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 so dot(∇, ∇*Weighted(Zernike(1))) still gives a diagonal operator?
Support tensor calculus a la Vasil et al. If I recall correctly this works on ∇_r and ∇_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 = 0 this seems straightforward via:
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 polynomialsreal(z^m)andimag(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:Zernike. But how do we incorporate the rotational invariance? And in a way sodot(∇, ∇*Weighted(Zernike(1)))still gives a diagonal operator?∇_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.m = 0this seems straightforward via: