PetscDTPKDEvalJet#

Evaluate the jet (function and derivatives) of the Proriol-Koornwinder-Dubiner (PKD) basis for the space of polynomials up to a given degree. The PKD basis is L2-orthonormal on the biunit simplex (which is used as the reference element for finite elements in PETSc), which makes it a stable basis to use for evaluating polynomials in that domain.

Synopsis#

#include "petscdt.h" 
PetscErrorCode PetscDTPKDEvalJet(PetscInt dim, PetscInt npoints, const PetscReal points[], PetscInt degree, PetscInt k, PetscReal p[])

Input Parameters#

dim - the number of variables in the multivariate polynomials
npoints - the number of points to evaluate the polynomials at
points - [npoints x dim] array of point coordinates
degree - the degree (sum of degrees on the variables in a monomial) of the polynomial space to evaluate. There are ((dim + degree) choose dim) polynomials in this space.
k - the maximum order partial derivative to evaluate in the jet. There are (dim + k choose dim) partial derivatives in the jet. Choosing k = 0 means to evaluate just the function and no derivatives

Output Parameter#

p - an array containing the evaluations of the PKD polynomials’ jets on the points. The size is ((dim + degree) choose dim) x ((dim + k) choose dim) x npoints, which also describes the order of the dimensions of this three-dimensional array: the first (slowest varying) dimension is basis function index; the second dimension is jet index; the third (fastest varying) dimension is the index of the evaluation point.

Notes#

The ordering of the basis functions, and the ordering of the derivatives in the jet, both follow the graded ordering of PetscDTIndexToGradedOrder() and PetscDTGradedOrderToIndex(). For example, in 3D, the polynomial with leading monomial x^2,y^0,z^1, which has degree tuple (2,0,1), which by PetscDTGradedOrderToIndex() has index 12 (it is the 13th basis function in the space); the partial derivative \(\partial_x \partial_z\) has order tuple (1,0,1), appears at index 6 in the jet (it is the 7th partial derivative in the jet).

The implementation uses Kirby’s singularity-free evaluation algorithm, https://doi.org/10.1145/1644001.1644006.