How can I get Gauss-Lobatto points on a quadrilateral or a triangle in $x$-$y$ plane?

I am only getting abscissa coordinates and weights by solving Lobatto polynomials using Lobatto quadrature. Please suggest a method to get the $x$-$y$ coordinates

  • $\begingroup$ What do you mean by Gauss-Lobatto points on a triangle? What properties of Gauss-Lobatto do you wish to preserve? $\endgroup$ – Jesse Chan Nov 18 '15 at 16:10
  • $\begingroup$ the distance between points and number of points i get on sides of a quadrilateral should be same on the sides of a triangle as well... $\endgroup$ – Nish Dec 14 '15 at 15:27
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    $\begingroup$ i see. if that's the only requirement, you can form the nodes a bunch of different ways. One explicit way is in uea.ac.uk/~h007/publications/lobatto.pdf. Others include Hesthaven and Teng's construction, or Warburton's Warp and Blend nodes. Many of these are described in the Nodal DG Methods book @GoHokies mentioned. $\endgroup$ – Jesse Chan Dec 15 '15 at 3:55

For domains that are logically square or cubic (like your quadrilateral), you can use the tensor product (dimension-by-dimension) approach. That is, generate the 2D Gauss-Lobatto point matrix as the tensor product of your 1D Gauss-Lobatto point vectors.

An example: if your 1D Gauss-Lobatto points are $(x_1,x_2)$, then in 2D you get the following four points: $(x_i,x_j)_{i,j=1,2}$.

Generating the Gauss-Lobatto points on triangular domains is a bit more complicated. Fortunately, there are good references. Here's a couple to get the ball rolling:

  1. The Nodal DG methods book by Hesthaven and Warburton, appendix A (Google Books)
  2. The Matlab code from said book (link).
  3. These lecture notes on multidimensional Gaussian quadrature.

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