Degree 1 Guzmán–Neilan (first kind) on a tetrahedron

• $R$ is the reference tetrahedron. The following numbering of the subentities of the reference is used:

• $\mathcal{V}$ is spanned by: $\left(\begin{array}{c}{\displaystyle 1}\\ {\displaystyle 0}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 1}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle 1}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle x}\\ {\displaystyle 0}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle x}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle x}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle y}\\ {\displaystyle 0}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle y}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle y}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle z}\\ {\displaystyle 0}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle z}\\ {\displaystyle 0}\end{array}\right)$, $\left(\begin{array}{c}{\displaystyle 0}\\ {\displaystyle 0}\\ {\displaystyle z}\end{array}\right)$, $\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle \frac{\sqrt{3}z(60x^2+180xy+132xz-96x-330yz-430z^2+192z+9)}{180}}\\ {\displaystyle \frac{\sqrt{3}z(180xy-330xz+60y^2+132yz-96y-430z^2+192z+9)}{180}}\\ {\displaystyle \frac{\sqrt{3}z(-150xz-150yz-88z^2+96z+9)}{180}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{\sqrt{3}y(60x^2+132xy+180xz-96x-430y^2-330yz+192y+9)}{180}}\\ {\displaystyle \frac{\sqrt{3}y(-150xy-88y^2-150yz+96y+9)}{180}}\\ {\displaystyle \frac{\sqrt{3}y(-330xy+180xz-430y^2+132yz+192y+60z^2-96z+9)}{180}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{\sqrt{3}x(-88x^2-150xy-150xz+96x+9)}{180}}\\ {\displaystyle \frac{\sqrt{3}x(-430x^2+132xy-330xz+192x+60y^2+180yz-96y+9)}{180}}\\ {\displaystyle \frac{\sqrt{3}x(-430x^2-330xy+132xz+192x+180yz+60z^2-96z+9)}{180}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{\sqrt{3}(112x^3+54x^{2}y+54x^{2}z-156x^2-168x{y}^2-216xyz+156xy-168x{z}^2+156xz+3x-110y^3-270y^{2}z+252y^2-270y{z}^2+444yz-183y-110z^3+252z^2-183z+41)}{180}}\\ {\displaystyle \frac{\sqrt{3}(-110x^3-168x^{2}y-270x^{2}z+252x^2+54x{y}^2-216xyz+156xy-270x{z}^2+444xz-183x+112y^3+54y^{2}z-156y^2-168y{z}^2+156yz+3y-110z^3+252z^2-183z+41)}{180}}\\ {\displaystyle \frac{\sqrt{3}(-110x^3-270x^{2}y-168x^{2}z+252x^2-270x{y}^2-216xyz+444xy+54x{z}^2+156xz-183x-110y^3-168y^{2}z+252y^2+54y{z}^2+156yz-183y+112z^3-156z^2+3z+41)}{180}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}$, $\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle \frac{z(-90xy+216xz+12x-60y^2+105yz+60y-55z^2-54z-3)}{60}}\\ {\displaystyle \frac{z(-30y^2-219yz+42y+325z^2-69z-3)}{60}}\\ {\displaystyle \frac{z(75yz+z^2-27z-3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{y(216xy-90xz+12x-55y^2+105yz-54y-60z^2+60z-3)}{60}}\\ {\displaystyle \frac{y(y^2+75yz-27y-3)}{60}}\\ {\displaystyle \frac{y(325y^2-219yz-69y-30z^2+42z-3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{43x^3}{30}+\frac{5x^{2}y}{4}+\frac{5x^{2}z}{4}-\frac{7x^2}{10}-\frac{x}{20}-y^{2}z-y{z}^2+yz}\\ {\displaystyle \frac{x(160x^2-129xy+165xz-69x-30y^2-90yz+42y-3)}{60}}\\ {\displaystyle \frac{x(160x^2+165xy-129xz-69x-90yz-30z^2+42z-3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{203x^3}{30}+\frac{369x^{2}y}{20}+\frac{369x^{2}z}{20}-\frac{173x^2}{10}+\frac{161x{y}^2}{10}+\frac{151xyz}{5}-\frac{151xy}{5}+\frac{161x{z}^2}{10}-\frac{151xz}{5}+\frac{283x}{20}+\frac{53y^3}{12}+\frac{45y^{2}z}{4}-\frac{62y^2}{5}+\frac{45y{z}^2}{4}-\frac{114yz}{5}+\frac{58y}{5}+\frac{53z^3}{12}-\frac{62z^2}{5}+\frac{58z}{5}-\frac{217}{60}}\\ {\displaystyle -\frac{8x^3}{3}-\frac{203x^{2}y}{20}-\frac{21x^{2}z}{4}+\frac{137x^2}{20}-\frac{59x{y}^2}{5}-\frac{133xyz}{10}+\frac{173xy}{10}-\frac{5x{z}^2}{2}+\frac{41xz}{5}-\frac{113x}{20}-\frac{259y^3}{60}-\frac{151y^{2}z}{20}+\frac{199y^2}{20}-\frac{63y{z}^2}{20}+\frac{103yz}{10}-\frac{71y}{10}+\frac{z^3}{12}+\frac{27z^2}{20}-\frac{29z}{10}+\frac{22}{15}}\\ {\displaystyle -\frac{8x^3}{3}-\frac{21x^{2}y}{4}-\frac{203x^{2}z}{20}+\frac{137x^2}{20}-\frac{5x{y}^2}{2}-\frac{133xyz}{10}+\frac{41xy}{5}-\frac{59x{z}^2}{5}+\frac{173xz}{10}-\frac{113x}{20}+\frac{y^3}{12}-\frac{63y^{2}z}{20}+\frac{27y^2}{20}-\frac{151y{z}^2}{20}+\frac{103yz}{10}-\frac{29y}{10}-\frac{259z^3}{60}+\frac{199z^2}{20}-\frac{71z}{10}+\frac{22}{15}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}$, $\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle \frac{z(30x^2+219xz-42x-325z^2+69z+3)}{60}}\\ {\displaystyle \frac{z(60x^2+90xy-105xz-60x-216yz-12y+55z^2+54z+3)}{60}}\\ {\displaystyle \frac{z(-75xz-z^2+27z+3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{y(30x^2+129xy+90xz-42x-160y^2-165yz+69y+3)}{60}}\\ {\displaystyle x^{2}z-\frac{5x{y}^2}{4}+x{z}^2-xz-\frac{43y^3}{30}-\frac{5y^{2}z}{4}+\frac{7y^2}{10}+\frac{y}{20}}\\ {\displaystyle \frac{y(-165xy+90xz-160y^2+129yz+69y+30z^2-42z+3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{x(-x^2-75xz+27x+3)}{60}}\\ {\displaystyle \frac{x(55x^2-216xy-105xz+54x+90yz-12y+60z^2-60z+3)}{60}}\\ {\displaystyle \frac{x(-325x^2+219xz+69x+30z^2-42z+3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{259x^3}{60}+\frac{59x^{2}y}{5}+\frac{151x^{2}z}{20}-\frac{199x^2}{20}+\frac{203x{y}^2}{20}+\frac{133xyz}{10}-\frac{173xy}{10}+\frac{63x{z}^2}{20}-\frac{103xz}{10}+\frac{71x}{10}+\frac{8y^3}{3}+\frac{21y^{2}z}{4}-\frac{137y^2}{20}+\frac{5y{z}^2}{2}-\frac{41yz}{5}+\frac{113y}{20}-\frac{z^3}{12}-\frac{27z^2}{20}+\frac{29z}{10}-\frac{22}{15}}\\ {\displaystyle -\frac{53x^3}{12}-\frac{161x^{2}y}{10}-\frac{45x^{2}z}{4}+\frac{62x^2}{5}-\frac{369x{y}^2}{20}-\frac{151xyz}{5}+\frac{151xy}{5}-\frac{45x{z}^2}{4}+\frac{114xz}{5}-\frac{58x}{5}-\frac{203y^3}{30}-\frac{369y^{2}z}{20}+\frac{173y^2}{10}-\frac{161y{z}^2}{10}+\frac{151yz}{5}-\frac{283y}{20}-\frac{53z^3}{12}+\frac{62z^2}{5}-\frac{58z}{5}+\frac{217}{60}}\\ {\displaystyle -\frac{x^3}{12}+\frac{5x^{2}y}{2}+\frac{63x^{2}z}{20}-\frac{27x^2}{20}+\frac{21x{y}^2}{4}+\frac{133xyz}{10}-\frac{41xy}{5}+\frac{151x{z}^2}{20}-\frac{103xz}{10}+\frac{29x}{10}+\frac{8y^3}{3}+\frac{203y^{2}z}{20}-\frac{137y^2}{20}+\frac{59y{z}^2}{5}-\frac{173yz}{10}+\frac{113y}{20}+\frac{259z^3}{60}-\frac{199z^2}{20}+\frac{71z}{10}-\frac{22}{15}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}$, $\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle \frac{z(-30x^2-90xy-129xz+42x+165yz+160z^2-69z-3)}{60}}\\ {\displaystyle \frac{z(-90xy+165xz-30y^2-129yz+42y+160z^2-69z-3)}{60}}\\ {\displaystyle -x^{2}y-x{y}^2+xy+\frac{5x{z}^2}{4}+\frac{5y{z}^2}{4}+\frac{43z^3}{30}-\frac{7z^2}{10}-\frac{z}{20}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{y(-30x^2-219xy+42x+325y^2-69y-3)}{60}}\\ {\displaystyle \frac{y(75xy+y^2-27y-3)}{60}}\\ {\displaystyle \frac{y(-60x^2+105xy-90xz+60x-55y^2+216yz-54y+12z-3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{x(x^2+75xy-27x-3)}{60}}\\ {\displaystyle \frac{x(325x^2-219xy-69x-30y^2+42y-3)}{60}}\\ {\displaystyle \frac{x(-55x^2+105xy+216xz-54x-60y^2-90yz+60y+12z-3)}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle -\frac{259x^3}{60}-\frac{151x^{2}y}{20}-\frac{59x^{2}z}{5}+\frac{199x^2}{20}-\frac{63x{y}^2}{20}-\frac{133xyz}{10}+\frac{103xy}{10}-\frac{203x{z}^2}{20}+\frac{173xz}{10}-\frac{71x}{10}+\frac{y^3}{12}-\frac{5y^{2}z}{2}+\frac{27y^2}{20}-\frac{21y{z}^2}{4}+\frac{41yz}{5}-\frac{29y}{10}-\frac{8z^3}{3}+\frac{137z^2}{20}-\frac{113z}{20}+\frac{22}{15}}\\ {\displaystyle \frac{x^3}{12}-\frac{63x^{2}y}{20}-\frac{5x^{2}z}{2}+\frac{27x^2}{20}-\frac{151x{y}^2}{20}-\frac{133xyz}{10}+\frac{103xy}{10}-\frac{21x{z}^2}{4}+\frac{41xz}{5}-\frac{29x}{10}-\frac{259y^3}{60}-\frac{59y^{2}z}{5}+\frac{199y^2}{20}-\frac{203y{z}^2}{20}+\frac{173yz}{10}-\frac{71y}{10}-\frac{8z^3}{3}+\frac{137z^2}{20}-\frac{113z}{20}+\frac{22}{15}}\\ {\displaystyle \frac{53x^3}{12}+\frac{45x^{2}y}{4}+\frac{161x^{2}z}{10}-\frac{62x^2}{5}+\frac{45x{y}^2}{4}+\frac{151xyz}{5}-\frac{114xy}{5}+\frac{369x{z}^2}{20}-\frac{151xz}{5}+\frac{58x}{5}+\frac{53y^3}{12}+\frac{161y^{2}z}{10}-\frac{62y^2}{5}+\frac{369y{z}^2}{20}-\frac{151yz}{5}+\frac{58y}{5}+\frac{203z^3}{30}-\frac{173z^2}{10}+\frac{283z}{20}-\frac{217}{60}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}$
• $\mathcal{L}=\{l_0,...,l_{15}\}$
• Functionals and basis functions:

${\displaystyle l_0:\mathit{v}\mapsto \mathit{v}(0,0,0)\cdot \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)}$

${\displaystyle {\mathit{\varphi }}_0=\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle 30xyz-72x{z}^2-4xz-x+20y^{2}z-35y{z}^2-20yz-y+\frac{55z^3}{3}+18z^2+1}\\ {\displaystyle \frac{z(30y^2+219yz-42y-325z^2+69z+3)}{3}}\\ {\displaystyle \frac{z(-75yz-z^2+27z+3)}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle -72x{y}^2+30xyz-4xy-x+\frac{55y^3}{3}-35y^{2}z+18y^2+20y{z}^2-20yz-z+1}\\ {\displaystyle \frac{y(-y^2-75yz+27y+3)}{3}}\\ {\displaystyle \frac{y(-325y^2+219yz+69y+30z^2-42z+3)}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle -\frac{86x^3}{3}-25x^{2}y-25x^{2}z+14x^2+20y^{2}z+20y{z}^2-20yz-y-z+1}\\ {\displaystyle \frac{x(-160x^2+129xy-165xz+69x+30y^2+90yz-42y+3)}{3}}\\ {\displaystyle \frac{x(-160x^2-165xy+129xz+69x+90yz+30z^2-42z+3)}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle -\frac{406x^3}{3}-369x^{2}y-369x^{2}z+346x^2-322x{y}^2-604xyz+604xy-322x{z}^2+604xz-284x-\frac{265y^3}{3}-225y^{2}z+248y^2-225y{z}^2+456yz-233y-\frac{265z^3}{3}+248z^2-233z+\frac{220}{3}}\\ {\displaystyle \frac{160x^3}{3}+203x^{2}y+105x^{2}z-137x^2+236x{y}^2+266xyz-346xy+50x{z}^2-164xz+113x+\frac{259y^3}{3}+151y^{2}z-199y^2+63y{z}^2-206yz+142y-\frac{5z^3}{3}-27z^2+58z-\frac{88}{3}}\\ {\displaystyle \frac{160x^3}{3}+105x^{2}y+203x^{2}z-137x^2+50x{y}^2+266xyz-164xy+236x{z}^2-346xz+113x-\frac{5y^3}{3}+63y^{2}z-27y^2+151y{z}^2-206yz+58y+\frac{259z^3}{3}-199z^2+142z-\frac{88}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}}$

This DOF is associated with vertex 0 of the reference element.

${\displaystyle l_1:\mathit{v}\mapsto \mathit{v}(0,0,0)\cdot \left(\begin{array}{c}0\\ 1\\ 0\end{array}\right)}$

${\displaystyle {\mathit{\varphi }}_1=\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle \frac{z(30x^2+219xz-42x-325z^2+69z+3)}{3}}\\ {\displaystyle 20x^{2}z+30xyz-35x{z}^2-20xz-x-72y{z}^2-4yz-y+\frac{55z^3}{3}+18z^2+1}\\ {\displaystyle \frac{z(-75xz-z^2+27z+3)}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{y(30x^2+129xy+90xz-42x-160y^2-165yz+69y+3)}{3}}\\ {\displaystyle 20x^{2}z-25x{y}^2+20x{z}^2-20xz-x-\frac{86y^3}{3}-25y^{2}z+14y^2-z+1}\\ {\displaystyle \frac{y(-165xy+90xz-160y^2+129yz+69y+30z^2-42z+3)}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{x(-x^2-75xz+27x+3)}{3}}\\ {\displaystyle \frac{55x^3}{3}-72x^{2}y-35x^{2}z+18x^2+30xyz-4xy+20x{z}^2-20xz-y-z+1}\\ {\displaystyle \frac{x(-325x^2+219xz+69x+30z^2-42z+3)}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{259x^3}{3}+236x^{2}y+151x^{2}z-199x^2+203x{y}^2+266xyz-346xy+63x{z}^2-206xz+142x+\frac{160y^3}{3}+105y^{2}z-137y^2+50y{z}^2-164yz+113y-\frac{5z^3}{3}-27z^2+58z-\frac{88}{3}}\\ {\displaystyle -\frac{265x^3}{3}-322x^{2}y-225x^{2}z+248x^2-369x{y}^2-604xyz+604xy-225x{z}^2+456xz-233x-\frac{406y^3}{3}-369y^{2}z+346y^2-322y{z}^2+604yz-284y-\frac{265z^3}{3}+248z^2-233z+\frac{220}{3}}\\ {\displaystyle -\frac{5x^3}{3}+50x^{2}y+63x^{2}z-27x^2+105x{y}^2+266xyz-164xy+151x{z}^2-206xz+58x+\frac{160y^3}{3}+203y^{2}z-137y^2+236y{z}^2-346yz+113y+\frac{259z^3}{3}-199z^2+142z-\frac{88}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}}$

This DOF is associated with vertex 0 of the reference element.

${\displaystyle l_2:\mathit{v}\mapsto \mathit{v}(0,0,0)\cdot \left(\begin{array}{c}0\\ 0\\ 1\end{array}\right)}$

${\displaystyle {\mathit{\varphi }}_2=\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle \frac{z(30x^2+90xy+129xz-42x-165yz-160z^2+69z+3)}{3}}\\ {\displaystyle \frac{z(90xy-165xz+30y^2+129yz-42y-160z^2+69z+3)}{3}}\\ {\displaystyle 20x^{2}y+20x{y}^2-20xy-25x{z}^2-x-25y{z}^2-y-\frac{86z^3}{3}+14z^2+1}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{y(30x^2+219xy-42x-325y^2+69y+3)}{3}}\\ {\displaystyle \frac{y(-75xy-y^2+27y+3)}{3}}\\ {\displaystyle 20x^{2}y-35x{y}^2+30xyz-20xy-x+\frac{55y^3}{3}-72y^{2}z+18y^2-4yz-z+1}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{x(-x^2-75xy+27x+3)}{3}}\\ {\displaystyle \frac{x(-325x^2+219xy+69x+30y^2-42y+3)}{3}}\\ {\displaystyle \frac{55x^3}{3}-35x^{2}y-72x^{2}z+18x^2+20x{y}^2+30xyz-20xy-4xz-y-z+1}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{259x^3}{3}+151x^{2}y+236x^{2}z-199x^2+63x{y}^2+266xyz-206xy+203x{z}^2-346xz+142x-\frac{5y^3}{3}+50y^{2}z-27y^2+105y{z}^2-164yz+58y+\frac{160z^3}{3}-137z^2+113z-\frac{88}{3}}\\ {\displaystyle -\frac{5x^3}{3}+63x^{2}y+50x^{2}z-27x^2+151x{y}^2+266xyz-206xy+105x{z}^2-164xz+58x+\frac{259y^3}{3}+236y^{2}z-199y^2+203y{z}^2-346yz+142y+\frac{160z^3}{3}-137z^2+113z-\frac{88}{3}}\\ {\displaystyle -\frac{265x^3}{3}-225x^{2}y-322x^{2}z+248x^2-225x{y}^2-604xyz+456xy-369x{z}^2+604xz-233x-\frac{265y^3}{3}-322y^{2}z+248y^2-369y{z}^2+604yz-233y-\frac{406z^3}{3}+346z^2-284z+\frac{220}{3}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}}$

This DOF is associated with vertex 0 of the reference element.

${\displaystyle l_3:\mathit{v}\mapsto \mathit{v}(1,0,0)\cdot \left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)}$

${\displaystyle {\mathit{\varphi }}_3=\{\begin{array}{ll}\left(\begin{array}{c}{\displaystyle -\frac{20x^{2}z}{3}-20xyz-\frac{44x{z}^2}{3}+\frac{32xz}{3}+x+\frac{110y{z}^2}{3}+\frac{430z^3}{9}-\frac{64z^2}{3}-z}\\ {\displaystyle \frac{z(-180xy+330xz-60y^2-132yz+96y+430z^2-192z-9)}{9}}\\ {\displaystyle \frac{z(150xz+150yz+88z^2-96z-9)}{9}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,1,0),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle -\frac{20x^{2}y}{3}-\frac{44x{y}^2}{3}-20xyz+\frac{32xy}{3}+x+\frac{430y^3}{9}+\frac{110y^{2}z}{3}-\frac{64y^2}{3}-y}\\ {\displaystyle \frac{y(150xy+88y^2+150yz-96y-9)}{9}}\\ {\displaystyle \frac{y(330xy-180xz+430y^2-132yz-192y-60z^2+96z-9)}{9}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(1,0,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle \frac{2x^2(44x+75y+75z-48)}{9}}\\ {\displaystyle \frac{x(430x^2-132xy+330xz-192x-60y^2-180yz+96y-9)}{9}}\\ {\displaystyle \frac{x(430x^2+330xy-132xz-192x-180yz-60z^2+96z-9)}{9}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((0,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\\ \left(\begin{array}{c}{\displaystyle -\frac{112x^3}{9}-6x^{2}y-6x^{2}z+\frac{52x^2}{3}+\frac{56x{y}^2}{3}+24xyz-\frac{52xy}{3}+\frac{56x{z}^2}{3}-\frac{52xz}{3}+\frac{2x}{3}+\frac{110y^3}{9}+30y^{2}z-28y^2+30y{z}^2-\frac{148yz}{3}+\frac{61y}{3}+\frac{110z^3}{9}-28z^2+\frac{61z}{3}-\frac{41}{9}}\\ {\displaystyle \frac{110x^3}{9}+\frac{56x^{2}y}{3}+30x^{2}z-28x^2-6x{y}^2+24xyz-\frac{52xy}{3}+30x{z}^2-\frac{148xz}{3}+\frac{61x}{3}-\frac{112y^3}{9}-6y^{2}z+\frac{52y^2}{3}+\frac{56y{z}^2}{3}-\frac{52yz}{3}-\frac{y}{3}+\frac{110z^3}{9}-28z^2+\frac{61z}{3}-\frac{41}{9}}\\ {\displaystyle \frac{110x^3}{9}+30x^{2}y+\frac{56x^{2}z}{3}-28x^2+30x{y}^2+24xyz-\frac{148xy}{3}-6x{z}^2-\frac{52xz}{3}+\frac{61x}{3}+\frac{110y^3}{9}+\frac{56y^{2}z}{3}-28y^2-6y{z}^2-\frac{52yz}{3}+\frac{61y}{3}-\frac{112z^3}{9}+\frac{52z^2}{3}-\frac{z}{3}-\frac{41}{9}}\end{array}\right)& \text{in }\mathrm{Tetrahedron}(((1,0,0),(0,1,0),(0,0,1),(1/4,1/4,1/4)))\end{array}}$

This DOF is associated with vertex 1 of the reference element.