JuliaApproximation / JuliaApproximation/MultivariateOrthogonalPolynomials.jl
Weak Laplacians on disks?
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- Dominant language
- Julia
- Stars
- 20
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- 6
- Avg merge
- 2d 17h
- Merged PRs (30d)
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Description
It would be nice to support weak laplacians. For example, it leads to a nice way of achieving Neumann condition via natural conditions: we would use Weighted(Zernike(1)) combined with the harmonic polynomials real(z^m) and imag(z^m) to pick up the rest of the polynomials. (see the new Neumann p-FEM test for an example in 1D) But it seems like there are a few options:
- Support gradients of
Zernike. But how do we incorporate the rotational invariance? And in a way sodot(∇, ∇*Weighted(Zernike(1)))still gives a diagonal operator? - Support tensor calculus a la Vasil et al. If I recall correctly this works on
∇_rand∇_θ. This will successfully capture rotational invariance, but technically leaves the world of polynomials. That is, we can't directly view it as a vector orthogonal polynomial basis. Though perhaps (1) can be built from these. - Do weak Laplacian's directly without constructing gradients. In the case of
m = 0this seems straightforward via:
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Research direction
Start with the new Neumann p-FEM example in test/test_odes.jl and review the three approaches described: gradients of Zernike, tensor calculus, or direct weak Laplacians. Compare how each handles rotational invariance and the stated diagonal-operator goal. Done requires choosing and specifying an implementation path for weak Laplacians on disks.
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Assessment
- Tech stack
- julia
- Domain
- backend
- Issue type
- Feature
- Difficulty
- 5/5
- Estimated time
- Over a week
- Activity status
- Stale
- Clarity
- Needs clarification
- Newbie friendliness
- 25/100