Gaussian Quadrature for an n-dimensional simplex
Author eschnett
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May 2020

# Gaussian Quadrature for an n-dimensional simplex

## Provenance of this package

This code was originally published by Greg von Winckel (Contact: gregvw(at)math(dot)unm(dot)edu, http://math.unm.edu/~gregvw) on the MathWorks File Exchange https://www.mathworks.com/matlabcentral/fileexchange/9435-n-dimensional-simplex-quadrature. (The given email and web addresses seems now defunct; however, a web search for Greg von Winckel easily finds up-to-date pointers.) The code in this package is a fairly literal translation from Matlab to Julia.

## Description

Construct Gauss points and weights for an `n`-dimensional simplex domain with vertices specified by the `n*(n-1)` matrix `vert`, where each row contains the coordinates `(x1,...,xn)` for a vertex. The order of the quadrature scheme in each dimension must be the same in this implementation.

## Sample usage

```X, W = simplexquad(n, vert)     # Specify the vertices
X, W = simplexquad(T, n, dim)   # Specify the dimension and use unit simplex```

`X` will be a matrix for which the `j`th column are the grid points in each coordinate `xj`.

To integrate a function `f`, use e.g.

`sum(W[i] * f(X[i,:]) for i in 1:length(W))`

I tested the package for up to `D=5` dimensions and order `N=10`, and found the integration error for polynomials of order `P≤N` (which should have only floating-point round-off error) to be less than `10eps`. This is tested by the test suite.

## Results for a 2d triangle

Integration points and weights for various numbers of points:

The area of the red disks corresponds to their integration weights. It is evident that the location of the integration points is not a tensor product and does not respect the symmetry of the domain.

## Trivia

The first four simplexes are

n Domain
1 Interval
2 Triangle
3 Tetrahedron
4 Pentatope

## TODO

Examine `JuAFEM.jl` and its `generate_quadrature.jl` file https://kristofferc.github.io/JuAFEM.jl/dev/reference/quadrature/.