Raviart–Thomas basis functions

In applied mathematics, Raviart–Thomas basis functions are

The space ${\displaystyle \mathrm {RT} _{q}}$ spanned by the Raviart–Thomas basis functions of order ${\displaystyle q}$ is the smallest polynomial space such that the divergence maps ${\displaystyle \mathrm {RT} _{q}}$ onto ${\displaystyle \mathrm {P} _{q}}$, the space of piecewise polynomials of order ${\displaystyle q}$.^[2]

Order 0 Raviart-Thomas Basis Functions in 2D

In two-dimensional space, the lowest order Raviart Thomas space, ${\displaystyle \mathrm {RT} _{0}}$, has degrees of freedom on the edges of the elements of the finite element mesh. The ${\displaystyle n}$th edge has an associated basis function defined by^[3]

${\displaystyle \mathbf {f} _{n}(\mathbf {r} )=\left\{{\begin{array}{ll}{\frac {l_{n}}{2A_{n}^{+}}}(\mathbf {r} -\mathbf {p} _{+})\quad &\mathrm {if\ \mathbf {r} \in \ T_{+}} \\-{\frac {l_{n}}{2A_{n}^{-}}}(\mathbf {r} -\mathbf {p} _{-})\quad &\mathrm {if\ \mathbf {r} \in \ T_{-}} \\\mathbf {0} \quad &\mathrm {otherwise} \end{array}}\right.}$

where ${\displaystyle l_{n}}$ is the length of the edge, ${\displaystyle T_{+}}$ and ${\displaystyle T_{-}}$ are the two triangles adjacent to the edge, ${\displaystyle A_{n}^{+}}$ and ${\displaystyle A_{n}^{-}}$ are the areas of the triangles and ${\displaystyle \mathbf {p} _{+}}$ and ${\displaystyle \mathbf {p} _{-}}$ are the opposite corners of the triangles.

Sometimes the basis functions are alternatively defined as

${\displaystyle \mathbf {f} _{n}(\mathbf {r} )=\left\{{\begin{array}{ll}{\frac {1}{2A_{n}^{+}}}(\mathbf {r} -\mathbf {p} _{+})\quad &\mathrm {if\ \mathbf {r} \in \ T_{+}} \\-{\frac {1}{2A_{n}^{-}}}(\mathbf {r} -\mathbf {p} _{-})\quad &\mathrm {if\ \mathbf {r} \in \ T_{-}} \\\mathbf {0} \quad &\mathrm {otherwise} \end{array}}\right.}$

with the length factor not included.

References