fem.projectors#

Functions#

`knot_insertion_projection_operator`(domain, ...)	Compute the projection operator based on the knot insertion technique.
`knots_to_insert`(coarse_grid, fine_grid[, tol])	Compute the point difference between the fine grid and coarse grid.

Details#

knots_to_insert(coarse_grid, fine_grid, tol=1e-14)[source]#: Compute the point difference between the fine grid and coarse grid.

knot_insertion_projection_operator(domain, codomain)[source]#

Compute the projection operator based on the knot insertion technique.

Return a linear operator which projects an element of the domain to an element of the codomain. Domain and codomain are scalar spline spaces over a cuboid, built as the tensor product of 1D spline spaces. In particular, domain and codomain have the same multi-degree (p1, p2, …).

This function returns a LinearOperator K working at the level of the spline coefficients, which are represented by StencilVector objects.

Thanks to the tensor-product structure of the spline spaces, the projection operator is the Kronecker product of 1D projection operators K[i] operating between 1D spaces. Each 1D operators is represented by a dense matrix:

K = K[0] x K[1] x …

For each dimension i the 1D grids defined by the breakpoints of the two spaces are assumed to be identical, or one nested into the other. Let nd[i] and nc[i] be the number of cells along dimension i for domain and codomain, respectively. We then have three different cases:

nd[i] == nc[i]: The two 1D grids are assumed identical, and K[i] is the identity matrix.
nd[i] < nc[i]: The 1D grid of the domain is assumed nested into the 1D grid of the codomain, hence the 1D spline space of the domain is a subspace of the 1D spline space of the codomain. In this case we build K[i] using the knot insertion algorithm.
nd[i] > nc[i]: The 1D grid of the codomain is assumed nested into the 1D grid of the domain, hence the 1D spline space of the codomain is a subspace of the 1D spline space of the domain. In this case we build K[i] as the transpose of the matrix obtained using the knot insertion algorithm from the codomain to the domain.

Parameters:

domainTensorFemSpace: Domain of the projection operator.
codomainTensorFemSpace: Codomain of the projection operator.

Returns:

KroneckerDenseMatrix: Matrix representation of the projection operator. This is a LinearOperator acting on the spline coefficients.