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Basis functions

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

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

The bases below differ by the meaning of the coefficients alpha_i. This choice is independent of whether the degrees of freedom are shared between cells (FEM) or not (DG).

1. Lagrange basis: coefficients are point values

Lagrange basis functions are nodal: phi_i equals 1 at node i and 0 at every other node of the cell, so alpha_i = u_h(node i). It is the usual choice for FEM, where the value at a node is shared between the cells that contain it. 

from scimba_jax.linear_approximation.basis.analytic_bases import local_lagrange_basis
# Nodal basis: 1 at its own node, 0 at the other nodes of the cell, so a DOF
# is a point value. Default choice for FEM and DG.
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 basis: coefficients are derivatives

Taylor basis functions are modal: they are powers of the centred, rescaled coordinate. Since there are no nodes to share, this basis is only used for DG. Each coefficient has a direct interpretation: the coefficients successively give the cell average, its gradient and its curvature. 

from scimba_jax.linear_approximation.basis.analytic_bases import local_taylor_basis
# Modal basis: powers of (x - x_center) / h, without nodes, so DG only.
# Mode 0 gives the cell average, mode 1 the 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",
)
# The powers use repeated multiplications, since x ** p with a traced p
# gives nan under vmap (derivative of 0 ** 0).

3. B-spline basis: coefficients are neither point values nor derivatives

This basis comes from isogeometric analysis. It is a smooth spline of degree p with continuity r across cell boundaries, evaluated on a sliding window of neighbouring cells instead of a single cell. For r = 0, the space is exactly the Lagrange space Q_p, written in a different basis. The two bases then give bit-identical solutions, and this property is used to test the B-spline basis. 

from scimba_jax.linear_approximation.basis.analytic_bases import (
    local_bspline_basis,
)
# B-spline basis: degree p, continuity r across cells (r = 0
# gives Lagrange Q_p).
# A DOF is a spline coefficient, read on a sliding window of
# cells.
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",
)

These three families do not depend on the type of mesh: they can be used in a FEM or DG scheme on structured, unstructured or block-structured meshes. Meshless bases (kernels and random features) are presented on the Collocation-based methods page.