Local and global dimensions of the mini element

On one $d$-simplex, $P_1(K)$ has the $d+1$ barycentric coordinate basis. The bubble $b_K$ is nonzero in the interior and vanishes at every vertex; because its polynomial degree is $d+1>1$, it is not in $P_1(K)$. Thus the scalar local velocity sum is direct with dimension $d+2$, and its $d$ vector components give $d(d+2)$. The local pressure dimension is $d+1$.

A continuous piecewise P1 scalar function is uniquely determined by its $N_v$ vertex values: affine uniqueness on each simplex proves injectivity, and assigning shared vertex values defines such a function by face-to-face conformity. If $w\in S_h^1\cap H_0^1(D)$, choose $w_n\in C_c^{\mathrm{\infty }}(D)$ with $w_n\to w$ in $H^1(D)$ by [F3]. On each boundary cell, affine pullback gives convergence in $H^1$ on the reference simplex; the reference traces of $w_n$ vanish there, so continuity of the trace gives zero trace for the pullback of $w$. Since that pullback is affine, its trace is its classical restriction, and the nodal values of $w$ vanish on every boundary facet. Conversely, if all boundary vertex values vanish, the affine trace of $w$ vanishes on each boundary facet, so [F4] gives $w\in H_0^1(D)$. Hence exactly the $N_v-N_{v,0}$ boundary nodal values are forced to zero, leaving $N_{v,0}$ independent velocity nodal values per component.

Each cell bubble is linearly independent of the nodal functions because its support interior is disjoint from every other cell bubble and it vanishes on the cell boundary, while a P1 polynomial vanishing on all vertices is zero. Hence the velocity dimension is $d{N}_{v,0}+d{N}_K$. The integral functional on $S_h^1$ is nonzero (it takes the positive value $\mathrm{\mid }D\mathrm{\mid }$ on $1$), so its zero-mean kernel has codimension one; therefore $\dim Q_h=N_v-1$.