Finite element

A finite element specifies a reference region, a finite-dimensional space of local trial functions, and degrees of freedom that uniquely determine each such function. Assembling compatible copies over a finite element mesh gives a global trial space. For a one-dimensional linear element, the region is an interval, the functions are affine functions, and the degrees of freedom are the endpoint values.

The quadratic Lagrange triangle uses degree-at-most-two polynomials on a nondegenerate triangle, with point evaluation at its three polygon vertices and three edge midpoints as the six finite element degrees of freedom. If ${\lambda }_i$ are the affine functions of its barycentric coordinate system, the cardinal shape functions areAt a polygon vertex, one coordinate in the barycentric coordinate system is one and the others vanish; at an edge midpoint, two are $1/2$ and the third is zero. These values prove the cardinal property. Since $\dim {\mathcal{P}}_2(K)=6$, this is an unisolvent finite element. Matching the two endpoint values and midpoint value on a shared edge of a polygon matches the whole quadratic restriction, producing a continuous global conforming finite element space.

A finite element with local shape space $P$ and $n=\dim P$ finite element degrees of freedom ${\ell }_1,\dots ,{\ell }_n$ is unisolvent when each possible list of values determines exactly one member of $P$. By the rank-nullity theorem, it suffices to show that ${\ell }_i(p)=0$ for every $i$ implies $p=0$. Equivalently, one may construct a cardinal basis $N_1,\dots ,N_n$ with ${\ell }_i(N_j)={\delta }_{ij}$. Such a basis is linearly independent by applying each ${\ell }_i$, and its size makes it span $P$; then $p={\sum }_i{\ell }_i(p)N_i$.

A finite element degree of freedom is one of the linear functionals chosen to specify a local shape function in a finite-dimensional vector space $P$. Examples are point evaluation functionals and edge or cell integrals. The finite element is an unisolvent finite element when the complete list gives a bijection from $P$ to the corresponding list of real values.