Degrees of Freedom
This document describes the degrees of freedom (DOFs) for each finite element space available in PyGeoN, and how they are laid out in the global solution vector.
Scalar spaces
Lagrange1 — \(H^1\), piecewise linear (source)
One DOF per mesh node.
DOF meaning. Each DOF is the function value at the corresponding node.
DOF ordering. DOF \(n\) has global index \(n\) (directly indexed by node number).
Lagrange2 — \(H^1\), piecewise quadratic (source)
One DOF per mesh node and one DOF per mesh edge.
DOF meaning. The two sets are heterogeneous: nodal DOFs are function values at nodes; edge DOFs are function values at edge midpoints.
DOF ordering. All nodal DOFs come first (index = node number), followed by all edge DOFs (global index of edge \(e\) = \(N_\text{nodes} + e\)).
PwConstants — \(L^2\), piecewise constant (source)
One DOF per mesh cell.
DOF meaning. Each DOF is the integral over the cell of the function.
DOF ordering. DOF \(c\) has global index \(c\) (directly indexed by cell number).
PwLinears — \(L^2\), piecewise linear (broken) (source)
\((d+1)\) DOFs per cell (one per local node), totalling:
DOF meaning. Each DOF is the coefficient of the corresponding local linear basis function, i.e. the function value at a local node of the cell.
DOF ordering. Cell index varies fastest: the \(k\)-th local basis function of cell \(c\) has global index \(k \cdot N_\text{cells} + c\).
PwQuadratics — \(L^2\), piecewise quadratic (broken) (source)
Piecewise degree-2 polynomials, discontinuous across cell boundaries. The local basis consists of all monomials of degree \(\leq 2\), giving \(\tfrac{(d+1)(d+2)}{2}\) DOFs per cell: one per vertex plus one per edge (midpoint) of each simplex. The total count is:
DOF meaning. The local DOFs are heterogeneous within each cell: the first \((d+1)\) are function values at the cell vertices, and the remaining \(\tfrac{d(d+1)}{2}\) are function values at the midpoints of the cell edges.
DOF ordering. Cell index varies fastest: the \(k\)-th local basis function of cell \(c\) has global index \(k \cdot N_\text{cells} + c\).
RT0 — \(H(\text{div})\), lowest-order Raviart–Thomas (source)
One DOF per mesh face.
DOF meaning. Each DOF is the integral of the normal component of the field across the face: \(\int_f \mathbf{u} \cdot \mathbf{n}_f \, ds\).
DOF ordering. DOF \(f\) has global index \(f\) (directly indexed by face number).
BDM1 — \(H(\text{div})\), first-order Brezzi–Douglas–Marini (source)
Like RT0 the normal flux across each face is the key quantity, but here the flux is allowed to vary linearly over the face. This requires one DOF per (face, node) pair. Since each face of a \(d\)-simplex has exactly \(d\) nodes:
DOF meaning. Each DOF is a weighted normal-flux moment \(\int_f (\mathbf{u} \cdot \mathbf{n}_f)\,\varphi_i^f \, ds\), where \(\varphi_i^f\) is the Lagrange basis function of node \(i\) restricted to face \(f\).
DOF ordering. The DOFs are grouped by node-position within the face: the first \(N_\text{faces}\) entries correspond to the first node of each face, the next \(N_\text{faces}\) entries to the second node, and so on. The global index of DOF \((f,i)\) is \(f + i\,N_\text{faces}\), \(i = 0,\ldots,d-1\).
RT1 — \(H(\text{div})\), first-order Raviart–Thomas (source)
\(d\) DOFs per face and \(d\) DOFs per cell.
DOF meaning. The two sets are heterogeneous: face DOFs are directional normal-flux moments (one per spatial direction per face); cell DOFs are the \(d\) components of the cell-average of the field.
DOF ordering. The global vector is split into a face block followed by a cell block, each further divided into \(d\) sub-blocks of uniform size.
Global index of face DOF \((f, k)\): \(f + k\,N_\text{faces}\); global index of cell DOF \((c, k)\): \(d\,N_\text{faces} + c + k\,N_\text{cells}\).
Nedelec0 — \(H(\text{curl})\), lowest-order Nédélec (source)
One DOF per mesh edge.
DOF meaning. Each DOF is the integral of the tangential component of the field along the edge: \(\int_e \mathbf{u} \cdot \boldsymbol{\tau}_e \, ds\).
DOF ordering. DOF \(e\) has global index \(e\) (directly indexed by edge number).
Nedelec1 — \(H(\text{curl})\), first-order Nédélec (source)
Analogous to BDM1 but for tangential circulations: the tangential component along each edge is allowed to vary linearly, requiring one DOF per (edge, node) pair. Each edge has exactly 2 endpoint nodes in any dimension, giving:
DOF meaning. Each DOF is a weighted tangential-circulation moment \(\int_e (\mathbf{u} \cdot \boldsymbol{\tau}_e)\,\varphi_i^e \, ds\), where \(\varphi_i^e\) is the Lagrange basis function of node \(i\) on edge \(e\).
DOF ordering. Two blocks of \(N_\text{edges}\) each (one per endpoint): the global index of DOF \((e, i)\) is \(e + i\,N_\text{edges}\), \(i \in \{0, 1\}\).
Vector-valued spaces
A vector-valued space Vec<X> wraps the corresponding scalar space <X> and
replicates its DOFs for each spatial component. The total DOF count is:
DOF layout
All vector spaces use a component-block ordering: first all DOFs of the \(x\)-component, then all DOFs of the \(y\)-component, and so on.
Matrix-valued spaces
The matrix-valued \(L^2\) spaces MatPwConstants, MatPwLinears, and
MatPwQuadratics replicate the DOFs of their underlying scalar piecewise
polynomial spaces for each entry of the \(d \times d\) tensor.
replicate their DOFs for each entry of the \(d \times d\) tensor.
DOF layout
The tensor is unrolled row-wise and stored in \(d^2\) consecutive blocks of size \(N_\text{dof}^{\text{scalar}}\).
Equivalently, the global DOF index for component \((i,j)\) of scalar DOF \(k\) is:
Note that VecBDM1 / VecRT0 classes serve as matrix-valued \(H(\text{div})\) spaces
for tensors but their degrees of freedom are vector-valued;
their DOF layout follows the same row-major block structure.
Symmetric a matrix-valued space
In the symmetric matrix-valued spaces (SymMatPwConstants, SymMatPwLinears,
SymMatPwQuadratics) only the upper triangular part of the DOF is stored following the
row-major block structure as in the matrix-valued spaces.