In applied mathematics, Raviart–Thomas basis functions are

vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

basis function In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represe ...

s used in

finite element Finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat tran ...

and

boundary element method The boundary element method (BEM) is a numerical computational method of solving linear partial differential equations which have been formulated as integral equations (i.e. in ''boundary integral'' form), including fluid mechanics, acoustics, ele ...

s. They are regularly used as basis functions when working in

electromagnetics In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

. They are sometimes called Rao-Wilton-Glisson basis functions. The

space Space is a three-dimensional continuum containing positions and directions. In classical physics, physical space is often conceived in three linear dimensions. Modern physicists usually consider it, with time, to be part of a boundless ...

\mathrm_q

spanned by the Raviart–Thomas basis functions of order

q

is the smallest polynomial space such that the

divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the rate that the vector field alters the volume in an infinitesimal neighborhood of each point. (In 2D this "volume" refers to ...

maps

\mathrm_q

onto

\mathrm_q

, the space of piecewise polynomials of order

q

Order 0 Raviart-Thomas Basis Functions in 2D

two-dimensional space A two-dimensional space is a mathematical space with two dimensions, meaning points have two degrees of freedom: their locations can be locally described with two coordinates or they can move in two independent directions. Common two-dimensiona ...

, the lowest order Raviart Thomas space,

\mathrm_0

, has degrees of freedom on the edges of the elements of the finite element mesh. The

n

th edge has an associated basis function defined by

\mathbf_n(\mathbf)=\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

l_n

is the length of the edge,

T_+

and

T_-

are the two triangles adjacent to the edge,

A_n^+

and

A_n^-

are the areas of the triangles and

\mathbf{p}_+

and

\mathbf{p}_-

are the opposite corners of the triangles. Sometimes the basis functions are alternatively defined as

\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

{{DEFAULTSORT:Raviart-Thomas basis functions Finite element method Numerical differential equations Partial differential equations