Discontinuous Galerkin methods Lecture 3 - Brown University

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