Discontinuous Galerkin methods Lecture 3 - Brown University

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),