1

Spring 2003

Prof. Tim Warburtontimwar@math.unm.edu

MA557/MA578/CS557Lecture 23

2

Basis Ordering

There has been some confusion about the numbering of the PKDO basis functions. There are now a number of basis functions. We will choose an ordering:

,0

1 0,02 0,1

2 1,03 1,1

3 2,0 0,

2 2 ,11 p

p

pM p

p pp

where M=(p+1)(p+2)/2 and we have normalized each of the { }i

3

Accuracy of Polynomial Interpolation

• We introduce the operator supremum-norm and the Linfinity norm:

for the p’th order interpolation operator.

• The interpolation error is bounded below by:

,0sup where max ,

p

px y Tf

I fI f f x y

f

*

*

, , , ,

where is the optimal polynomial approximation

in the . norm

pf x y f x y f x y I f x y

f

4

cont

• This tells us that the interpolating polynomial which approximates a function can only be less or equal in accuracy to the best polynomial approximation.

• Finding the optimal polynomial which approximates a function is an unsolved problem.

• However, there is a theorem by Lebesgue which allows us to bound the interpolation error in terms of the error one would attain by choosing the best possible polynomial approximation.

5

Theorem (Lebesgue)

• Assume that and we consider a set of M interpolation points then:

• i.e. the Lebesgue constant gives us an idea of the optimality of the M points. The smaller is the better.

f C T

*

,1

1

where max , is

termed the Lebesgue constant

p p

i Mp p

ix y T

i

f I f f f

I h x y

p

6

Proof

* *

* *

1

* *

1 1

* p

* *

1

*

triangle inequality

, , definition of interpolant

, , , ,

ok since P T

, , ,

p p

n M

n n nn

n M n M

n n n n n nn n

n M

n n n n nn

f I f f f f I f

f f f f x y h x y

f f f x y h x y f x y h x y

f

f f f x y f x y h x y

f f

*

1

* *

1

*

1

, , ,

,

1 ,

n M

n n n n nn

n M

nn

n M

nn

f x y f x y h x y

f f f f h x y

h x y f f

7

Comments

• Note – the Lebesgue number only depends on the distribution of nodes. The only condition on the function being approximated is that it is continuous.

• Here’s the tricky part. It is difficult to compute exactly as it requires us to search the continuum of points in the triangle.

• However, we can approximate this number by randomly sampling a large number of points in the triangle. i.e. for some, large, N compute the following:

p

1,..,

1

1,..,

max ,

for some finite subset ,

i Mp

i j jj N

i

j jj N

h x y

x y T

8

Open Problem

• It is still an open problem to find the set of nodes for the triangle which minimize the Lebesgue constant.

• Various attempts have been made:• Hesthaven:• http://epubs.siam.org/sam-bin/getfile/SINUM/articles/30587.pdf

(You will need access permission – so download on on campus)

• Wingate and Taylor:• http://citeseer.nj.nec.com/taylor98fekete.html

• And others.. See comparison tables in Hesthaven paper.

9

Fekete Nodes For the Triangle

1) Read the Wingate and Taylor paper over the break.

2) The Fekete nodes are chosen according to:the condition that they are the nodes which maximize the determinant of the VDM matrix.

3) Wingate and Taylor suggest a method for calculating the Fekete nodes based on the method of steepest ascent.

4) Nodes are moved in the direction which most increase the log of the determinant of the generalized Vandermonde matrix.

10

Steepest Descent Approach

Suppose we start with a list of M initial node locations we willsolve the following ODE to steady state. We denote thedeterminant of the Vandermonde matrix by:

We will now show that this is equivalent to solving:

1 1, , log , , , ,n n

i i

M Mr st r s

r s r sV

V

, , ,,

n n n n n nn nr s

tr s r s

h h

r s

11

We need to compute the gradient of log(V)

1 1 11

1 ,M

n

n nn

r sV V

Step 1: expand the determinant of the Vandermonde matrix with respect to the co-determinants:

1 1, , log , , , ,n n

i i

M Mr st r s

r s r sV

12

Step 2: as an example of how to compute the gradient of |V| with respect to the node coordinates we first differentiate with respect to r1

1 1 11

1 1 11

1 1 1 1 1

1 2 2 2 2

1

11 ,

1 ,

, ,

, ,

, ,

Mn

r nn

Mn

r n nn

r r M

M

M M M M M

r r s

r s

r s r s

r s r s

r s r s

V V

V

13

Step 3: Expand derivative terms with respect to the Lagrange interpolating basis for the M nodes

1

1

, , ,

, , ,

m M

n m n m mm

m M

r n r m n m mm

r s h r s r s

r s h r s r s

14

Step 4: replace derivative terms in differentiation of V:

1 1 1 1 11 1

1 2 2 2 2

1

1 1 1 1 1

1 2 2 2 2

1

, , , ,

, ,

, ,

1

, ,

, ,

, ,

m M m M

r m m m r m M m mm m

M

M M M M M

r r M

M

M M M M M

r

h r s r s h r s r s

r s r s

r s r s

r s r s

r s r s

r s r s

V

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Step 5: factorize matrix

1 1 1 2 1 1 1 1

1 1 1 1 1

1 2 2 2 2

1

1 1 1

1

, , ,0 1 0

0 0 1

, ,

, ,

, ,

,

r r r M

M

M

M M M M M

r

r

h r s h r s h r s

r s r s

r s r s

r s r s

h r s

V

V

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Conclusion:

1 11

1

,r sh

r r

VV

• This is not exactly as reported in the Wingate & Taylor paper.

• However, since |V| simply scales the node velocities uniformly it will not make very much difference as we are integrating in a “pseudo-time” manner.

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Interpretation of Algorithm

At each node, the algorithm pushes the node in the direction of the slope of the interpolating polynomial at that node:

At steady state, the interpolating polynomial associated with a node will have (approximately)zero slope at that node..

This is a property shared by the Gauss-Lobatto- Legendre nodes in 1D !.

,n

n n nn

rh r s

st

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Download The Code

• Download the Fekete node routine computing routine from the class web page.

• Over the break use this code to compute the Fekete nodes for p=1 to p=10.

• Save the node coordinates in a file for each polynomial order..

• These will be the nodes we use for future computations.

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To Run:

• Unzip the files and set the current directory to the umFEKETE directory.

• In Matlab:

p = 7;[R,S] = am282fekete2d(p);scatter(R,S);

• This might take quite a while. The code will output the approximate Lebesgue constant as the nodes move.

• Run now with p=4

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P=7 nodes

• Note there are 8 nodes on each edge and a total of 36 nodes.

• The approximate Lebesgue number is: 4.92707087069006

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am282fekete2d

am282ab

am282jacobideriv2dab

am282derivmatrix2d ˆ

r r1D D V

am282vdm2dab

,a bV

am282jacobi2dab ,n a b

am282jacobi1d, ( )nP x

am282jacobderiv1d,

( )ndPx

dx

, , ,n na b a br s

2 11

1

ra

sb s

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Surface Mass Matrices

• Recall the DG scheme:

• We now have defined the set of nodes which we can build the hn Lagrange interpolating polynomials.

• The only remaining task is to discretize the boundary terms.

1 1 1

Find , 1,..., such that:

.....

0

where ,

, ,

2

,

,

m M m M m Mm m m

n m n m n mTm m mT T

T

m

n

dC h hh h u h C v h C

dt

C t m M

C C

x y

C

C

h

u n

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Surface Term

• We now break the boundary integral into the three edge components:

• Last step due to the face normals being constant

• The last step is to compute:

3

1 1

3

1 1

, ,2 2

,2

f

f

fm Mf f

n n m mm f

T T

fm Mf f

n m mTm f

h C h h C

h h C

,f

n m Th h

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• Starting with edge 1 (-1<=a<=1,b=-1), we will work with the PKDO basis:

,f

n m Th h

1̂

1

1

10,0 2 1,0 0,0 2 1,0

1

12 1,0 2 1,0 0,0 0,0

1

2 1,0 2 1,0

,

, 1 , 1

1 ( 1) 1 ( 1)1 1

2 2

1 1

21 1

2 1

ij klT

ij kl

i ki k

i j k l

i kj l i k

i kj l ik

a a da

P a P P a P da

P P P a P a da

P Pi

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1

1

1

1

2 1,0 2 1,0

ˆ

1

1 0 0

1 1ˆ

0 0 0 0 ˆ

1 1

ˆ0

2, 1 1

2 1

, , , ,

, ,

,

i kj l ikij kl

T

i p j p in M

n n n n ijij ij n ijn i j

i p j p i k p l p k

n m ij klT n ij n kli j k l T

l

ij kln ij n kl Tl

P Pi

r s r s h r s h r s

h h

V

V V

V V

0 0 0

1 1 2 1,0 2 1,0

0 0 0 0

1 1 2 1,0 2 1,0

0 0 0

21 1

2 1

21 1

2 1

i p j p i k p p k

i j k

i p j p i k p l p ki kj l ikn ij n kl

i j k l

i p j p i l p ii ij ln ij n il

i j l

P Pi

P Pi

V V

V V

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Edge 2

• If we try the same thing with the second edge it gets more complicated:

2̂

1

1

10,0 2 1,0 0,0 2 1,0

1

12 1,0 2 1,0

1

,

1, 1,

1 11 1

2 2

1...?

2

ij klT

ij kl

i ki k

i j k l

i ki kj l

b b db

b bP P b P P b db

bP b P b db

27

Edges 2 & 3

• We can use the surface mass matrix from edge 1 by noting that the Fekete nodes have the same distribution on edge 1 as they do on edge 2.

• Identify which nodes correspond and permute the edge 1 matrix.

28

Computing Jump Terms

• Note we need [Cm] which only makes sense for the nodes which lie on the f’th edge.

29

Next Lecture

• Next class we will start examining the consistency of the DG scheme

• Also – finally the consistency analysis.

• Review the Hesthaven-Warburton paper for details.