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.

An isoparametric finite element uses the same reference shape functions to interpolate its geometry and its unknown field. With reference functions $N_i$ and physical nodes ${\mathbf{x}}_i$, the geometry map is $F={\sum }_i{\mathbf{x}}_i{N}_i$ and the physical functions are $N_i\circ F^{-1}$. The map must be one-to-one and have a nonvanishing Jacobian determinant. For bilinear quadrilaterals, compatible common edges have affine geometry maps and identical linear field traces, giving a continuous conforming finite element space. On a general quadrilateral the pulled-back functions need not be polynomials in physical coordinates.

A shape function is a local basis function associated with a degree of freedom of a finite element. For nodal degrees of freedom it is one at its own node and zero at the other nodes, allowing local interpolation from the nodal values. For a bilinear square finite element, the four products of linear coordinate functions provide the nodal shape functions.

A bilinear square finite element uses polynomials of degree at most one separately in each coordinate and takes values at the four corners as degrees of freedom. On the unit square its nodal shape functions are $(1-\xi )(1-\eta )$, $\xi (1-\eta )$, $\xi \eta$, and $(1-\xi )\eta$, in anticlockwise corner order. Each is a tensor product of one-dimensional linear shape functions. A function in this space need not be affine, since $\xi \eta$ has total degree two. On a physical square of side $h$, its Laplacian stiffness matrix has diagonal entries $2/3$, adjacent-corner entries $-1/6$ and opposite-corner entries $-1/3$. Its Robin boundary condition edge matrix has the two-node block $(h/6)\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)$ for a unit boundary coefficient.

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.