Spring 2003

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MA557/MA578/CS557Lecture 23

• Spring 2003
• Prof. Tim Warburton
• timwar_at_math.unm.edu

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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
where M(p1)(p2)/2 and we have normalized each
of the
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Accuracy of Polynomial Interpolation

• We introduce the operator supremum-norm and the
  Linfinity norm
• for the pth order interpolation operator.
• The interpolation error is bounded below by

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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.

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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.

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Proof
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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.
• Heres 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

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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/articl
  es/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.

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Fekete Nodes For the Triangle

1. Read the Wingate and Taylor paper over the break.
2. The Fekete nodes are chosen according tothe
   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.

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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
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We need to compute the gradient of log(V)
Step 1 expand the determinant of the
Vandermonde matrix with respect to the
co-determinants
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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
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Step 3 Expand derivative terms with respect to
the Lagrange interpolating basis for
the M nodes
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Step 4 replace derivative terms in
differentiation of V
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Step 5 factorize matrix
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Conclusion

• 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 !.
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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 p1 to p10.
• 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 p4

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P7 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
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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.

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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

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• Starting with edge 1 (-1ltalt1,b-1), we will
  work with the PKDO basis

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Edge 2

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

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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.

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Computing Jump Terms

• Note we need Cm which only makes sense for the
  nodes which lie on the fth edge.

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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.