= Bilinear square finite element
{title2=$Q_1=\operatorname{span}\{1,x,y,xy\}$}
= Q1 square element
{c}
{synonym}
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 $(h/6)\begin{pmatrix}2&1\\1&2\end{pmatrix}$ for a unit boundary coefficient.
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