of conservation of u across the point of
nuity. They To appreciate must hold this, consider across the each scalar wave equation
wave and can be used to
connect across the interface
∂u
∂t
+ λ∂u
∂x
=0, x ∈ [a, b].
ing over the interval, we have
Fig. 2.3. Sketch of the characteristic wave speeds of
u –
λ 2
u +
λ 3

Linear systems and fluxes
2.4 Interlude on linear hyperbolic problems 37
36 2 The key ideas
So for the 3-wave problem we have
turning to the problem in Fig. 2.3, we have the system of equations
λQ − (u ∗ − u − )+ (Πu) ∗ − (Πu) − =0,
[(Πu) ∗ − (Πu) ∗∗ ]=0,
−λQ + (u ∗∗ − u + )+ (Πu) ∗∗ − (Πu) + =0,
(u
and the numerical flux is given as
∗ , u∗∗ ) represents the intermediate states.
e numerical flux can then be obtained by realizing that
2.4 Interlude on linear hyperbolic problems 3
eturning to the problem in Fig. 2.3, we have the system of equations
λQ − (u ∗ − u − )+ (Πu) ∗ − (Πu) − =0,
[(Πu) ∗ − (Πu) ∗∗ ]=0,
−λQ + (u ∗∗ − u + )+ (Πu) ∗∗ − (Πu) + =0,
e (u∗ , u∗∗ ) represents the intermediate states.
he numerical flux can then be obtained by realizing that
(ˆn ·F) ∗ =(Πu) ∗ =(Πu) ∗∗ ,
λ 1
u –
λ 2
u * u **
Fig. 2.3. Sketch of the characteristic wave speeds of a
boundary between two states, u − and u + . The two interme
are used to derive the upwind flux. In this case, λ1 < 0, λ2
(ˆn ·F) ∗ =(Πu) ∗ =(Πu) ∗∗ ,
one can attempt to express using (u− , u + ) through the jump conditions
. This leads to an upwind flux for the general discontinuous case.
appreciate this approach, we consider a few examples.
ple 2.5. Consider first the linear hyperbolic problem
∂q
∂ u a(x) 0 ∂ u
+ A∂q = +
=0,
∂t ∂x ∂t v 0 −a(x) ∂x v
To derive the proper numerical upwind flux for s
be more careful. One should keep in mind that the m
Friedrichs flux also works in this case, albeit most
dissipation than if the upwind flux is used.
For simplicity, we assume that we have only three
one can attempt to express using (u− , u + ) through the jump condition
e. This This leads approach to an upwind is general flux for and theyields general the discontinuous exact case.
o appreciate this approach, we consider a few examples.
upwind fluxes -- but requires that the system
can be solved !
ple 2.5. Consider first the linear hyperbolic problem
∂q ∂q ∂
∂
u +
λ 3
λ1 = −λ, λ2 =0, λ3 = λ,
with λ > 0; that is, the wave corresponding to λ1 is en
wave corresponding to λ3 is leaving, and λ2 correspond

which one can attempt to express using (u
Linear systems and fluxes -- an example
Consider
− , u + ) through the jump conditions
above. This leads to an upwind flux for the general discontinuous case.
To appreciate this approach, we consider a few examples.
Example 2.5. Consider first the linear hyperbolic problem
∂q
∂ u a(x) 0 ∂ u
+ A∂q = +
=0,
∂t ∂x ∂t v 0 −a(x) ∂x v
with a(x) being piecewise constant. For this simple equation, it is clear that
u(x, t) propagates right while v(x, t) propagates left and we could use this to
form a simple upwind flux.
However, let us proceed using the Riemann jump conditions. If we introduce
a ± as the values of a(x) on two sides of the interface, we recover the
conditions
a − (q ∗ − q − )+(Πq) ∗ − (Πq) − =0,
−a + (q ∗ − q + )+(Πq) ∗ − (Πq) + =0,
where q∗ refers to the intermediate state, (Πq) ∗ is the numerical flux along
ˆn, and
(Πq) ± = ˆn · (Aq) ±
± a 0
= ˆn ·
0 −a ±
± u
v ±
± ± a u
= ˆn ·
−a ± v ±
Example 2.5. Consider first the linear hyperbolic problem
∂q
∂ u a(x) 0 ∂ u
+ A∂q = +
=0,
∂t ∂x ∂t v 0 −a(x) ∂x v
with a(x) being piecewise constant. For this simple equation, it is clear that
u(x, t) propagates right while v(x, t) propagates left and we could use this to
form a simple upwind flux.
However, let us proceed using the Riemann jump conditions. If we introduce
a
.
± as the values of a(x) on two sides of the interface, we recover the
conditions
a − (q ∗ − q − )+(Πq) ∗ − (Πq) − =0,
−a + (q ∗ − q + )+(Πq) ∗ − (Πq) + =0,
where q∗ refers to the intermediate state, (Πq) ∗ is the numerical flux along
ˆn, and
(Πq) ± = ˆn · (Aq) ±
± a 0
= ˆn ·
0 −a ±
± u
v ±
± ± a u
= ˆn ·
−a ± v ±
.
A bit of manipulation yields
(Πq) ∗ 1
=
a + + a− ∂q
∂ u a(x) 0 ∂ u
+ A∂q = +
∂t ∂x ∂t v 0 −a(x) ∂x v
+ − − + + − − +
a (Πq) + a (Πq) + a a (q − q ) ,
which simplifies as
=0,
with a(x) being piecewise constant. For this simple equation, it is clear that
u(x, t) propagates right while v(x, t) propagates left and we could use this to
form a simple upwind flux.
However, let us proceed using the Riemann jump conditions. If we introduce
a ± as the values of a(x) on two sides of the interface, we recover the
conditions
a − (q ∗ − q − )+(Πq) ∗ − (Πq) − =0,
−a + (q ∗ − q + )+(Πq) ∗ − (Πq) + =0,
where q∗ refers to the intermediate state, (Πq) ∗ is the numerical flux along
ˆn, and
(Πq) ± = ˆn · (Aq) ±
± a 0
= ˆn ·
0 −a ±
± u
v ±
± ± a u
= ˆn ·
−a ± v ±
.
A bit of manipulation yields
(Πq) ∗ 1
=
a + + a− + − − + + − − +
a (Πq) + a (Πq) + a a (q − q ) ,
which simplifies as
(Πq) ∗ = 2a+ a− a +
{{u}}
ˆn ·
+
+ a− −{{v}}
1
propagates right while v(x, t) propagates left and we could use this to
simple upwind flux.
ever, let us proceed using the Riemann jump conditions. If we introas
the values of a(x) on two sides of the interface, we recover the
ns
a
[[u]]
,
2 [[v]]
− (q ∗ − q − )+(Πq) ∗ − (Πq) − =0,
−a + (q ∗ − q + )+(Πq) ∗ − (Πq) + =0,
∗ ∗ refers to the intermediate state, (Πq) is the numerical flux along
(Πq) ± = ˆn · (Aq) ±
± a 0
= ˆn ·
0 −a ±
± u
v ±
± ± a u
= ˆn ·
−a ± v ±
.
f manipulation yields
(Πq) ∗ 1
=
a + + a− + − − + + − − +
a (Πq) + a (Πq) + a a (q − q ) ,
implifies as
(Πq) ∗ = 2a+ a− a +
{{u}}
ˆn ·
+
+ a− −{{v}}
1
[[u]]
,
2 [[v]]
numerical flux (Aq) ∗ follows directly from the definition of (Πq) ∗
erve that if a(x) is smooth (i.e., a− = a + 38 2 The key ideas
simply upwinding. Furthermore, for the general case, t
to defining an intermediate wave speed, a
) then the numerical flux is
∗ , as
a ∗ = 2a−a +
a + Following the general approach, we have
with
Solving this yields Intermediate
velocity
,
+ a− which is the harmonic average.

Let us also consider a slightly more complicated problem, originating
Linear systems and fluxes -- an example
electromagnetics.
2.5 Exercises 39
Example Consider 2.6. Consider Maxwell’s the equations
one-dimensional Maxwell’s equations
1
ε(x) 0 ∂ E 01 ∂ E
{{ZH}} + +
=0. (2.2
{{Z}}
0 µ(x) ∂t H 10 ∂x H
1
2 [[E]]
, E ∗ = 1
{{YE}} +
{{Y }}
1
2 [[H]]
or
H
,
∗ = 1
{{ZH}} +
{{Z}}
1
2 [[E]]
, E ∗ = 1
{{Y }}
where
Z ±
µ ±
=
ε ± =(Y ± ) −1 ,
2.5 Exercises 39
Here (E, H) = (E(x, t),H(x, t)) represent the electric and magnetic fiel
respectively, while ε and µ are the electric and magnetic material propertie
known as permittivity and permeability, respectively.
To simplify the notation, let us write this as
Q ∂q
+ A∂q
∂t ∂x =0,
or
H
where
ε(x) 0
01 E
Q =
, A = , q = ,
0 µ(x) 10 H
reflect the spatially varying material coefficients, the one-dimensional rotatio
operator, and the vector of state variables, respectively.
∗ = 1
{{ZH}} +
{{Z}}
1
2 [[E]]
, E ∗ = 1
{{YE}} +
{{Y }}
1
2 [[H]]
,
where
Z ±
µ ±
=
ε ± =(Y ± ) −1 ,
represents the impedance of the medium.
If we again consider the simplest case of a continuous medium, things
simplify considerably as
H ∗ = {{H}} + Y
2 [[E]], E∗ = {{E}} + Z
2 [[H]],
or
H
which we recognize as the Lax-Friedrichs flux since
∗ = 1
{{ZH}} +
{{Z}}
1
2 [[E]]
, E ∗ = 1
{{Y }}
where
Z ±
µ ±
=
ε ± =(Y ± ) −1 ,
represents the impedance of the medium.
If we again consider the simplest case of a cont
simplify considerably as
H ∗ = {{H}} + Y
2 [[E]], E∗ Z
= {{E}} +
±
µ ±
=
ε ± =(Y ± ) −1 ,
the impedance of the medium.
again consider the simplest case of a continuous medium, things
onsiderably as
H ∗ = {{H}} + Y
2 [[E]], E∗ = {{E}} + Z
2 [[H]],
recognize as the Lax-Friedrichs flux since
Y Z
=
ε µ = 1
represents the impedance of the medium.
If we again consider the simplest case of a cont
simplify considerably as
H
√ = c
εµ ∗ = {{H}} + Y
2 [[E]], E∗ = {{E}} +
which we recognize as the Lax-Friedrichs flux since
Y Z
=
ε µ = 1
The exact same approach leads to
Now assume smooth materials:
√ = c
εµ
We have recovered isthe the LF speed flux!
of light (i.e., the fastest wave speed in th
−1 −1/2

x ∈ D
Lets move on
At this point we have a good understanding of
stability for linear problems -- through the flux.
Lets now look at accuracy in more detail.
k : u k Np
h(x, t) = û
n=1
k Np
n(t)ψn(x) = u
i=1
k h(x k i ,t)ℓ k i (x).
Here, we have introduced two complementary expressions for the local solution
known as the modal form, we use a local polynomial basis, ψn(x). A simple exa
be ψn(x) =xn−1 . In the alternative form, known as the nodal representation, w
N + 1 local grid points, xk i ∈ Dk , and express the polynomial through the associ
Lagrange polynomial, ℓk i (x). The connection between these two forms is through
the expansion coefficients, ûk n. We return to a discussion of these choices in much
for now it suffices to assume that we have chosen one of these representations.
The global solution u(x, t) is then assumed to be approximated by the piec
polynomial approximation uh(x, t),
K
u(x, t) uh(x, t) = u
k=1
k y are d-dimensional simplexes.
e local inner product and L
h(x, t),
2 (D k ) norm
(u, v) k
D =
D k
uv dx, u 2
D k =(u, u) k
D ,
lobal broken inner product and norm
K
(u, v) Ω,h = (u, v) k
D , u
k=1
2 Ω,h =(u, u) Ω,h .
ects that Ω is only approximated by the union of D k , that is
K
Ω Ωh = D
k=1
k this slightly more general approach will pay off when we begin to consider
more complex nonlinear and/or higher-dimensional problems.
3.1 Legendre polynomials and nodal elements
In Chapter 2 we started out by assuming that one can represent the global
solution as a the direct sum of local piecewise polynomial solution as
K
u(x, t) uh(x, t) = u
k=1
Recall
,
ll not distinguish the two domains unless needed.
we assume the local solution to be
ne has local information as well as information from the neighalong
an intersection between two elements. Often we will refer
these intersections in an element as the trace of the element.
s we discuss here, we will have two or more solutions or boundt
the same physical location along the trace of the element.
k h(x k ,t).
The careful reader will note that this notation is a bit careless, as we do not
address what exactly happens at the overlapping interfaces. However, a more
careful definition does not add anything essential at this point and we will use
this notation to reflect that the global solution is obtained by combining the
K local solutions as defined by the scheme.
The local solutions are assumed to be of the form
x ∈ D k =[x k l ,x k r]: u k Np
h(x, t) = û
n=1
k Np
n(t)ψn(x) = u
i=1
k h(x k i ,t)ℓ k i (x).
modal basis nodal basis

choosing this.
rtant from a theoretical point of view. The results in Example 2.4 clearly
rate, Local however, approximation
that the accuracy of the method is closely linked to the
r of the local polynomial representation and some care is warranted when
sing this.
et us To beginsimplify by introducing matters, the affine introduce mapping local affine mapping
x ∈ D k : x(r) =x k l + 1+r
2 hk , h k = x k r − x k l , (3.1)
the reference variable r ∈ I = [−1, 1]. We consider local polynomial
sentations of the form
x ∈ D k : u k Np
h(x(r),t)= û
n=1
k Np
n(t)ψn(r) = u
i=1
k h(x k i ,t)ℓ k i (r).
s first discuss the local modal expansion,
Np
uh(r) = ûnψn(r).
n=1
e we have dropped the superscripts for element k and the explicit time
ndence, t, for clarity of notation.
s a first choice, one could consider ψn(r) =rn−1 Let us begin by introducing the affine mapping
x ∈ D
r ∈ [−1, 1]
Consider first the modal expansion
(i.e., the simple monobasis).
This leaves only the question of how to recover ûn. A natural way
k : x(r) =x k l + 1+r
2 hk , h k = x k r − x k l , (3.1)
with the reference variable r ∈ I = [−1, 1]. We consider local polynomial
representations of the form
x ∈ D k : u k Np
h(x(r),t)= û
n=1
k Np
n(t)ψn(r) = u
i=1
k h(x k i ,t)ℓ k i (r).
Let us first discuss the local modal expansion,
Np
uh(r) = ûnψn(r).
n=1
where we have dropped the superscripts for element k and the explicit time
dependence, t, for clarity of notation.
As a first choice, one could consider ψn(r) =rn−1 (i.e., the simple monomial
basis). This leaves only the question of how to recover ûn. A natural way
is by an L2 x ∈ D
-projection; that is by requiring that
Np
(u(r), ψm(r)) I = ûn (ψn(r), ψm(r)) I ,
n=1
for each of the Np basis functions ψn. We have introduced the inner product
on the interval I as 1
k : u k Np
h(x(r),t)= û
n=1
k Np
n(t)ψn(r) = u
i=1
k h(x k i ,t)ℓ k i (r).
et us first discuss the local modal expansion,
Np
uh(r) = ûnψn(r).
n=1
here we have dropped the superscripts for element k and the explicit time
ependence, t, for clarity of notation.
As a first choice, one could consider ψn(r) =rn−1 (i.e., the simple monoial
basis). This leaves only the question of how to recover ûn. A natural way
by an L2 Let us first discuss the local modal expansion,
Np
uh(r) = ûnψn(r).
n=1
where we have dropped the superscripts for element k
dependence, t, for clarity of notation.
As a first choice, one could consider ψn(r) =r
-projection; that is by requiring that
Np
(u(r), ψm(r)) I = ûn (ψn(r), ψm(r)) I ,
n=1
r each of the Np basis functions ψn. We have introduced the inner product
the interval I as
1
(u, v) I = uv dx.
−1
his yields
n−1 (
mial basis). This leaves only the question of how to reco
is by an L2 Np
uh(r) = ûnψn(r).
n=1
where we have dropped the superscripts for element k and the explicit time
dependence, t, for clarity of notation.
As a first choice, one could consider ψn(r) =r
-projection; that is by requiring that
Np
(u(r), ψm(r)) I = ûn (ψn(r), ψm(r
n=1
for each of the Np basis functions ψn. We have introdu
on the interval I as
1
(u, v) I = uv dx.
−1
This yields
Mû = u,
where
leading to Np equations for the Np unknown expansion
n−1 (i.e., the simple monomial
basis). This leaves only the question of how to recover ûn. A natural way
is by an L2-projection; that is by requiring that
Np
(u(r), ψm(r)) I = ûn (ψn(r), ψm(r)) I ,
n=1
for each of the Np basis functions ψn. We have introduced the inner product
on the interval I as
1
(u, v) I = uv dx.
−1
This yields
Mû = u,
where
Mij =(ψi, ψj) I , û = [û1, . . . , ûNp ]T We immediately have
or
, ui =(u, ψi) I ,
leading to Np equations for the Np unknown expansion coefficients, ûi. How-
Mij =(ψi, ψj) I , û = [û1, . . . , ûNp ]T , ui =

mial basis). This leaves only the question of how to recover ûn. A natural
is by an L2-projection; that is by requiring that
Local approximation
(u(r), ψm(r)) I =
Np
Consider the most straightforward choice
... and look at the condition number of M
Table 3.1. Condition number of M based on the monomial basis for increasing
value of N.
(u, v) I = uv dx.
N 2 4 8 16
κ(M) 1.4e+1 3.6e+2 3.1e+5 3.0e+11
This does not look like a robust choice for high N !
(i + j − 1) −1 implies an increasing close to linear dependence of the basis as
(i, j) increases, resulting in the ill-conditioning. The impact of this is that,
even at moderately high order (e.g., N>10), one cannot accurately compute
û and therefore not a good polynomial representation, uh.
A natural solution to this problem is to seek an orthonormal basis as
a more suitable and computationally stable approach. We can simply take
the monomial basis, rn , and recover an orthonormal basis from this through
an L2 But ever, why note? that - note that
1 i+j
Mij = 1+(−1)
i + j − 1
-based Gram-Schmidt orthogonalization approach. This results in the
orthonormal basis [296]
,
Linear dependence for high N (i,j)
n=1
ûn (ψn(r), ψm(r)) I ,
ψn(r) =rn−1 for each of the Np basis 3.1 Legendre functions polynomials ψn. Weand have nodal introduced elements the45 inner pro
on the interval I as
1
This yields
where
−1
Mû = u,
Mij =(ψi, ψj) I , û = [û1, . . . , ûNp ]T , ui =(u, ψi) I ,
leading to Np equations for the Np unknown expansion coefficients, ûi. H
which resembles a Hilbert matrix, known to be very poorly conditioned.
compute the condition number, κ(M), for M for increasing order of app
imation, N, we observe in Table 3.1 the very rapidly deteriorating condi

− 1) Local approximation
With this understanding, look for a better basis
Idea - make M diagonal - an orthonormal basis
−1 implies an increasing close to linear dependence of the basis as
increases, resulting in the ill-conditioning. The impact of this is that,
t moderately high order (e.g., N>10), one cannot accurately compute
therefore not a good polynomial representation, uh.
natural solution to this problem is to seek an orthonormal basis as
e suitable and computationally stable approach. We can simply take
onomial basis, rn , and recover an orthonormal basis from this through
-based Gram-Schmidt orthogonalization approach. This results in the
ormal basis [296]
ψn(r) = ˜ Pn−1(r) = Pn−1(r)
√ ,
γn−1
Pn(r) are the classic Legendre polynomials of order n and
2
γn =
2n +1
normalization. An easy way to compute this new basis is through the
ence
r ˜ Pn(r) =an ˜ Pn−1(r)+an+1 ˜ Pn+1(r),
n
an =
2
even at moderately high order (e.g., N>10), one cannot accurately
û and therefore not a good polynomial representation, uh.
A natural solution to this problem is to seek an orthonorma
a more suitable and computationally stable approach. We can si
the monomial basis, r
,
n , and recover an orthonormal basis from th
an L2-based Gram-Schmidt orthogonalization approach. This resu
orthonormal basis [296]
ψn(r) = ˜ Pn−1(r) = Pn−1(r)
√ ,
γn−1
where Pn(r) are the classic Legendre polynomials of order n and
2
γn =
2n +1
is the normalization. An easy way to compute this new basis is th
recurrence
r ˜ Pn(r) =an ˜ Pn−1(r)+an+1 ˜ Pn+1(r),
n
an =
2
(2n + 1)(2n − 1) ,
with
ψ1(r) = ˜ P0(r) = 1
√ , ψ2(r) =
2 ˜
3
P1(r) =
2 r.
h
A natural solution to this problem is to seek an orthonormal basis
a more suitable and computationally stable approach. We can simply tak
the monomial basis, r
Appendix A discusses how to implement these recurrences and a
n , and recover an orthonormal basis from this throug
an L2-based Gram-Schmidt orthogonalization approach. This results in th
orthonormal basis [296]
ψn(r) = ˜ Pn−1(r) = Pn−1(r)
√ ,
γn−1
where Pn(r) are the classic Legendre polynomials of order n and
2
γn =
2n +1
is the normalization. An easy way to compute this new basis is through th
recurrence
r ˜ Pn(r) =an ˜ Pn−1(r)+an+1 ˜ Pn+1(r),
n
an =
2
(2n + 1)(2n − 1) ,
with
ψ1(r) = ˜ P0(r) = 1
√ , ψ2(r) =
2 ˜
3
P1(r) =
2 r.
even at moderately high order (e.g., N>10), one cannot accurately compute
û and therefore not a good polynomial representation, uh.
A natural solution to this problem is to seek an orthonormal basis as
a more suitable and computationally stable approach. We can simply take
the monomial basis, r
Appendix A discusses how to implement these recurrences and a number
n , and recover an orthonormal basis from this through
an L2-based Gram-Schmidt orthogonalization approach. This results in the
orthonormal basis [296]
ψn(r) = ˜ Pn−1(r) = Pn−1(r)
√ ,
γn−1
where Pn(r) are the classic Legendre polynomials of order n and
2
γn =
2n +1
is the normalization. An easy way to compute this new basis is through the
recurrence
r ˜ Pn(r) =an ˜ Pn−1(r)+an+1 ˜ Pn+1(r),
n
an =
2
(2n + 1)(2n − 1) ,
with
ψ1(r) = ˜ P0(r) = 1
√ , ψ2(r) =
2 ˜
3
P1(r) =
2 r.
The Legendre Polynomials
Evaluated as
Appendix A discusses how to implement these recurrences and a number of
˜

erty quadrature ˜Pn(r) of wesuch use can the quadratures library be computed routine is that
46
JacobiP.m using theyJacobiGQ.m, are
3 Making
asexact,
provided as discussed u is in Appendix
it work in one dimension
fTo order compute 2Np −the 1 [90]. Np = The N +1 weights nodes and andnodes weights for (r, thew) Gaussian for an (2N +1)-th-or
Local approximation
obiP(r,0,0,n);
s, the mass matrix, M, is the identity and the problem of
resolved. So we It leaves could openevaluate the question these of how asto
compute ûn,
s
ûn =(u, ψn) I =(u, ˜ For a general function u, the evaluation of this inner produ
only practical way is to approximate the integral with a
one can use a Gaussian quadrature of the form
Np
Pn−1) I .
ûn u(ri) ˜ accurate be computed quadrature, using JacobiGQ.m, one executes as discussed the command in Appendix A.
ature, >> one [r,w] executes =JacobiGQ(0,0,N);
the command
For JacobiGQ(0,0,N);
the purpose of later generalizations, we consider a slightly different
Pn−1(ri)wi,
e Np = N +1 nodes and weights (r, w) for an (2N +1)-th-order
proach and define ûn such that the approximation is interpolatory; tha
e of later generalizations, we consider a slightly different i=1 ap-
fine ûn such that the approximation
is interpolatory; that is
u(ξi) = ûn ˜ We instead require that
Pn−1(ξi),
u(ξi) =
Np
Np
where ri are the quadrature points and wi are the quad
important property of such quadratures is that they are e
a polynomialn=1 of order 2Np − 1 [90]. The weights and nod
quadrature canfor be computed some using ξi JacobiGQ.m, as discus
To compute the Np = N +1 nodes and weights (r, w) for a
accurate quadrature, one executes the command
ûn
n=1
˜ Pn−1(ξi),
where ξi represents a set of Np distinct grid points. Note that these need
ents
be associated
a set of
with
Np distinct
any
grid
quadrature
points.
points.
Note that
We
these
can now
need
write
not
ith any This quadrature yields points. We >> can [r,w] Vû now =JacobiGQ(0,0,N);
= write u,
where
Vû = u,
For the purpose of later generalizations, we consider a sl
proach and define ûn such that the approximation is int
Vij = ˜ Pj−1(ξi), ûi =ûi, ui = u(ξi).
This matrix, V, is recognized as a generalized Vandermonde matrix and
play a pivotal role in all of the subsequent developments of the scheme. T
matrix establishes the connection between the modes, û, and the nodal val
u(ξi) = ûn
n=1
˜ Vij =
Pn−1(ξi),
˜ Pj−1(ξi), ûi =ûi, ui = u(ξi).
V is a Vandermonde matrix
, is recognized as a generalized Vandermonde matrix and will
or
Np

imating polynomial of order N. Hence, the Lebesque constant indicates h
o appreciate how this relates to the conditioning of V, recognize t
Since we have already determined that
Local approximation
˜ Pn is the optimal
Np with the need to choose the grid points ξi to define the Vand
u(r) uh(r) = ûn There is significant freedom in this choice, so let us try to
n=1
intuition for a reasonable criteria.
We first recognize that if
˜ re · ∞ is the usual maximum norm and u
Pn−1(r),
olant (i.e., u(ξi) =uh(ξi) at the grid points, ξi), then we can write
∗ far away the interpolation may be from the best
represents
possible
the
polynomial
best appr
rep
tingsentation polynomial u of order N. Hence, the Lebesque constant indicates h
away the interpolation may be from the best possible polynomial rep
∗ nsequence of uniqueness . Note that Λofisthe determined polynomial solelyinterpolation, by the grid points, we have ξi. To
an optimal approximation, we should therefore aim to identify those poin
Recall first that we have
Np
u(r) uh(r) = ûn
n=1
˜ ation u
Np
u(r) uh(r) = u(ξi)ℓi(r).
Pn−1(r),
i=1
∗ ξi, that . Note minimize thatthe Λ is Lebesque determined constant. solely by the grid points, ξi. To
optimalTo approximation, appreciate how we thisshould relates to therefore the conditioning aim to identify of V, recognize those that poi
hat aminimize consequence theof Lebesque uniqueness constant. of the polynomial interpolation, we have
To appreciate how this relates to the conditioning of V, recognize that
V
nsequence of uniqueness of the polynomial interpolation, we have
T ℓ(r) = ˜ V
P (r),
T ℓ(r) = ˜ P (r),
re ℓ =[ℓ1(r),...,ℓNp (r)]T and ˜ P (r) =[ ˜ P0(r),..., ˜ PN (r)] T and
. We are
d in the particular solution, ℓ, which minimizes the Lebesque const
ecall Cramer’s rule for solving linear systems of equations
This immediately implies
is an interpolant (i.e., u(ξi) =uh(ξi) at the grid points, ξi),
it as
Np
u(r) uh(r) = u(ξi)ℓi(r).
where ℓ =[ℓ1(r),...,ℓNp (r)]T and ˜ P (r) =[ ˜ P0(r),..., ˜ PN (r)] T . We are int
ested in the particular solution, ℓ, which minimizes the Lebesque constant
V i=1
T ℓ(r) = ˜ P (r),
ℓi(r) = Det[VT (:, 1), VT (:, 2), . . . , ˜ P (r), VT (:,i+ 1),...,VT (:,Np)]
or
re ℓ =[ℓ1(r),...,ℓNp (r)]T and ˜ P (r) =[ ˜ P0(r),..., ˜ PN (r)] T . We are in
d in the particular solution, ℓ, which minimizes the Lebesque constant
recall Cramer’s rule for solving linear systems of equations
ℓi(r) = Det[VT (:, 1), VT (:, 2), . . . , ˜ P (r), VT (:,i+ 1),...,VT (:,Np)]
Det(VT we recall Cramer’s rule for solving linear systems of equations
ℓi(r) =
.
)
Det[VT (:, 1), VT (:, 2), . . . , ˜ P (r), VT (:,i+ 1),...,VT (:,Np)]
Det(VT Det(V
.
)
It suggests that it is reasonable to seek ξi such that the denominator (i.e.,
determinant of V), is maximized.
T )
ggests that it is reasonable to seek ξi such that the denominator (i.
rminant of V), is maximized.
or this
so we
one
should
dimensional
choose
case,
the
the
point
solution
to maximize
to this problem
the
is know
tively simple form as the Np zeros of [151, 159]
determinant -- the solution is
For this one dimensional case, the solution to this problem is known i
relatively simple form as the Np zeros of [151, 159]
zeros of
f(r) = (1 − r 2 ) ˜ P ′ N (r).
uggests that it is reasonable to seek ξi such that the denominator (i.e.,
2 ˜′

It suggests that it is reasonable to seek ξi such that the denomin
determinant Local approximation
of V), is maximized.
For this one dimensional case, the solution to this problem
relatively simple form as the Np zeros of [151, 159]
This problem has an exact solution
zeros of
f(r) = (1 − r 2 ) ˜ P ′ N (r).
also known as the Gauss-Lobatto quadrature points
These are closely related to the normalized Legendre polynom
46 3 Making it work in one dimension
known as the Legendre-Gauss-Lobatto (LGL) quadrature poin
library For a general routine function JacobiGL.m u, the evaluation in Appendix of this inner A, product theseisnodes nontrivial. canThe be c
Note: These happen to also be quadrature points
which we could use to evaluate
only practical way is to approximate the integral with a sum. In particular,
one can use a Gaussian quadrature of the form
>> [r] = JacobiGL(0,0,N);
ûn
Np
i=1
u(ri) ˜ Pn−1(ri)wi,
However, they were only chosen to be good for
interpolation -- this is essential in 2D/3D
where ri are the quadrature points and wi are the quadrature weights. An
important property of such quadratures is that they are exact, provided u is
a polynomial of order 2Np − 1 [90]. The weights and nodes for the Gaussian
quadrature can be computed using JacobiGQ.m, as discussed in Appendix A.
To compute the Np = N +1 nodes and weights (r, w) for an (2N +1)-th-order

determinant continues to increase in size with N, reflecting a well-behaved
interpolation where ℓi(r) takes small values in between the nodes. For the
Local approximation - an example
equidistant nodes, the situation is entirely different.
While the determinant has comparable values for small values of N, the
determinant for the equidistant nodes begins to decay exponentially fast once
N exceeds So does 16, caused it really by the matter? close to linear -- dependence consider of Vthe
last columns
a)
c)
1.00 −1.00 1.00 −1.00 1.00 −1.00 1.00
1.00 −0.67 0.44 −0.30 0.20 −0.13 0.09
1.00 −0.33 0.11 −0.04 0.01 −0.00 0.00
1.00 0.00 0.00 0.00 0.00 0.00 0.00
1.00 0.33 0.11 0.04 0.01 0.00 0.00
1.00 0.67 0.44 0.30 0.20 0.13 0.09
1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.71 −1.22 1.58 −1.87 2.12 −2.35 2.55
0.71 −0.82 0.26 0.49 −0.91 0.72 −0.04
0.71 −0.41 −0.53 0.76 0.03 −0.78 0.49
0.71 0.00 −0.79 0.00 0.80 0.00 −0.80
0.71 0.41 −0.53 −0.76 0.03 0.78 0.49
0.71 0.82 0.26 −0.49 −0.91 −0.72 −0.04
0.71 1.22 1.58 1.87 2.12 2.35 2.55
b)
d)
1.00 −1.00 1.00 −1.00 1.00 −1.00 1.00
1.00 −0.83 0.69 −0.57 0.48 −0.39 0.33
1.00 −0.47 0.22 −0.10 0.05 −0.02 0.01
1.00 0.00 0.00 0.00 0.00 0.00 0.00
1.00 0.47 0.22 0.10 0.05 0.02 0.01
1.00 0.83 0.69 0.57 0.48 0.39 0.33
1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.71 −1.22 1.58 −1.87 2.12 −2.35 2.55
0.71 −1.02 0.84 −0.35 −0.28 0.81 −1.06
0.71 −0.57 −0.27 0.83 −0.50 −0.37 0.85
0.71 0.00 −0.79 0.00 0.80 0.00 −0.80
0.71 0.57 −0.27 −0.83 −0.50 0.37 0.85
0.71 1.02 0.84 0.35 −0.29 −0.81 −1.06
0.71 1.22 1.58 1.87 2.12 2.35 2.55
Bad points Good points
Fig. 3.1. Entries of V for N = 6 and different choices of the basis, ψn(r), and
evaluation points, ξi. For (a) and (c) we use equidistant points and (b) and (d) are
based on LGL points. Furthermore, (a) and (b) are based on V computed using a
simple monomial basis, ψn(r) =r n−1 , while (c) and (d) are based on the orthonormal
Bad
basis
Good
basis

Local approximation - an example
Determinant of V
10 25
10 20
10 15
10 10
10 5
10 0
3.1 Legendre polynomials and nodal elements 49
Equidistant nodes
Legendre Gauss
Lobatto nodes
2 6 10 14 18
N
22 26 30 34
50 3 Making it work in one dimension
Fig. 3.2. Growth of the determinant of the Vandermonde matrix, V, when computed
based on LGL nodes and the simple equidistant nodes.
It is the basic
in V. For the interpolation, this manifests itself as the well-known Runge
phenomenon [90] for interpolation at equidistant grids. The rapid decay of
the determinant enables an exponentially fast growth of ℓi(r) between the
grid points, resulting in a very poorly behaved interpolation. This is also
reflected in the corresponding Lebesque constant, which grows like 2Np 7
6
5
4
structure that 3
for
the equidistant nodes while it grows like2 log Np in the optimal case of the
LGLreally nodes (see [151, matters
307] and references 1therein).
The main lesson here is that it is generally 0 not a good idea to use equidistant
grid points when representing the local −1 interpolating polynomial solution.
This point is particularly important when the interest is in the development
of methods performing robustly at higher values of N.
ξi One can ask, however, whether there is anything special about the LGL
α
10
9
8
−1.25 −1 −0.75 −0.5 −0.25 0 0.25
It matters a great deal
Determinant of V
10 25
10 20
10 15
10 10
10 5
10 0
N=4
N=8
N=16
N=24
N=32
0 2 4 6 8 10
α

where To summarize ℓ =[ℓ1(r),...,ℓNp matters, we have local approximations of the form
(r)]T and ˜ P (r) =[ ˜ P0(r),..., ˜ PN (r)] T . W
ested in the particular solution, ℓ, which minimizes the Lebesque
This highlights that it is the overall structure of the nodes rather than
the details of the individual node position that is important; for example, one
could optimize these nodal sets Npfor
various applications. Np
Lets summarize this
u(r) uh(r) = ûn ˜ we recall Cramer’s rule for solving linear systems of equations
Pn−1(r) = u(ri)ℓi(r), (3.3
So we have the local approximations
n=1
To summarize matters, we have local approximations i=1 of the form
ℓi(r) = Det[VT (:, 1), VT (:, 2), . . . , ˜ P (r), VT (:,i+ 1),...,VT (:
Det(VT )
where ξi = ri are the Legendre-Gauss-Lobatto quadrature points. A centr
Np
u(r) uh(r) = ûn ˜ Np
component of this construction is thePn−1(r) Vandermonde = u(ri)ℓi(r), matrix, V, which (3.3) estab
lishes the connections
n=1
i=1
It suggests that it is reasonable to seek ξi such that the denomina
where ξi = ri are the Legendre-Gauss-Lobatto quadrature points. A central
component of this u = construction Vû, V is the Vandermonde matrix, V, which establishes
the connections
T ℓ(r) = ˜ P (r), Vij = ˜ determinant of V), is maximized.
Pj(ri).
For this one dimensional case, the solution to this problem is
relatively simple form as the Np zeros of [151, 159]
and are the Legendre Gauss Lobatto points:
u = Vû, V T ℓ(r) = ˜ P (r), Vij = ˜ By carefully choosing the orthonormal Legendre basis,
Pj(ri).
˜ ri
Pn(r), and the nod
points, ri, we have ensured that V is a well-conditioned object and that th
resulting interpolation zeros of is well behaved. A script for initializing V is given i
Vandermonde1D.m.
By carefully choosing the orthonormal Legendre basis, ˜ f(r) = (1 − r Pn(r), and the nodal
points, ri, we have ensured that V is a well-conditioned object and that the
resulting interpolation is well behaved. A script for initializing V is given in
Vandermonde1D.m.
2 ) ˜ P ′ N (r).
These This areleads closely to a related robust toway theof normalized computing/evaluating
Legendre polynomi
known a high-order as the Legendre-Gauss-Lobatto polynomial approximation. (LGL) quadrature points
library routine JacobiGL.m in Appendix A, these nodes can be co
but is it accurate ?
>> [r] = JacobiGL(0,0,N);

e notation A second look at approximation
k on a more rigorous discussion, we need to introduce some
ion. InWe particular, will need both global a little norms, more defined notation on Ω, and broed
over Ωh as sums of K elements D
Regular energy norms
k , need to be introduced.
ion, u(x,t), we define the global L2-norm over the domain
u 2
Ω = u
Ω
2 dx
oken norms
u 2 K
Ω,h = u
k=1
2
k
D , u2 k
D =
D k
u 2 dx.
we define the associated Sobolev norms
u (α) 2 Ω, u 2 K
Ω,q,h = u 2
D k , u2
,q D k ,q =
q
u (α) 2
additional notation. In particular, both global norms, defined on Ω, a
ken norms, defined over Ωh as sums of K elements D k , need to be intr
For the solution, u(x,t), we define the global L2-norm over the
Ω ∈ R d as
u 2
Ω = u
Ω
2 dx
as well as the broken norms
u 2 K
Ω,h = u
k=1
2
k
D , u2 k
D =
D k
u 2 dx.
In a similar way, we define the associated Sobolev norms
u 2 q
Ω,q = u
|α|=0
(α) 2 Ω, u 2 K
Ω,q,h = u
k=1
2
D k , u2
,q D k ,q =
q
u
|α|=0
where α is a multi-index of length d. For the one-dimensional cas
often considered in this chapter, this norm is simply the L2 Before we embark on a more rigorous discussion, we need to introduce some
additional notation. In particular, both global norms, defined on Ω, and broken
norms, defined over Ωh as sums of K elements D
-norm of
k , need to be introduced.
For the solution, u(x,t), we define the global L2-norm over the domain
Ω ∈ R d as
u 2
Ω = u
Ω
2 dx
as well as the broken norms
u 2 K
Ω,h = u
k=1
2
k
D , u2 k
D =
D k
u 2 dx.
In a similar way, we define the associated Sobolev norms
u 2 q
Ω,q = u
|α|=0
(α) 2 Ω, u 2 K
Ω,q,h = u
k=1
2
D k , u2
,q D k ,q =
q
u
|α|=0
(α) 2
D k,
where α is a multi-index of length d. For the one-dimensional case, most
often considered in this chapter, this norm is simply the L2 Sobolev norms
76 4 Insight through theory
derivative. We will also define the space of functions, u ∈ H
Semi-norms
-norm of the q-th
q (Ω), as those
functions for which uΩ,q or uΩ,q,h is bounded.
Finally, we will need the semi-norms
k=1
|u| 2 Ω,q,h =
|α|=0
D k ,q
D k,
u (α) 2
D k.
ulti-index of length d. For the one-dimensional case, most
in this chapter, this norm is simply the L2-norm of the q-th
4.2 Briefly on convergence
K
k=1
|u| 2
D k ,q
, |u|2
=
|α|=q

Lagrange polynomial, ℓ
Approximation theory
k i (x). The connection between these two forms is through
the expansion coefficients, ûk n. We return to a discussion of these choices in much m
for now it suffices to assume that we have chosen one of these representations.
The global solution u(x, t) is then assumed to be approximated by the piecew
polynomial approximation uh(x, t),
K
u(x, t) uh(x, t) = u
k=1
k K
, v) Ω,h = (u, v) k
D , u
k=1
h(x, t),
2 Ω,h =(u, u) Ω,h .
cts that Ω is only approximated by the union of D k , that is
K
Ω Ωh = D
k=1
k K
u(x, t) uh(x, t) = u
k=1
Recall
,
not distinguish we assume the twothe domains local unless solution needed. to be
has local information as well as information from the neighong
an intersection between two elements. Often we will refer
ese intersections in an element as the trace of the element.
e discuss here, we will have two or more solutions or boundthe
same physical location along the trace of the element.
k h(x k ,t).
The careful reader will note that this notation is a bit careless, as we do not
address what exactly happens at the overlapping interfaces. However, a more
careful definition does not add anything essential at this point and we will use
this notation to reflect that the global solution is obtained by combining the
K local solutions as defined by the scheme.
The local solutions are assumed to be of the form
x ∈ D k =[x k l ,x k r]: u k Np
h(x, t) = û
n=1
k Np
n(t)ψn(x) = u
i=1
k h(x k i ,t)ℓ k Here, we have introduced two complementary expressions
known as the modal form, we use a local polynomial basis
be ψn(x) =x
i (x).
n−1 . In the alternative form, known as the n
N + 1 local grid points, xk i ∈ Dk , and express the polynom
Lagrange polynomial, ℓk i (x). The connection between thes
the expansion coefficients, ûk n. We return to a discussion of
for now it suffices to assume that we have chosen one of th
The global solution u(x, t) is then assumed to be app
polynomial approximation uh(x, t),
K
The question is in what sense is
u(x, t) uh(x, t) =
We have observed improved accuracy in two ways
• Increase K/decrease h
• Increase N
k=1
u

simplicity, we assume that all elements have length h (i.e., D k = x k + r
˜Pn(r) = Pn(r)
√ , γn =
γn
simplicity, we assume that all elements have length h (i.e., D
Approximation theory
k = xk +
where xk = 1
2 (xkr +xk where x
l ) represents the cell center and r ∈ [−1, 1] is the refere
k = 1
2 (xk r +xk l ) represents the cell center and r ∈ vh(r) [−1, 1] = is the reference ˆvn
coordinate).
n=0
˜ Pn(r),
coordinate). We begin by considering the standard interval and introduce the new vari-
We able Let begin us assume by considering all elements the standard have interval size h and introduce consider the new va
able
v(r) =u(hr) =u(x);
that is, v is defined on thev(r) standard =u(hr) interval, =u(x); I =[−1, 1], and xk represents v ∈ L
= 0, x ∈
2 (I). Note, that for simplicity of the n
with standard notation, we now have the sum runn
than from 1 to N + 1 = Np, as used previously.
d
r
[−h, h]. We discussed in Chapter 3 the advantage of using a local orthonormal
(1 − r2 ) d
dr ˜ Pn + n(n + 1) ˜ Pn =0. (4.2)
Prior to discussing interpolation, used throughou
sic question of how well the properties of the projection where we utilize the
that is, v is defined on the standard interval, I =[−1, 1], and xk basis – in this case, the normalized Legendre polynomials
= 0, x
[−h, h]. We discussed in Chapter 3 the advantage of using a local orthonorm
basis – in this case, the normalized ˜Pn(r) = Legendre polynomials
Pn(r)
to find ˜vn as
N
2
√ γn
, γn =
˜Pn(r) = Pn(r)
√ , γn =
γn
2
2n +1 .
Here, Pn(r) are the classic Legendre polynomials of order n. A key prope
dr
of these polynomials is that they satisfy a singular Sturm-Liouville proble
˜ Pn + n(n + 1) ˜ Pn interpolation, used throughout this text, we consider =0. (4.2)
d
dr (1 − r2 ) d
dr ˜ Pn + n(n + 1) ˜ N Pn =0. (4
n=0
Let us consider the basic question of how well
2n +1 .
Here, Pn(r) are the classic Legendre polynomials of order n. A key property
of these polynomials is that they satisfy a singular Sturm-Liouville problem
d
dr (1 − r2 ) d
Let us consider the basic question of how well
vh(r) = ˆvn ˜ ote, that for simplicity of the notation and to conform
n, we now have the sum running from 0 to N rather
= Np, as used previously.
projection where we utilize the orthonomality of
Pn(r),
˜ Pn(r)
˜vn = v(r)
I
˜ Pn(r) dr.
2
2
2n +1 .
assic Legendre polynomials of order n. A key property
s that they satisfy a singular Sturm-Liouville problem
We consider expansions as
vh(r) =
n=0
ˆvn ˜ Pn(r),
˜vn =
I
N
2 h
v(r) ˜ Pn(r) dr.

simplicity, we assume that all elements have length h (i.e., D k = x k + r
˜Pn(r) = Pn(r)
√ , γn =
γn
2
2n +1 .
simplicity, we assume that all elements have length h (i.e., D
Approximation theory
k = xk +
where xk = 1
2 (xkr +xk where x
l ) represents the cell center and r ∈ [−1, 1] is the refere
k = 1
2 (xk r +xk l ) represents the cell center and r ∈ vh(r) [−1, 1] = is the reference ˆvn
coordinate).
n=0
˜ Pn(r),
assic Legendre polynomials of order n. A key property
s that they satisfy a singular Sturm-Liouville problem
coordinate). We begin by considering the standard interval and introduce the new vari-
We able Let begin us assume by considering all elements the standard have interval size h and introduce consider the new va
able
v(r) =u(hr) =u(x);
that is, v is defined on thev(r) standard =u(hr) interval, =u(x); I =[−1, 1], and xk represents v ∈ L
4.3 Approximations by orthogonal polynomials and consistency 79
= 0, x ∈
2 (I). Note, that for simplicity of the n
with standard notation, we now have the sum runn
than from 1 to N + 1 = Np, as used previously.
d
r
[−h, h]. We discussed in Chapter 3 the advantage of using a local orthonormal
(1 − r2 ) d
dr ˜ Pn + n(n + 1) ˜ Pn =0. (4.2)
An immediate consequence of
Prior
the orthonormality
to discussing
of
interpolation,
the basis is that
used throughou
sic question of how well the properties∞ of the projection where we utilize the
We consider expansions as
that is, v is defined on the standard interval, I =[−1, 1], and xk basis – in this case, the normalized Legendre polynomials
= 0, x
[−h, h]. We discussed in Chapter 3 the advantage of using a local orthonorm
basis – in this case, the normalized ˜Pn(r) = Legendre polynomials
Pn(r)
v − vh
N
2
2
I = |˜vn|
n=N+1
2 to find ,
˜vn as
˜Pn(r) = Pn(r) 2
√ , γn =
γn 2n +1 .
√ , γn =
γn 2n +1 .
Here, Pn(r) are the classic Legendre polynomials of order n. A key property
of these polynomials is that they satisfy a singular Sturm-Liouville problem
d
dr (1 − r2 ) d
Let us consider the basic question of how well
vh(r) = ˆvn ˜ vh(r) = ˆvn
n=0
Pn(r),
˜ Pn(r),
ote, that for simplicity of the notation and to conform
n, we now have the sum running from 0 to N rather
= Np, as used previously.
projection where we utilize the orthonomality of ˜ Pn(r)
˜vn = v(r)
I
˜ recognized as Parseval’s identity. A basic result for this approximation error
follows directly.
Theorem 4.1. Assume that v ∈ H
Pn(r) dr.
p ˜vn = v(r)
I
(I) and that vh represents a polynomial
projection of order N. Then
where
3q,
0 ≤ q ≤ 1
and 0 ≤ q ≤ p.
˜ Pn(r) dr.
Here, Pn(r) are the classic Legendre polynomials of order n. A key prope
dr
of these polynomials is that they satisfy a singular Sturm-Liouville proble
˜ Pn + n(n + 1) ˜ Pn interpolation, used throughout v −this vhtext, I,q ≤ we N consider =0. (4.2)
ρ−p |v| I,p ,
d
dr (1 − r2 ) d
dr ˜ Pn + n(n + 1) ˜ ρ = 2
2q − N Pn =0. (4
1
2 , q ≥ 1
n=0
Let us consider the basic question of how well
Proof. We will just sketch the proof; the details can be found in [43]. Com-
bining the definition 2 of ˜vn and Eq. (4.2), we recover
N
2 h

(N +1+σ)! n=N+1
v − vh
Approximation theory
2
|˜vn| I,0 (n − σ)!
n=N+1
,
hich, combined with Lemma 4.2 and σ = min(N +1,p), gives the result.
(N +1− σ)!
≤
(N +1+σ)!
A sharper result can be obtained by using
∞
2 (n + σ)!
which, combined with Lemma 4.2 and σ = min(N +1,p), gives the result.
A generalization of this is the following result.
A generalization of this is the following result.
emma 4.4. If v ∈ H p (I), p ≥ 1 then
Lemma 4.4. If v ∈ H p (I), p ≥ 1 then
v (q) − v (q)
h I,0 ≤
v (q) − v (q)
h I,0 ≤
where σ = min(N +1,p) and q ≤ p.
here σ = min(N +1,p) and q ≤ p.
1/2 (N +1− σ)!
|v| I,σ |v| , I,σ ,
(N + 1 + σ − 4q)!
The proof follows the one above and is omitted. Note that in the limit
N ≫ p, the Stirling formula gives
in agreement with Theorem 4.1.
(N +1− σ)!
(N + 1 + σ − 4q)!
1/2 v (q) − v (q)
h I,0 ≤ N 2q−p |v| I,p ,
(n − σ)!
The proof follows the one above and is omitted. Note that in the limit
≫ p, Note the Stirling that in formula the limit gives of N>>p we recover
agreement with Theorem 4.1.
v (q) − v (q)
h I,0 ≤ N 2q−p |v| I,p ,
A minor issues arises -- these results are based on
projections and we are using interpolations ?

The above estimates is based on areN all + n=0 related 1 points. to The projections. difference However, mayn=0 beasminor, we disbut
it is essential to
cussedisinbased Chapter appreciate on N 3, + we1 are points. it. Consider oftenThe concerned difference withmay the be interpolations minor, butofitvis essential to
where
Approximation theory
4.3 Approximations by orthogonal polynomials a
v = V ˆv,
The above estimates are all related to projections.
We consider cussed in Chapter 3, we are often concerned with th
where N
vh(r) = ˆvn v = V ˆv,
˜ N
Pn(r), ˜vh(r) = ˜vn ˜ appreciate it. Consider
N
vh(r) = ˆvn
n=0
Pn(r),
˜ N
Pn(r), ˜vh(r) = ˜vn
n=0
˜ N
vh(r) = ˆvn
n=0
Pn(r),
where vh(x) is the usual approximation based on interpolation and ˜vh(r) refers
˜ N
Pn(r), ˜vh(r) = ˜vn
n=0
˜ ere vh(x) is the usual approximation based on interpolation and ˜vh(r) ref
the approximation based on projection.
Pn(r),
By the interpolation property we have
where vh(x) is the usual approximation based on interpolation and ˜vh(r) refers
∞
to the approximation based on projection.
(V ˆv)i = vh(ri) = ˜vn ˜ N
Pn(ri) = ˜vn ˜ ∞
Pn(ri)+ ˜vn ˜ interpolation projection
Pn(ri),
is based on N + 1 points. The difference may be minor, but it is essential to
appreciate it. Consider
to the approximation By the interpolation based on projection. property we have
n=0
n=0
By the interpolation is based property on N ∞ + we1 have points. The difference may be minor,
appreciate (V ˆv)i = vh(ri)
∞ it. = ˜vn Consider N
∞
˜ N
Pn(ri) = ˜vn ˜ ∞
Pn(ri)+ ˜vn ˜ n=0
n=0
n=N+1
Compare the two
Pn(ri),
m which we recover
where vh(x) is the usual approximation based on interpolation and ˜vh(r) refers
to the approximation based on projection.
By the interpolation property we haven=0
n=0
N
vh(r) = ˆvn ˜ ∞
˜vn Pn(r), ˜vh(r) =
˜ n=0
n=N+1
˜vn Pn(ri),
˜ Pn(r), r =(r0,...,rN ) T .
∞
(V ˆv)i = vh(ri) = ˜vn
n=0
n=0
˜ N
Pn(ri) = ˜vn
n=0
˜ from which we recover n=0
Pn(ri)+
from which we recover
∞
n=N+1
V ˆv = V ˜v + ˜vn
∞
from which we recover
n=N+1
˜ Pn(r), r =(r0,...,rN ) T V ˆv = V ˜v +
n=N+1
.
is implies
This implies
This implies
(V ˆv)i = vh(ri) =
V ˆv = V ˜v +
˜vn ˜ Pn(ri) =
∞
N
n=0
˜vn ˜ Pn
where∞ vh(x) is the usual approximation based on interpol
˜vn to the approximation based on projection.
n=N+1
By the interpolation property we have
˜ Pn(r), r =(r0,...,rN ) T n=N+1
.
vh(r) = ˜vh(r)+ ˜ P T (r)V −1
∞
˜vn ˜ ∞
˜vn Pn(r). ˜ Pn(r).
This implies
V ˆv = V ˜v +
vh(r) = ˜vh(r)+ ˜ P T (r)V −1
vh(r) = ˜vh(r)+ ∞
˜ P T (r)V −1
˜vn ˜ n=N+1
Now, consider the additional term on the right-hand
Pn(r).
side,
∞
(V ˆv)i = vh(ri) = ˜vn ˜ N
Pn(ri) = ˜vn ˜ vh(r) = ˜vh(r)+
Pn(ri)+
˜ P T (r)V −1
˜vn
n=N+1
˜ Pn(r). n=N+1
Now, consider the˜P additional term on the right-hand side,
T (r)V −1
∞
˜vn ˜ w, consider the additional term on the right-hand side,
∞
T
Pn(r) = ˜vn
˜P −1
(r)V Pn(r) ˜
∞ ∞
,
Now, consider the additional term on the right-hand side,
˜vn ˜ Pn(ri)+
˜vn ˜ Pn(r), r =(r0,...,rN ) T .
∞
n=N+1
n=N+1
˜vn ˜ Pn(ri),

h h
This implies
vh(r) = ˜vh(r)+ ˜ P T (r)V −1
∞
˜vn ˜ n=N+1
Now, consider the additional term on the right-handPn(r). side,
Approximation theory
n=N+1
Now, Consider consider the this additional term term on the right-hand side,
˜P T (r)V −1
∞
˜vn ˜ ˜P
∞
T
Pn(r) = ˜vn
˜P −1
(r)V Pn(r) ˜ ,
T (r)V −1
∞
˜vn
n=N+1
˜ ∞
T
Pn(r) = ˜vn
˜P −1
(r)V Pn(r) ˜ ,
n=N+1
which is allowed provided v ∈ Hp P1: FQF/GQE (I), p>1/2 P2: FQF/GQE[280]. QC: FQF/GQE Since T1: FQF
n=N+1
n=N+1
which is allowed provided v ∈ Hp (I), p>1/2 [280]. Since
˜P T (r)V −1 N
Pn(r) ˜ = ˜pl
l=0
˜ Pl(r), V ˜p = ˜ ˜P
Pn(r),
we can interpret the additional term as those high-order modes (recall n>N)
that look like lower order modes on the grid. It is exactly this phenomenon
that is known as aliasing and which is the fundamental difference between an
interpolation and a projection.
The final statement, the proof of which is technical and given in [27], yields
the following
T (r)V −1 N
Pn(r) ˜ = ˜pl
l=0
˜ Pl(r), V ˜p = ˜ Pn(r),
we Caused can interpret by interpolation the additional termof ashigh those high-order modes (recall n>N)
that look like lower order modes on the grid. It is exactly this phenomenon
frequency unresolved modes
n = 6
that is known as aliasing and which is the fundamental difference between an
interpolation and a projection.
The final statement, Aliasing the proof of which is technical and given in [27], yields
n = −2
the following
n = 10
Caused by the grid
n n
CUUK702-Hesthaven June 6, 2006 18:54
2.2 Discrete trigonometric polynomials 3

Approximation theory
This 82 has 4 Insight a some through impact theory on the accuracy
Theorem 4.5. Assume that v ∈ Hp (I), p > 1
Theorem 4.5. Assume that v ∈ H
polynomial interpolation of order N. Then
p (I), p > 1
polynomial interpolation of order N. Then
where 0 ≤ q ≤ p.
v − vhI,q ≤ N 2q−p+1/2 v − vhI,q ≤ N |v| I,p ,
2q−p+1/2 |v| I,p ,
2 , and that vh 2 represents a
, and that vh represents a
Note in particular that the order of convergence can be up to one order
lower due to the aliasing errors, but not worse than that.
Example 4.6. Let us revisit Example 3.2 where we considered the error when
computing the discrete derivative. Using Theorem 4.5, we get a crude estimate
as
v ′ − DrvhI,0 ≤v− vhI,1 ≤ N 5/2−p Example 4.6. Let us revisit Example 3.2 where we considered the error when
computing the discrete derivative. Using Theorem 4.5, we get a crude estimate
as
v |v| I,p .
′ − DrvhI,0 ≤v− vhI,1 ≤ N 5/2−p |v| I,p .
For the first example, that is,
v(r) = exp(sin(πr)),

the situation is different. Note that v (i) ∈ Hi (I). In this case, the computations
indicate
Approximation theory
This 82 has 4 Insight a some through impact theory on the accuracy
Theorem 4.5. Assume that v ∈ Hp (I), p > 1
which is considerably better than indicated by the general result. An improved
estimate, valid only for interpolation of Gauss-Lobatto grid points, is [27]
polynomial interpolation v of order N. Then
′ − DrvhI,0 ≤ N 1−p Theorem 4.5. Assume that v ∈ H
|v| I,p ,
p (I), p > 1
polynomial interpolation of order N. Then
2 , and that vh 2 represents a
, and that vh represents a
v − vhI,q ≤ N 2q−p+1/2 and is much closer to the observed v − vh behavior but suboptimal. |v| I,p , This is, however,
I,q ≤ N
a sharp result for the general case, reflecting that special cases may show
better whereresults. 0 ≤ q ≤ p.
2q−p+1/2 |v| I,p ,
where 0 ≤ q ≤ p.
Note in particular that the order of convergence can be up to one order
We return to the behavior for the general element of length h to recover
lower due to the aliasing errors, but not worse than that.
the estimates for u(x) rather than v(r). We have the following bound:
To also account for the cell size we have
Example 4.6. Let us revisit Example 3.2 where we considered the error when
computing the discrete derivative. Using Theorem 4.5, we get a crude estimate
as
v ′ − DrvhI,0 ≤v− vhI,1 ≤ N 5/2−p Theorem 4.7. Assume that u ∈ H
|v| I,p .
p (D k ) and that uh represents a piecewise
polynomial approximation of order N. Then
u − uhΩ,q,h ≤ Ch σ−q Example 4.6. Let us revisit Example 3.2 where we considered the error when
computing the discrete derivative. Using Theorem 4.5, we get a crude estimate
as
v |u|Ω,σ,h,
′ − DrvhI,0 ≤v− vhI,1 ≤ N 5/2−p |v| I,p .
for For 0 ≤ the q ≤ first σ, example, and σ = min(N that is, +1,p).
(v (i) )r − Drv (i)
h I,0 ≤ CN1/2−i ,
v(r) = exp(sin(πr)),

u − uh 2
D k ,q ≤ h1−2qv − vh 2
Similarly fashion, we have
Approximation theory
I,q ≤ h1−2q |v| 2
I,σ = h2σ−2q |u| 2
D k ,σ ,
where we have used Lemma 4.4 and defined σ = min(N +1,p). Summing over
all elements yields the result.
For a more general grid where the element length is variable, it is natural to
use h = maxk hk u
(i.e., the maximum interval length). Combining this with
Theorem 4.7 yields the main approximation result:
2
D k ,q =
q
|u| 2
D k ,p =
q
h 1−2p |v| 2
p=0
p=0
Combining everything, we have the general result
We combine these estimates to obtain
Theorem 4.8. Assume that u ∈ Hp (D k ), p>1/2, and that uh represents a
piecewise polynomial interpolation of order N. Then
h
u − uhΩ,q,h ≤ C
σ−q
u − uh
|u|Ω,σ,h,
N p−2q−1/2 2
D k ,q ≤ h1−2qv − vh 2
I,q ≤ h1−2q |v| 2
I,σ
for 0 ≤ q ≤ σ, and σ = min(N +1,p).
This result gives the essence of the approximation properties; that is, it shows
clearly under which conditions on u we can expect consistency and what
convergence rate to expect depending on the regularity of u and the norm in
which Theorem the error4.7 is measured. yields the main approximation result:
4.4 Stability
I,p ≤ h1−2q
= h2σ−
where we have used Lemma 4.4 and defined σ = min(N +1,p
all elements yields the result.
For a more general grid where the element length is variable
use h = maxk hk (i.e., the maximum interval length). Com
with
Theorem 4.8. Assume that u ∈ Hp (D k ), p>1/2, and tha
piecewise polynomial interpolation of order N. Then

Lets summarize
Fluxes:
• For linear systems, we can derive exact upwind fluxes
using Rankine-Hugonoit conditions.
Accuracy:
• Legendre polynomials are the right basis
• We must use good interpolation points
• Local accuracy depends on elementwise smoothness
• Aliasing appears due to the grid but is under control
• For smooth problems, we have a spectral method
• Convergence can be recovered in two ways
• Increase N
• Decrease h