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| 01-Apr-2020 |
Toby Isaac <tisaac@cc.gatech.edu> |
Add PetscDTPDKEvalJet(), PetscDTJacobiEvalJet(), PetscDTIndexToGradedOrder(), and PetscDTGradedOrderToIndex()
The Proriol-Koordwinder-Dubiner basis (PKD) is orthonormal on the
reference simplex, so
Add PetscDTPDKEvalJet(), PetscDTJacobiEvalJet(), PetscDTIndexToGradedOrder(), and PetscDTGradedOrderToIndex()
The Proriol-Koordwinder-Dubiner basis (PKD) is orthonormal on the
reference simplex, so we should make it available as a stable basis of
evaluation.
The recursive definition allows for a straightforward and stable
calculation of all derivatives, not just first and second.
`src/dm/dt/tests/ex9.c` tests that the basis is orthonormal
and compares the derivatives against the derivatives computed by
tensor-product Legendre polynomials.
I thought about how to make an interface that would handle that,
and I decided returning the jet in the same kind of graded
representation that makes sense for multivariate polynomial spaces was a
good choice.
To facilitate this, I also added indexing functions for figuring out
where a tuple (either of exponents in a monomial or orders in a partial
derivatives) is in this ordering.
Along the way I made PetscDTStroudConicalQuadrature() handle arbitrary
dimensions instead of just 1--3.
We have a variety of dimensional orderings in PetscDT, related to how we
order points of evaluation, components of the jet, and basis functions.
I thought that:
- downstream users concerned with performance might prefer points of
evaluation as the fastest varying index to facilitate vectorization
of numerical integration, and
- having basis functions as one of the outer dimensions (first or last)
would simplify computing the matrix-matrix product of
a [basis function x evaluation point] matrix with a Vandermonde
inverse matrix
So I ordered the output dimensions of PetscDTPKDEvalJet() as
[basis function, jet component (derivative), evaluation point]. I am
seeking comment as to whether this is good, and whether it might be a
good convention to adopt elsewhere in PetscDT.
undo extern
Fix complex type issue in dm/dt/tests/ex9.c
squashme: fix scaling in PetscDTStroudConicalQuadrature()
squashme: loosen tolerance for single precision
squashme: cast to double before printing
squashme: remove unused static functions
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