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Basis functions

FEM and DG both represent the solution, cell by cell, as a combination of local polynomials:

\(u_h(x) = \sum_i \alpha_i\, \varphi_i(x)\)

What changes from one basis to the next is what a coefficient alpha_i actually MEANS, and that choice is independent of whether the degrees of freedom end up shared (FEM) or not (DG).

1. Lagrange — a coefficient is a point value

The nodal family: phi_i equals 1 at node i and 0 at every other node of the cell, so alpha_i = u_h(node i) directly. The natural choice for FEM, where a node's value is exactly what gets shared between cells. 

from scimba_jax.linear_approximation.basis.analytic_bases import local_lagrange_basis

# NODAL: 1 at its own node, 0 at every other node of the cell -- a DOF IS
# a point value. The natural choice for FEM, and the default for DG too.
basis = AnalyticBasis(
    nb_basis=(order + 1) ** 2,  # Q_p tensor-product
    out_dim=1,
    mesh=mesh,
    local_basis=lambda y, i, m: local_lagrange_basis(y, i, m, order=order, out_dim=1),
    basis_type="scalar",
)

2. Taylor — a coefficient is a derivative

The modal family: basis functions are powers of the centred, rescaled coordinate. There is no node to share, so this one only makes sense on DG. The upside is that reading a coefficient off the vector already tells you something — the cell average, then its gradient, then its curvature. 

from scimba_jax.linear_approximation.basis.analytic_bases import local_taylor_basis

# MODAL: powers of (x - x_center) / h -- no nodes, so DG only. The zeroth
# mode is the cell average, the first its gradient, and so on.
basis = AnalyticBasis(
    nb_basis=(order + 1) ** 2,
    out_dim=1,
    mesh=mesh,
    local_basis=lambda y, i, m: local_taylor_basis(y, i, m, order=order, out_dim=1),
    basis_type="scalar",
)
# Never raise a vmapped exponent to a traced power (0**0 = nan under vmap):
# built by repeated multiplication instead.

3. B-spline — a coefficient is neither

Isogeometric analysis: a smooth spline of degree p and continuity r across cell boundaries, read by a sliding window of neighbouring cells rather than a single one. At r = 0 the space is exactly Lagrange Q_p again, written in a different basis — the two must give bit-identical solutions, and that identity is exactly how this basis is tested. 

from scimba_jax.linear_approximation.basis.analytic_bases import local_bspline_basis

# B-SPLINE: degree p, continuity r across cells. r = 0 is Lagrange Q_p in
# another basis (tested for bit-identical solutions). A DOF is a spline
# coefficient, read by a sliding window of cells rather than a single one.
basis = AnalyticBasis(
    nb_basis=(order + 1) ** 2,
    out_dim=1,
    mesh=mesh,
    local_basis=lambda y, i, m: local_bspline_basis(
        y, i, m, order=order, regularity=order - 1, out_dim=1
    ),
    spline_regularity=order - 1,
    basis_type="scalar",
)

None of this is specific to a mesh shape — the same three families plug into a FEM or DG scheme on a structured, unstructured or block-structured mesh alike. A different family entirely — kernels and random features, with no mesh at all — is on the Collocation-based methods page.