Discontinuous Galerkin methods Lecture 3 - Brown University

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 ˜′