Degree 1 Guzmán–Neilan on a tetrahedron

• \(R\) is the reference tetrahedron. The following numbering of the subentities of the reference is used:

• \(\mathcal{V}\) is spanned by: \(\begin{cases} \left(\begin{array}{c}\displaystyle - x - y - 2 z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - x - 2 y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 2 x - y - z + 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - y - 2 z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x - 2 y - z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 2 x - y - z + 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x - y - 2 z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x - 2 y - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 2 x - y - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle x - z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x - y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 2 x + y + z - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle x - z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x - y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 2 x + y + z - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x - z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x - y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 2 x + y + z - 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle y - z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - x + y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x + 2 y + z - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle y - z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x + y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x + 2 y + z - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle y - z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x + y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x + 2 y + z - 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - y + z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - x + z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x + y + 2 z - 1\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - y + z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - x + z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle x + y + 2 z - 1\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - y + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - x + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle x + y + 2 z - 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 4 x - 4 y - 4 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x - 4 y - 4 z + 4\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x - 4 y - 4 z + 4\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 183750 x^{2} - 143936 x y - 222439 x z + 174750 x + 221968 y z + 428657 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \tfrac{z \left(- 163564 x y + 231782 x z - 183750 y^{2} - 202811 y z + 174750 y + 418843 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \tfrac{2 z \left(132766 x z + 127859 y z + 70875 z^{2} - 87375 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 183750 x^{2} - 109061 x y - 257314 x z + 174750 x + 371968 y^{2} + 278657 y z - 208875 y + 2250\right)}{375}\\\displaystyle \tfrac{y \left(68218 x y + 1125 y^{2} + 31157 y z - 34125 y + 2250\right)}{375}\\\displaystyle \tfrac{2 y \left(17234 x y + 115532 x z + 97141 y^{2} + 52843 y z - 34125 y + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(1125 x^{2} + 78032 x y + 21343 x z - 34125 x + 2250\right)}{375}\\\displaystyle \tfrac{x \left(381782 x^{2} - 128689 x y + 268843 x z - 208875 x - 183750 y^{2} - 237686 y z + 174750 y + 2250\right)}{375}\\\displaystyle \tfrac{2 x \left(92234 x^{2} + 22141 x y + 62657 x z - 34125 x + 105718 y z + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 263 x^{3} + \tfrac{91468 x^{2} y}{125} + \tfrac{148157 x^{2} z}{125} - 699 x^{2} + \tfrac{277747 x y^{2}}{375} + 2166 x y z - \tfrac{515122 x y}{375} + \tfrac{504503 x z^{2}}{375} - \tfrac{741878 x z}{375} + 627 x + \tfrac{101968 y^{3}}{375} + \tfrac{377593 y^{2} z}{375} - \tfrac{277811 y^{2}}{375} + \tfrac{434282 y z^{2}}{375} - 1824 y z + \tfrac{247468 y}{375} + \tfrac{158657 z^{3}}{375} - \tfrac{391189 z^{2}}{375} + \tfrac{304157 z}{375} - 191\\\displaystyle \tfrac{111782 x^{3}}{375} + \tfrac{317003 x^{2} y}{375} + \tfrac{387407 x^{2} z}{375} - \tfrac{297439 x^{2}}{375} + \tfrac{101282 x y^{2}}{125} + 2166 x y z - \tfrac{554378 x y}{375} + \tfrac{424468 x z^{2}}{375} - 1824 x z + \tfrac{257282 x}{375} + 263 y^{3} + \tfrac{138343 y^{2} z}{125} - 699 y^{2} + \tfrac{465247 y z^{2}}{375} - \tfrac{702622 y z}{375} + 627 y + \tfrac{148843 z^{3}}{375} - \tfrac{371561 z^{2}}{375} + \tfrac{294343 z}{375} - 191\\\displaystyle - \tfrac{85532 x^{3}}{375} - \tfrac{231782 x^{2} y}{375} - \tfrac{612878 x^{2} z}{375} + \tfrac{237814 x^{2}}{375} - \tfrac{221968 x y^{2}}{375} - 3084 x y z + 1176 x y - \tfrac{283532 x z^{2}}{125} + \tfrac{1078628 x z}{375} - \tfrac{221282 x}{375} - \tfrac{75718 y^{3}}{375} - \tfrac{573622 y^{2} z}{375} + \tfrac{218186 y^{2}}{375} - \tfrac{273718 y z^{2}}{125} + \tfrac{1039372 y z}{375} - \tfrac{211468 y}{375} - 862 z^{3} + 1926 z^{2} - 1248 z + 184\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 20250 x^{2} - 59434 x y - 39791 x z + 22050 x - 9000 y^{2} + 16217 y z + 9000 y + 50158 z^{2} - 13725 z - 450\right)}{75}\\\displaystyle \tfrac{z \left(68434 x y - 7217 x z + 38250 y^{2} + 37091 y z - 36450 y - 45658 z^{2} + 15525 z - 450\right)}{75}\\\displaystyle \tfrac{z \left(- 13967 x z - 8533 y z + 900 z^{2} + 7200 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 20250 x^{2} - 133909 x y + 34684 x z + 22050 x + 97217 y^{2} - 30842 y z - 13725 y - 9000 z^{2} + 9000 z - 450\right)}{75}\\\displaystyle \tfrac{y \left(61217 x y + 29025 y^{2} + 658 y z - 20925 y - 450\right)}{75}\\\displaystyle \tfrac{y \left(67967 x y - 81934 x z - 36467 y^{2} + 46834 y z - 12600 y - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 303 x^{3} - \tfrac{34217 x^{2} y}{75} + \tfrac{12842 x^{2} z}{75} + 111 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z\\\displaystyle \tfrac{x \left(- 43217 x^{2} + 95209 x y - 9658 x z + 15525 x + 38250 y^{2} + 10316 y z - 36450 y - 450\right)}{75}\\\displaystyle \tfrac{x \left(31967 x^{2} - 467 x y - 27034 x z - 12600 x - 8066 y z - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1923 x^{3} - \tfrac{107683 x^{2} y}{25} - \tfrac{154742 x^{2} z}{25} + 4719 x^{2} - \tfrac{243607 x y^{2}}{75} - 9246 x y z + \tfrac{520582 x y}{75} - \tfrac{431843 x z^{2}}{75} + \tfrac{708818 x z}{75} - 3687 x - \tfrac{64783 y^{3}}{75} - \tfrac{259408 y^{2} z}{75} + \tfrac{196841 y^{2}}{75} - \tfrac{306467 y z^{2}}{75} + 6744 y z - \tfrac{198883 y}{75} - \tfrac{111842 z^{3}}{75} + \tfrac{290959 z^{2}}{75} - \tfrac{245942 z}{75} + 891\\\displaystyle \tfrac{28783 x^{3}}{75} + \tfrac{51907 x^{2} y}{75} + \tfrac{83908 x^{2} z}{75} - \tfrac{78041 x^{2}}{75} + \tfrac{15883 x y^{2}}{25} + 2094 x y z - \tfrac{119182 x y}{75} + \tfrac{81467 x z^{2}}{75} - 2016 x z + \tfrac{70183 x}{75} + 327 y^{3} + \tfrac{34442 y^{2} z}{25} - 951 y^{2} + \tfrac{105143 y z^{2}}{75} - \tfrac{172418 y z}{75} + 903 y + \tfrac{26342 z^{3}}{75} - \tfrac{73159 z^{2}}{75} + \tfrac{67742 z}{75} - 279\\\displaystyle \tfrac{103967 x^{3}}{75} + \tfrac{243467 x^{2} y}{75} + \tfrac{380768 x^{2} z}{75} - \tfrac{256534 x^{2}}{75} + \tfrac{175033 x y^{2}}{75} + 7344 x y z - 5016 x y + \tfrac{128567 x z^{2}}{25} - \tfrac{588668 x z}{75} + \tfrac{201617 x}{75} + \tfrac{35533 y^{3}}{75} + \tfrac{170032 y^{2} z}{75} - \tfrac{119666 y^{2}}{75} + \tfrac{81133 y z^{2}}{25} - \tfrac{377932 y z}{75} + \tfrac{133183 y}{75} + 1452 z^{3} - 3576 z^{2} + 2778 z - 654\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle 6 z \left(40 x^{2} + 72 x y + 6 x z - 44 x - 96 y z + 26 z^{2} + 28 z + 1\right)\\\displaystyle \tfrac{2 z \left(22500 x^{2} - 24262 x y + 45881 x z - 22500 x - 10762 y^{2} + 118857 y z + 6262 y - 119738 z^{2} + 3619 z + 1125\right)}{375}\\\displaystyle \tfrac{2 z \left(- 32869 x z - 29738 y z - 41869 z^{2} + 21619 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 y \left(40 x^{2} - 50 x y + 128 x z - 44 x - 16 y^{2} - 54 y z + 28 y + 1\right)\\\displaystyle 120 x^{2} z - \tfrac{1762 x y^{2}}{375} + 120 x z^{2} - 120 x z + \tfrac{52238 y^{3}}{375} - \tfrac{75524 y^{2} z}{375} + \tfrac{19762 y^{2}}{375} + 6 y\\\displaystyle \tfrac{2 y \left(55369 x y - 88238 x z - 15262 y^{2} - 22107 y z - 8119 y - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 x \left(- 2 x^{2} - 24 x y + 74 x z - 16 x + 1\right)\\\displaystyle \tfrac{2 x \left(- 156619 x^{2} + 192619 x y + 82762 x z + 3619 x - 10762 y^{2} - 98024 y z + 6262 y + 22500 z^{2} - 22500 z + 1125\right)}{375}\\\displaystyle \tfrac{2 x \left(145369 x^{2} - 105262 x y - 185869 x z - 8119 x + 75524 y z - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1452 x^{3} - 4152 x^{2} y - 4236 x^{2} z + 3576 x^{2} - 3756 x y^{2} - 7344 x y z + 6528 x y - 3588 x z^{2} + 6360 x z - 2778 x - 1056 y^{3} - 2916 y^{2} z + 2760 y^{2} - 2664 y z^{2} + 5016 y z - 2358 y - 804 z^{3} + 2256 z^{2} - 2106 z + 654\\\displaystyle \tfrac{496762 x^{3}}{375} + \tfrac{394162 x^{2} y}{75} + \tfrac{1654048 x^{2} z}{375} - \tfrac{463762 x^{2}}{125} + \tfrac{2379334 x y^{2}}{375} + \tfrac{4326668 x y z}{375} - \tfrac{732524 x y}{75} + \tfrac{345562 x z^{2}}{75} - \tfrac{3020096 x z}{375} + \tfrac{430012 x}{125} + \tfrac{301762 y^{3}}{125} + \tfrac{520124 y^{2} z}{75} - \tfrac{2199334 y^{2}}{375} + \tfrac{755286 y z^{2}}{125} - \tfrac{3957668 y z}{375} + \tfrac{337912 y}{75} + \tfrac{570524 z^{3}}{375} - \tfrac{307762 z^{2}}{75} + \tfrac{1363798 z}{375} - \tfrac{395512}{375}\\\displaystyle - \tfrac{69262 x^{3}}{375} - \tfrac{529048 x^{2} y}{375} - \tfrac{67462 x^{2} z}{75} + \tfrac{100762 x^{2}}{125} - \tfrac{170062 x y^{2}}{75} - \tfrac{1644668 x y z}{375} + \tfrac{1247096 x y}{375} - \tfrac{574834 x z^{2}}{375} + \tfrac{196124 x z}{75} - \tfrac{133012 x}{125} - \tfrac{390524 y^{3}}{375} - \tfrac{435786 y^{2} z}{125} + \tfrac{188962 y^{2}}{75} - \tfrac{244724 y z^{2}}{75} + \tfrac{1950668 y z}{375} - \tfrac{720298 y}{375} - \tfrac{102262 z^{3}}{125} + \tfrac{786334 z^{2}}{375} - \tfrac{129112 z}{75} + \tfrac{166012}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\), \(\begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(343124 x^{2} + 421874 x y + 485436 x z - 334124 x - 233437 y z - 258749 z^{2} + 153562 z - 2250\right)}{375}\\\displaystyle 3 z \left(- 230 x y + 95 x z - 80 y^{2} + 74 y z + 88 y + 45 z^{2} - 56 z - 2\right)\\\displaystyle - 120 x^{2} y - 120 x y^{2} + 120 x y - \tfrac{213749 x z^{2}}{375} - \tfrac{120937 y z^{2}}{375} - \tfrac{189562 z^{3}}{375} + \tfrac{117562 z^{2}}{375} - 6 z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(343124 x^{2} + 479812 x y + 427498 x z - 334124 x - 413437 y^{2} - 78749 y z + 153562 y - 2250\right)}{375}\\\displaystyle 3 y \left(- 135 x y + 4 y^{2} + 35 y z + 32 y - 2\right)\\\displaystyle \tfrac{y \left(- 45000 x^{2} + 123749 x y - 382498 x z + 45000 x + 435937 y^{2} - 493312 y z - 144562 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(221062 x^{2} + 188437 x y + 348749 x z - 180562 x - 2250\right)}{375}\\\displaystyle 3 x \left(- 65 x^{2} + 14 x y + 205 x z - 56 x - 80 y^{2} - 170 y z + 88 y - 2\right)\\\displaystyle \tfrac{x \left(528749 x^{2} + 30937 x y - 678936 x z - 144562 x - 45000 y^{2} - 196874 y z + 45000 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{35062 x^{3}}{125} - \tfrac{342559 x^{2} y}{375} - \tfrac{38699 x^{2} z}{75} + \tfrac{174934 x^{2}}{375} - \tfrac{58162 x y^{2}}{75} - \tfrac{277868 x y z}{375} + \tfrac{66824 x y}{75} + \tfrac{4314 x z^{2}}{125} + \tfrac{30368 x z}{375} - \tfrac{8212 x}{75} - \tfrac{53437 y^{3}}{375} - \tfrac{5623 y^{2} z}{375} + \tfrac{26812 y^{2}}{125} + \tfrac{29813 y z^{2}}{75} - \tfrac{148504 y z}{375} + \tfrac{563 y}{125} + \tfrac{101251 z^{3}}{375} - \tfrac{45788 z^{2}}{75} + \tfrac{156377 z}{375} - \tfrac{28688}{375}\\\displaystyle 765 x^{3} + 3432 x^{2} y + 2625 x^{2} z - 2178 x^{2} + 4119 x y^{2} + 7344 x y z - 6204 x y + 2955 x z^{2} - 5016 x z + 2067 x + 1452 y^{3} + 4269 y^{2} z - 3576 y^{2} + 3912 y z^{2} - 6684 y z + 2778 y + 1095 z^{3} - 2838 z^{2} + 2397 z - 654\\\displaystyle - \tfrac{281251 x^{3}}{375} - \tfrac{205313 x^{2} y}{75} - \tfrac{323814 x^{2} z}{125} + \tfrac{164588 x^{2}}{75} - \tfrac{1119377 x y^{2}}{375} - \tfrac{2404132 x y z}{375} + \tfrac{1921504 x y}{375} - \tfrac{236701 x z^{2}}{75} + \tfrac{1976632 x z}{375} - \tfrac{799877 x}{375} - \tfrac{374063 y^{3}}{375} - \tfrac{268538 y^{2} z}{75} + \tfrac{336188 y^{2}}{125} - \tfrac{1461941 y z^{2}}{375} + \tfrac{469576 y z}{75} - \tfrac{297563 y}{125} - \tfrac{164438 z^{3}}{125} + \tfrac{1238066 z^{2}}{375} - \tfrac{200588 z}{75} + \tfrac{258188}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)
• \(\mathcal{L}=\{l_0,...,l_{18}\}\)
• Functionals and basis functions:

\(\displaystyle l_{0}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{0} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{245 x^{2} z}{3} + \tfrac{71968 x y z}{1125} + \tfrac{222439 x z^{2}}{2250} - \tfrac{233 x z}{3} + x - \tfrac{110984 y z^{2}}{1125} - \tfrac{428657 z^{3}}{2250} + \tfrac{557 z^{2}}{6} - 2 z\\\displaystyle \tfrac{z \left(163564 x y - 231782 x z + 183750 y^{2} + 202811 y z - 174750 y - 418843 z^{2} + 208875 z - 2250\right)}{2250}\\\displaystyle \tfrac{z \left(- 132766 x z - 127859 y z - 70875 z^{2} + 87375 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{245 x^{2} y}{3} + \tfrac{109061 x y^{2}}{2250} + \tfrac{128657 x y z}{1125} - \tfrac{233 x y}{3} + x - \tfrac{185984 y^{3}}{1125} - \tfrac{278657 y^{2} z}{2250} + \tfrac{557 y^{2}}{6} - 2 y\\\displaystyle \tfrac{y \left(- 68218 x y - 1125 y^{2} - 31157 y z + 34125 y - 2250\right)}{2250}\\\displaystyle \tfrac{y \left(- 17234 x y - 115532 x z - 97141 y^{2} - 52843 y z + 34125 y - 48750 z^{2} + 53250 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 1125 x^{2} - 78032 x y - 21343 x z + 34125 x - 2250\right)}{2250}\\\displaystyle \tfrac{x \left(- 381782 x^{2} + 128689 x y - 268843 x z + 208875 x + 183750 y^{2} + 237686 y z - 174750 y - 2250\right)}{2250}\\\displaystyle \tfrac{x \left(- 92234 x^{2} - 22141 x y - 62657 x z + 34125 x - 105718 y z - 48750 z^{2} + 53250 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{233 x^{2}}{2} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{257561 x y}{1125} - \tfrac{504503 x z^{2}}{2250} + \tfrac{370939 x z}{1125} - \tfrac{205 x}{2} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277811 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + 304 y z - \tfrac{122609 y}{1125} - \tfrac{158657 z^{3}}{2250} + \tfrac{391189 z^{2}}{2250} - \tfrac{301907 z}{2250} + \tfrac{185}{6}\\\displaystyle - \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{297439 x^{2}}{2250} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{277189 x y}{1125} - \tfrac{212234 x z^{2}}{1125} + 304 x z - \tfrac{128641 x}{1125} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{233 y^{2}}{2} - \tfrac{465247 y z^{2}}{2250} + \tfrac{351311 y z}{1125} - \tfrac{209 y}{2} - \tfrac{148843 z^{3}}{2250} + \tfrac{371561 z^{2}}{2250} - \tfrac{294343 z}{2250} + \tfrac{191}{6}\\\displaystyle \tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{118907 x^{2}}{1125} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - 196 x y + \tfrac{141766 x z^{2}}{375} - \tfrac{539314 x z}{1125} + \tfrac{110641 x}{1125} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{109093 y^{2}}{1125} + \tfrac{136859 y z^{2}}{375} - \tfrac{519686 y z}{1125} + \tfrac{105734 y}{1125} + \tfrac{431 z^{3}}{3} - 321 z^{2} + 208 z - \tfrac{92}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{1}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{1} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(183750 x^{2} + 143936 x y + 222439 x z - 174750 x - 221968 y z - 428657 z^{2} + 208875 z - 2250\right)}{2250}\\\displaystyle \tfrac{81782 x y z}{1125} - \tfrac{115891 x z^{2}}{1125} + \tfrac{245 y^{2} z}{3} + \tfrac{202811 y z^{2}}{2250} - \tfrac{233 y z}{3} + y - \tfrac{418843 z^{3}}{2250} + \tfrac{557 z^{2}}{6} - 2 z\\\displaystyle \tfrac{z \left(- 132766 x z - 127859 y z - 70875 z^{2} + 87375 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(183750 x^{2} + 109061 x y + 257314 x z - 174750 x - 371968 y^{2} - 278657 y z + 208875 y - 2250\right)}{2250}\\\displaystyle \tfrac{y \left(- 68218 x y - 1125 y^{2} - 31157 y z + 34125 y - 2250\right)}{2250}\\\displaystyle \tfrac{y \left(- 17234 x y - 115532 x z - 97141 y^{2} - 52843 y z + 34125 y - 48750 z^{2} + 53250 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 1125 x^{2} - 78032 x y - 21343 x z + 34125 x - 2250\right)}{2250}\\\displaystyle - \tfrac{190891 x^{3}}{1125} + \tfrac{128689 x^{2} y}{2250} - \tfrac{268843 x^{2} z}{2250} + \tfrac{557 x^{2}}{6} + \tfrac{245 x y^{2}}{3} + \tfrac{118843 x y z}{1125} - \tfrac{233 x y}{3} - 2 x + y\\\displaystyle \tfrac{x \left(- 92234 x^{2} - 22141 x y - 62657 x z + 34125 x - 105718 y z - 48750 z^{2} + 53250 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{233 x^{2}}{2} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{257561 x y}{1125} - \tfrac{504503 x z^{2}}{2250} + \tfrac{370939 x z}{1125} - \tfrac{209 x}{2} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277811 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + 304 y z - \tfrac{123734 y}{1125} - \tfrac{158657 z^{3}}{2250} + \tfrac{391189 z^{2}}{2250} - \tfrac{304157 z}{2250} + \tfrac{191}{6}\\\displaystyle - \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{297439 x^{2}}{2250} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{277189 x y}{1125} - \tfrac{212234 x z^{2}}{1125} + 304 x z - \tfrac{127516 x}{1125} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{233 y^{2}}{2} - \tfrac{465247 y z^{2}}{2250} + \tfrac{351311 y z}{1125} - \tfrac{205 y}{2} - \tfrac{148843 z^{3}}{2250} + \tfrac{371561 z^{2}}{2250} - \tfrac{292093 z}{2250} + \tfrac{185}{6}\\\displaystyle \tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{118907 x^{2}}{1125} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - 196 x y + \tfrac{141766 x z^{2}}{375} - \tfrac{539314 x z}{1125} + \tfrac{110641 x}{1125} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{109093 y^{2}}{1125} + \tfrac{136859 y z^{2}}{375} - \tfrac{519686 y z}{1125} + \tfrac{105734 y}{1125} + \tfrac{431 z^{3}}{3} - 321 z^{2} + 208 z - \tfrac{92}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{2}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\1\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{2} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(183750 x^{2} + 143936 x y + 222439 x z - 174750 x - 221968 y z - 428657 z^{2} + 208875 z - 2250\right)}{2250}\\\displaystyle \tfrac{z \left(163564 x y - 231782 x z + 183750 y^{2} + 202811 y z - 174750 y - 418843 z^{2} + 208875 z - 2250\right)}{2250}\\\displaystyle \tfrac{z \left(- 132766 x z - 127859 y z - 70875 z^{2} + 87375 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(183750 x^{2} + 109061 x y + 257314 x z - 174750 x - 371968 y^{2} - 278657 y z + 208875 y - 2250\right)}{2250}\\\displaystyle \tfrac{y \left(- 68218 x y - 1125 y^{2} - 31157 y z + 34125 y - 2250\right)}{2250}\\\displaystyle - \tfrac{17234 x y^{2}}{1125} - \tfrac{115532 x y z}{1125} - \tfrac{97141 y^{3}}{1125} - \tfrac{52843 y^{2} z}{1125} + \tfrac{91 y^{2}}{3} - \tfrac{130 y z^{2}}{3} + \tfrac{142 y z}{3} - 2 y + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 1125 x^{2} - 78032 x y - 21343 x z + 34125 x - 2250\right)}{2250}\\\displaystyle \tfrac{x \left(- 381782 x^{2} + 128689 x y - 268843 x z + 208875 x + 183750 y^{2} + 237686 y z - 174750 y - 2250\right)}{2250}\\\displaystyle - \tfrac{92234 x^{3}}{1125} - \tfrac{22141 x^{2} y}{1125} - \tfrac{62657 x^{2} z}{1125} + \tfrac{91 x^{2}}{3} - \tfrac{105718 x y z}{1125} - \tfrac{130 x z^{2}}{3} + \tfrac{142 x z}{3} - 2 x + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{263 x^{3}}{6} - \tfrac{45734 x^{2} y}{375} - \tfrac{148157 x^{2} z}{750} + \tfrac{233 x^{2}}{2} - \tfrac{277747 x y^{2}}{2250} - 361 x y z + \tfrac{257561 x y}{1125} - \tfrac{504503 x z^{2}}{2250} + \tfrac{370939 x z}{1125} - \tfrac{209 x}{2} - \tfrac{50984 y^{3}}{1125} - \tfrac{377593 y^{2} z}{2250} + \tfrac{277811 y^{2}}{2250} - \tfrac{217141 y z^{2}}{1125} + 304 y z - \tfrac{123734 y}{1125} - \tfrac{158657 z^{3}}{2250} + \tfrac{391189 z^{2}}{2250} - \tfrac{304157 z}{2250} + \tfrac{191}{6}\\\displaystyle - \tfrac{55891 x^{3}}{1125} - \tfrac{317003 x^{2} y}{2250} - \tfrac{387407 x^{2} z}{2250} + \tfrac{297439 x^{2}}{2250} - \tfrac{50641 x y^{2}}{375} - 361 x y z + \tfrac{277189 x y}{1125} - \tfrac{212234 x z^{2}}{1125} + 304 x z - \tfrac{128641 x}{1125} - \tfrac{263 y^{3}}{6} - \tfrac{138343 y^{2} z}{750} + \tfrac{233 y^{2}}{2} - \tfrac{465247 y z^{2}}{2250} + \tfrac{351311 y z}{1125} - \tfrac{209 y}{2} - \tfrac{148843 z^{3}}{2250} + \tfrac{371561 z^{2}}{2250} - \tfrac{294343 z}{2250} + \tfrac{191}{6}\\\displaystyle \tfrac{42766 x^{3}}{1125} + \tfrac{115891 x^{2} y}{1125} + \tfrac{306439 x^{2} z}{1125} - \tfrac{118907 x^{2}}{1125} + \tfrac{110984 x y^{2}}{1125} + 514 x y z - 196 x y + \tfrac{141766 x z^{2}}{375} - \tfrac{539314 x z}{1125} + \tfrac{111766 x}{1125} + \tfrac{37859 y^{3}}{1125} + \tfrac{286811 y^{2} z}{1125} - \tfrac{109093 y^{2}}{1125} + \tfrac{136859 y z^{2}}{375} - \tfrac{519686 y z}{1125} + \tfrac{106859 y}{1125} + \tfrac{431 z^{3}}{3} - 321 z^{2} + 210 z - \tfrac{95}{3}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{3}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{3} = \begin{cases} \left(\begin{array}{c}\displaystyle 45 x^{2} z + \tfrac{29717 x y z}{225} + \tfrac{39791 x z^{2}}{450} - 49 x z - x + 20 y^{2} z - \tfrac{16217 y z^{2}}{450} - 20 y z - y - \tfrac{25079 z^{3}}{225} + \tfrac{61 z^{2}}{2} - z + 1\\\displaystyle \tfrac{z \left(- 68434 x y + 7217 x z - 38250 y^{2} - 37091 y z + 36450 y + 45658 z^{2} - 15525 z + 450\right)}{450}\\\displaystyle \tfrac{z \left(13967 x z + 8533 y z - 900 z^{2} - 7200 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 45 x^{2} y + \tfrac{133909 x y^{2}}{450} - \tfrac{17342 x y z}{225} - 49 x y - x - \tfrac{97217 y^{3}}{450} + \tfrac{15421 y^{2} z}{225} + \tfrac{61 y^{2}}{2} + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle \tfrac{y \left(- 61217 x y - 29025 y^{2} - 658 y z + 20925 y + 450\right)}{450}\\\displaystyle \tfrac{y \left(- 67967 x y + 81934 x z + 36467 y^{2} - 46834 y z + 12600 y + 18000 z^{2} - 19800 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{37 x^{2}}{2} - x + 20 y^{2} z + 20 y z^{2} - 20 y z - y - z + 1\\\displaystyle \tfrac{x \left(43217 x^{2} - 95209 x y + 9658 x z - 15525 x - 38250 y^{2} - 10316 y z + 36450 y + 450\right)}{450}\\\displaystyle \tfrac{x \left(- 31967 x^{2} + 467 x y + 27034 x z + 12600 x + 8066 y z + 18000 z^{2} - 19800 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{1573 x^{2}}{2} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{260291 x y}{225} + \tfrac{431843 x z^{2}}{450} - \tfrac{354409 x z}{225} + \tfrac{1229 x}{2} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{196841 y^{2}}{450} + \tfrac{306467 y z^{2}}{450} - 1124 y z + \tfrac{198883 y}{450} + \tfrac{55921 z^{3}}{225} - \tfrac{290959 z^{2}}{450} + \tfrac{122971 z}{225} - \tfrac{297}{2}\\\displaystyle - \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{78041 x^{2}}{450} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{59591 x y}{225} - \tfrac{81467 x z^{2}}{450} + 336 x z - \tfrac{70183 x}{450} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{317 y^{2}}{2} - \tfrac{105143 y z^{2}}{450} + \tfrac{86209 y z}{225} - \tfrac{301 y}{2} - \tfrac{13171 z^{3}}{225} + \tfrac{73159 z^{2}}{450} - \tfrac{33871 z}{225} + \tfrac{93}{2}\\\displaystyle - \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{128267 x^{2}}{225} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + 836 x y - \tfrac{128567 x z^{2}}{150} + \tfrac{294334 x z}{225} - \tfrac{201617 x}{450} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{59833 y^{2}}{225} - \tfrac{81133 y z^{2}}{150} + \tfrac{188966 y z}{225} - \tfrac{133183 y}{450} - 242 z^{3} + 596 z^{2} - 463 z + 109\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{4}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{4} = \begin{cases} \left(\begin{array}{c}\displaystyle 45 x^{2} z + \tfrac{29717 x y z}{225} + \tfrac{39791 x z^{2}}{450} - 49 x z + 20 y^{2} z - \tfrac{16217 y z^{2}}{450} - 20 y z + y - \tfrac{25079 z^{3}}{225} + \tfrac{61 z^{2}}{2}\\\displaystyle - \tfrac{34217 x y z}{225} + \tfrac{7217 x z^{2}}{450} - 85 y^{2} z - \tfrac{37091 y z^{2}}{450} + 81 y z - y + \tfrac{22829 z^{3}}{225} - \tfrac{69 z^{2}}{2} + 2 z\\\displaystyle \tfrac{z \left(13967 x z + 8533 y z - 900 z^{2} - 7200 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(20250 x^{2} + 133909 x y - 34684 x z - 22050 x - 97217 y^{2} + 30842 y z + 13725 y + 9000 z^{2} - 9000 z + 450\right)}{450}\\\displaystyle \tfrac{y \left(- 61217 x y - 29025 y^{2} - 658 y z + 20925 y + 450\right)}{450}\\\displaystyle \tfrac{y \left(- 67967 x y + 81934 x z + 36467 y^{2} - 46834 y z + 12600 y + 18000 z^{2} - 19800 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{37 x^{2}}{2} + 20 y^{2} z + 20 y z^{2} - 20 y z + y\\\displaystyle \tfrac{43217 x^{3}}{450} - \tfrac{95209 x^{2} y}{450} + \tfrac{4829 x^{2} z}{225} - \tfrac{69 x^{2}}{2} - 85 x y^{2} - \tfrac{5158 x y z}{225} + 81 x y + 2 x - y\\\displaystyle \tfrac{x \left(- 31967 x^{2} + 467 x y + 27034 x z + 12600 x + 8066 y z + 18000 z^{2} - 19800 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{1573 x^{2}}{2} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{260291 x y}{225} + \tfrac{431843 x z^{2}}{450} - \tfrac{354409 x z}{225} + \tfrac{1231 x}{2} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{196841 y^{2}}{450} + \tfrac{306467 y z^{2}}{450} - 1124 y z + \tfrac{199783 y}{450} + \tfrac{55921 z^{3}}{225} - \tfrac{290959 z^{2}}{450} + \tfrac{123196 z}{225} - \tfrac{299}{2}\\\displaystyle - \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{78041 x^{2}}{450} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{59591 x y}{225} - \tfrac{81467 x z^{2}}{450} + 336 x z - \tfrac{70633 x}{450} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{317 y^{2}}{2} - \tfrac{105143 y z^{2}}{450} + \tfrac{86209 y z}{225} - \tfrac{305 y}{2} - \tfrac{13171 z^{3}}{225} + \tfrac{73159 z^{2}}{450} - \tfrac{34096 z}{225} + \tfrac{95}{2}\\\displaystyle - \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{128267 x^{2}}{225} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + 836 x y - \tfrac{128567 x z^{2}}{150} + \tfrac{294334 x z}{225} - \tfrac{201617 x}{450} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{59833 y^{2}}{225} - \tfrac{81133 y z^{2}}{150} + \tfrac{188966 y z}{225} - \tfrac{133183 y}{450} - 242 z^{3} + 596 z^{2} - 463 z + 109\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{5}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{5} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(20250 x^{2} + 59434 x y + 39791 x z - 22050 x + 9000 y^{2} - 16217 y z - 9000 y - 50158 z^{2} + 13725 z + 450\right)}{450}\\\displaystyle \tfrac{z \left(- 68434 x y + 7217 x z - 38250 y^{2} - 37091 y z + 36450 y + 45658 z^{2} - 15525 z + 450\right)}{450}\\\displaystyle \tfrac{z \left(13967 x z + 8533 y z - 900 z^{2} - 7200 z + 450\right)}{450}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 45 x^{2} y + \tfrac{133909 x y^{2}}{450} - \tfrac{17342 x y z}{225} - 49 x y - \tfrac{97217 y^{3}}{450} + \tfrac{15421 y^{2} z}{225} + \tfrac{61 y^{2}}{2} + 20 y z^{2} - 20 y z + z\\\displaystyle \tfrac{y \left(- 61217 x y - 29025 y^{2} - 658 y z + 20925 y + 450\right)}{450}\\\displaystyle - \tfrac{67967 x y^{2}}{450} + \tfrac{40967 x y z}{225} + \tfrac{36467 y^{3}}{450} - \tfrac{23417 y^{2} z}{225} + 28 y^{2} + 40 y z^{2} - 44 y z + 2 y - z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{101 x^{3}}{2} + \tfrac{34217 x^{2} y}{450} - \tfrac{6421 x^{2} z}{225} - \tfrac{37 x^{2}}{2} + 20 y^{2} z + 20 y z^{2} - 20 y z + z\\\displaystyle \tfrac{x \left(43217 x^{2} - 95209 x y + 9658 x z - 15525 x - 38250 y^{2} - 10316 y z + 36450 y + 450\right)}{450}\\\displaystyle - \tfrac{31967 x^{3}}{450} + \tfrac{467 x^{2} y}{450} + \tfrac{13517 x^{2} z}{225} + 28 x^{2} + \tfrac{4033 x y z}{225} + 40 x z^{2} - 44 x z + 2 x - z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{641 x^{3}}{2} + \tfrac{107683 x^{2} y}{150} + \tfrac{77371 x^{2} z}{75} - \tfrac{1573 x^{2}}{2} + \tfrac{243607 x y^{2}}{450} + 1541 x y z - \tfrac{260291 x y}{225} + \tfrac{431843 x z^{2}}{450} - \tfrac{354409 x z}{225} + \tfrac{1231 x}{2} + \tfrac{64783 y^{3}}{450} + \tfrac{129704 y^{2} z}{225} - \tfrac{196841 y^{2}}{450} + \tfrac{306467 y z^{2}}{450} - 1124 y z + \tfrac{199333 y}{450} + \tfrac{55921 z^{3}}{225} - \tfrac{290959 z^{2}}{450} + \tfrac{123421 z}{225} - \tfrac{299}{2}\\\displaystyle - \tfrac{28783 x^{3}}{450} - \tfrac{51907 x^{2} y}{450} - \tfrac{41954 x^{2} z}{225} + \tfrac{78041 x^{2}}{450} - \tfrac{15883 x y^{2}}{150} - 349 x y z + \tfrac{59591 x y}{225} - \tfrac{81467 x z^{2}}{450} + 336 x z - \tfrac{70183 x}{450} - \tfrac{109 y^{3}}{2} - \tfrac{17221 y^{2} z}{75} + \tfrac{317 y^{2}}{2} - \tfrac{105143 y z^{2}}{450} + \tfrac{86209 y z}{225} - \tfrac{301 y}{2} - \tfrac{13171 z^{3}}{225} + \tfrac{73159 z^{2}}{450} - \tfrac{33871 z}{225} + \tfrac{93}{2}\\\displaystyle - \tfrac{103967 x^{3}}{450} - \tfrac{243467 x^{2} y}{450} - \tfrac{190384 x^{2} z}{225} + \tfrac{128267 x^{2}}{225} - \tfrac{175033 x y^{2}}{450} - 1224 x y z + 836 x y - \tfrac{128567 x z^{2}}{150} + \tfrac{294334 x z}{225} - \tfrac{202067 x}{450} - \tfrac{35533 y^{3}}{450} - \tfrac{85016 y^{2} z}{225} + \tfrac{59833 y^{2}}{225} - \tfrac{81133 y z^{2}}{150} + \tfrac{188966 y z}{225} - \tfrac{133633 y}{450} - 242 z^{3} + 596 z^{2} - 465 z + 110\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{6}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{6} = \begin{cases} \left(\begin{array}{c}\displaystyle z \left(- 40 x^{2} - 72 x y - 6 x z + 44 x + 96 y z - 26 z^{2} - 28 z - 1\right)\\\displaystyle - 20 x^{2} z + \tfrac{24262 x y z}{1125} - \tfrac{45881 x z^{2}}{1125} + 20 x z + x + \tfrac{10762 y^{2} z}{1125} - \tfrac{39619 y z^{2}}{375} - \tfrac{6262 y z}{1125} + y + \tfrac{119738 z^{3}}{1125} - \tfrac{3619 z^{2}}{1125} + z - 1\\\displaystyle \tfrac{z \left(32869 x z + 29738 y z + 41869 z^{2} - 21619 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle y \left(- 40 x^{2} + 50 x y - 128 x z + 44 x + 16 y^{2} + 54 y z - 28 y - 1\right)\\\displaystyle - 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - 20 x z^{2} + 20 x z + x - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{9881 y^{2}}{1125} + y + z - 1\\\displaystyle \tfrac{y \left(- 55369 x y + 88238 x z + 15262 y^{2} + 22107 y z + 8119 y + 34238 z^{2} - 29738 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x \left(2 x^{2} + 24 x y - 74 x z + 16 x - 1\right)\\\displaystyle \tfrac{156619 x^{3}}{1125} - \tfrac{192619 x^{2} y}{1125} - \tfrac{82762 x^{2} z}{1125} - \tfrac{3619 x^{2}}{1125} + \tfrac{10762 x y^{2}}{1125} + \tfrac{98024 x y z}{1125} - \tfrac{6262 x y}{1125} - 20 x z^{2} + 20 x z + x + y + z - 1\\\displaystyle \tfrac{x \left(- 145369 x^{2} + 105262 x y + 185869 x z + 8119 x - 75524 y z + 34238 z^{2} - 29738 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 242 x^{3} + 692 x^{2} y + 706 x^{2} z - 596 x^{2} + 626 x y^{2} + 1224 x y z - 1088 x y + 598 x z^{2} - 1060 x z + 463 x + 176 y^{3} + 486 y^{2} z - 460 y^{2} + 444 y z^{2} - 836 y z + 393 y + 134 z^{3} - 376 z^{2} + 351 z - 109\\\displaystyle - \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{231881 x^{2}}{375} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{366262 x y}{225} - \tfrac{172781 x z^{2}}{225} + \tfrac{1510048 x z}{1125} - \tfrac{215006 x}{375} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{1099667 y^{2}}{1125} - \tfrac{125881 y z^{2}}{125} + \tfrac{1978834 y z}{1125} - \tfrac{168956 y}{225} - \tfrac{285262 z^{3}}{1125} + \tfrac{153881 z^{2}}{225} - \tfrac{681899 z}{1125} + \tfrac{197756}{1125}\\\displaystyle \tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{50381 x^{2}}{375} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{623548 x y}{1125} + \tfrac{287417 x z^{2}}{1125} - \tfrac{98062 x z}{225} + \tfrac{66506 x}{375} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{94481 y^{2}}{225} + \tfrac{122362 y z^{2}}{225} - \tfrac{975334 y z}{1125} + \tfrac{360149 y}{1125} + \tfrac{51131 z^{3}}{375} - \tfrac{393167 z^{2}}{1125} + \tfrac{64556 z}{225} - \tfrac{83006}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{7}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{7} = \begin{cases} \left(\begin{array}{c}\displaystyle - 40 x^{2} z - 72 x y z - 6 x z^{2} + 44 x z + x + 96 y z^{2} - 26 z^{3} - 28 z^{2} - 2 z\\\displaystyle - 20 x^{2} z + \tfrac{24262 x y z}{1125} - \tfrac{45881 x z^{2}}{1125} + 20 x z - x + \tfrac{10762 y^{2} z}{1125} - \tfrac{39619 y z^{2}}{375} - \tfrac{6262 y z}{1125} + \tfrac{119738 z^{3}}{1125} - \tfrac{3619 z^{2}}{1125}\\\displaystyle \tfrac{z \left(32869 x z + 29738 y z + 41869 z^{2} - 21619 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 40 x^{2} y + 50 x y^{2} - 128 x y z + 44 x y + x + 16 y^{3} + 54 y^{2} z - 28 y^{2} - 2 y\\\displaystyle - 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - 20 x z^{2} + 20 x z - x - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{9881 y^{2}}{1125}\\\displaystyle \tfrac{y \left(- 55369 x y + 88238 x z + 15262 y^{2} + 22107 y z + 8119 y + 34238 z^{2} - 29738 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x \left(2 x^{2} + 24 x y - 74 x z + 16 x - 1\right)\\\displaystyle \tfrac{x \left(156619 x^{2} - 192619 x y - 82762 x z - 3619 x + 10762 y^{2} + 98024 y z - 6262 y - 22500 z^{2} + 22500 z - 1125\right)}{1125}\\\displaystyle \tfrac{x \left(- 145369 x^{2} + 105262 x y + 185869 x z + 8119 x - 75524 y z + 34238 z^{2} - 29738 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 242 x^{3} + 692 x^{2} y + 706 x^{2} z - 596 x^{2} + 626 x y^{2} + 1224 x y z - 1088 x y + 598 x z^{2} - 1060 x z + 465 x + 176 y^{3} + 486 y^{2} z - 460 y^{2} + 444 y z^{2} - 836 y z + 394 y + 134 z^{3} - 376 z^{2} + 352 z - 110\\\displaystyle - \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{231881 x^{2}}{375} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{366262 x y}{225} - \tfrac{172781 x z^{2}}{225} + \tfrac{1510048 x z}{1125} - \tfrac{215756 x}{375} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{1099667 y^{2}}{1125} - \tfrac{125881 y z^{2}}{125} + \tfrac{1978834 y z}{1125} - \tfrac{169181 y}{225} - \tfrac{285262 z^{3}}{1125} + \tfrac{153881 z^{2}}{225} - \tfrac{683024 z}{1125} + \tfrac{198881}{1125}\\\displaystyle \tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{50381 x^{2}}{375} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{623548 x y}{1125} + \tfrac{287417 x z^{2}}{1125} - \tfrac{98062 x z}{225} + \tfrac{66506 x}{375} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{94481 y^{2}}{225} + \tfrac{122362 y z^{2}}{225} - \tfrac{975334 y z}{1125} + \tfrac{360149 y}{1125} + \tfrac{51131 z^{3}}{375} - \tfrac{393167 z^{2}}{1125} + \tfrac{64556 z}{225} - \tfrac{83006}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{8}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,1)\cdot\left(\begin{array}{c}0\\-1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{8} = \begin{cases} \left(\begin{array}{c}\displaystyle z \left(- 40 x^{2} - 72 x y - 6 x z + 44 x + 96 y z - 26 z^{2} - 28 z - 1\right)\\\displaystyle \tfrac{z \left(- 22500 x^{2} + 24262 x y - 45881 x z + 22500 x + 10762 y^{2} - 118857 y z - 6262 y + 119738 z^{2} - 3619 z - 1125\right)}{1125}\\\displaystyle \tfrac{z \left(32869 x z + 29738 y z + 41869 z^{2} - 21619 z - 1125\right)}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle y \left(- 40 x^{2} + 50 x y - 128 x z + 44 x + 16 y^{2} + 54 y z - 28 y - 1\right)\\\displaystyle - 20 x^{2} z + \tfrac{881 x y^{2}}{1125} - 20 x z^{2} + 20 x z - \tfrac{26119 y^{3}}{1125} + \tfrac{37762 y^{2} z}{1125} - \tfrac{9881 y^{2}}{1125} - z\\\displaystyle - \tfrac{55369 x y^{2}}{1125} + \tfrac{88238 x y z}{1125} + \tfrac{15262 y^{3}}{1125} + \tfrac{7369 y^{2} z}{375} + \tfrac{8119 y^{2}}{1125} + \tfrac{34238 y z^{2}}{1125} - \tfrac{29738 y z}{1125} - 2 y + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle x \left(2 x^{2} + 24 x y - 74 x z + 16 x - 1\right)\\\displaystyle \tfrac{156619 x^{3}}{1125} - \tfrac{192619 x^{2} y}{1125} - \tfrac{82762 x^{2} z}{1125} - \tfrac{3619 x^{2}}{1125} + \tfrac{10762 x y^{2}}{1125} + \tfrac{98024 x y z}{1125} - \tfrac{6262 x y}{1125} - 20 x z^{2} + 20 x z - z\\\displaystyle - \tfrac{145369 x^{3}}{1125} + \tfrac{105262 x^{2} y}{1125} + \tfrac{185869 x^{2} z}{1125} + \tfrac{8119 x^{2}}{1125} - \tfrac{75524 x y z}{1125} + \tfrac{34238 x z^{2}}{1125} - \tfrac{29738 x z}{1125} - 2 x + z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 242 x^{3} + 692 x^{2} y + 706 x^{2} z - 596 x^{2} + 626 x y^{2} + 1224 x y z - 1088 x y + 598 x z^{2} - 1060 x z + 463 x + 176 y^{3} + 486 y^{2} z - 460 y^{2} + 444 y z^{2} - 836 y z + 393 y + 134 z^{3} - 376 z^{2} + 351 z - 109\\\displaystyle - \tfrac{248381 x^{3}}{1125} - \tfrac{197081 x^{2} y}{225} - \tfrac{827024 x^{2} z}{1125} + \tfrac{231881 x^{2}}{375} - \tfrac{1189667 x y^{2}}{1125} - \tfrac{2163334 x y z}{1125} + \tfrac{366262 x y}{225} - \tfrac{172781 x z^{2}}{225} + \tfrac{1510048 x z}{1125} - \tfrac{215381 x}{375} - \tfrac{150881 y^{3}}{375} - \tfrac{260062 y^{2} z}{225} + \tfrac{1099667 y^{2}}{1125} - \tfrac{125881 y z^{2}}{125} + \tfrac{1978834 y z}{1125} - \tfrac{169181 y}{225} - \tfrac{285262 z^{3}}{1125} + \tfrac{153881 z^{2}}{225} - \tfrac{684149 z}{1125} + \tfrac{198881}{1125}\\\displaystyle \tfrac{34631 x^{3}}{1125} + \tfrac{264524 x^{2} y}{1125} + \tfrac{33731 x^{2} z}{225} - \tfrac{50381 x^{2}}{375} + \tfrac{85031 x y^{2}}{225} + \tfrac{822334 x y z}{1125} - \tfrac{623548 x y}{1125} + \tfrac{287417 x z^{2}}{1125} - \tfrac{98062 x z}{225} + \tfrac{66881 x}{375} + \tfrac{195262 y^{3}}{1125} + \tfrac{72631 y^{2} z}{125} - \tfrac{94481 y^{2}}{225} + \tfrac{122362 y z^{2}}{225} - \tfrac{975334 y z}{1125} + \tfrac{361274 y}{1125} + \tfrac{51131 z^{3}}{375} - \tfrac{393167 z^{2}}{1125} + \tfrac{65006 z}{225} - \tfrac{84131}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{9}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{9} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 343124 x^{2} - 421874 x y - 485436 x z + 334124 x + 233437 y z + 258749 z^{2} - 153562 z + 2250\right)}{2250}\\\displaystyle \tfrac{z \left(230 x y - 95 x z + 80 y^{2} - 74 y z - 88 y - 45 z^{2} + 56 z + 2\right)}{2}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y + \tfrac{213749 x z^{2}}{2250} - x + \tfrac{120937 y z^{2}}{2250} - y + \tfrac{94781 z^{3}}{1125} - \tfrac{58781 z^{2}}{1125} - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 343124 x^{2} - 479812 x y - 427498 x z + 334124 x + 413437 y^{2} + 78749 y z - 153562 y + 2250\right)}{2250}\\\displaystyle \tfrac{y \left(135 x y - 4 y^{2} - 35 y z - 32 y + 2\right)}{2}\\\displaystyle 20 x^{2} y - \tfrac{123749 x y^{2}}{2250} + \tfrac{191249 x y z}{1125} - 20 x y - x - \tfrac{435937 y^{3}}{2250} + \tfrac{246656 y^{2} z}{1125} + \tfrac{72281 y^{2}}{1125} + \tfrac{126562 y z^{2}}{1125} - \tfrac{131062 y z}{1125} - y - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 221062 x^{2} - 188437 x y - 348749 x z + 180562 x + 2250\right)}{2250}\\\displaystyle \tfrac{x \left(65 x^{2} - 14 x y - 205 x z + 56 x + 80 y^{2} + 170 y z - 88 y + 2\right)}{2}\\\displaystyle - \tfrac{528749 x^{3}}{2250} - \tfrac{30937 x^{2} y}{2250} + \tfrac{113156 x^{2} z}{375} + \tfrac{72281 x^{2}}{1125} + 20 x y^{2} + \tfrac{98437 x y z}{1125} - 20 x y + \tfrac{126562 x z^{2}}{1125} - \tfrac{131062 x z}{1125} - x - y - z + 1\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{87467 x^{2}}{1125} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{33412 x y}{225} - \tfrac{719 x z^{2}}{125} - \tfrac{15184 x z}{1125} + \tfrac{4106 x}{225} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{13406 y^{2}}{375} - \tfrac{29813 y z^{2}}{450} + \tfrac{74252 y z}{1125} - \tfrac{563 y}{750} - \tfrac{101251 z^{3}}{2250} + \tfrac{22894 z^{2}}{225} - \tfrac{156377 z}{2250} + \tfrac{14344}{1125}\\\displaystyle - \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + 363 x^{2} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + 1034 x y - \tfrac{985 x z^{2}}{2} + 836 x z - \tfrac{689 x}{2} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + 596 y^{2} - 652 y z^{2} + 1114 y z - 463 y - \tfrac{365 z^{3}}{2} + 473 z^{2} - \tfrac{799 z}{2} + 109\\\displaystyle \tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{82294 x^{2}}{225} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{960752 x y}{1125} + \tfrac{236701 x z^{2}}{450} - \tfrac{988316 x z}{1125} + \tfrac{799877 x}{2250} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{168094 y^{2}}{375} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{234788 y z}{225} + \tfrac{297563 y}{750} + \tfrac{82219 z^{3}}{375} - \tfrac{619033 z^{2}}{1125} + \tfrac{100294 z}{225} - \tfrac{129094}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 3 of the reference element.

\(\displaystyle l_{10}:\boldsymbol{v}\mapsto\boldsymbol{v}(1,0,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{10} = \begin{cases} \left(\begin{array}{c}\displaystyle - \tfrac{171562 x^{2} z}{1125} - \tfrac{210937 x y z}{1125} - \tfrac{80906 x z^{2}}{375} + \tfrac{167062 x z}{1125} - x + \tfrac{233437 y z^{2}}{2250} + \tfrac{258749 z^{3}}{2250} - \tfrac{76781 z^{2}}{1125} + 2 z\\\displaystyle \tfrac{z \left(230 x y - 95 x z + 80 y^{2} - 74 y z - 88 y - 45 z^{2} + 56 z + 2\right)}{2}\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y + \tfrac{213749 x z^{2}}{2250} + x + \tfrac{120937 y z^{2}}{2250} + \tfrac{94781 z^{3}}{1125} - \tfrac{58781 z^{2}}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{171562 x^{2} y}{1125} - \tfrac{239906 x y^{2}}{1125} - \tfrac{213749 x y z}{1125} + \tfrac{167062 x y}{1125} - x + \tfrac{413437 y^{3}}{2250} + \tfrac{78749 y^{2} z}{2250} - \tfrac{76781 y^{2}}{1125} + 2 y\\\displaystyle \tfrac{y \left(135 x y - 4 y^{2} - 35 y z - 32 y + 2\right)}{2}\\\displaystyle 20 x^{2} y - \tfrac{123749 x y^{2}}{2250} + \tfrac{191249 x y z}{1125} - 20 x y + x - \tfrac{435937 y^{3}}{2250} + \tfrac{246656 y^{2} z}{1125} + \tfrac{72281 y^{2}}{1125} + \tfrac{126562 y z^{2}}{1125} - \tfrac{131062 y z}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 221062 x^{2} - 188437 x y - 348749 x z + 180562 x + 2250\right)}{2250}\\\displaystyle \tfrac{x \left(65 x^{2} - 14 x y - 205 x z + 56 x + 80 y^{2} + 170 y z - 88 y + 2\right)}{2}\\\displaystyle \tfrac{x \left(- 528749 x^{2} - 30937 x y + 678936 x z + 144562 x + 45000 y^{2} + 196874 y z - 45000 y + 253124 z^{2} - 262124 z + 2250\right)}{2250}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{87467 x^{2}}{1125} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{33412 x y}{225} - \tfrac{719 x z^{2}}{125} - \tfrac{15184 x z}{1125} + \tfrac{3656 x}{225} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{13406 y^{2}}{375} - \tfrac{29813 y z^{2}}{450} + \tfrac{74252 y z}{1125} - \tfrac{1313 y}{750} - \tfrac{101251 z^{3}}{2250} + \tfrac{22894 z^{2}}{225} - \tfrac{158627 z}{2250} + \tfrac{15469}{1125}\\\displaystyle - \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + 363 x^{2} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + 1034 x y - \tfrac{985 x z^{2}}{2} + 836 x z - \tfrac{689 x}{2} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + 596 y^{2} - 652 y z^{2} + 1114 y z - 463 y - \tfrac{365 z^{3}}{2} + 473 z^{2} - \tfrac{799 z}{2} + 109\\\displaystyle \tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{82294 x^{2}}{225} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{960752 x y}{1125} + \tfrac{236701 x z^{2}}{450} - \tfrac{988316 x z}{1125} + \tfrac{804377 x}{2250} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{168094 y^{2}}{375} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{234788 y z}{225} + \tfrac{298313 y}{750} + \tfrac{82219 z^{3}}{375} - \tfrac{619033 z^{2}}{1125} + \tfrac{100519 z}{225} - \tfrac{130219}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 3 of the reference element.

\(\displaystyle l_{11}:\boldsymbol{v}\mapsto\boldsymbol{v}(0,1,0)\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{11} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 343124 x^{2} - 421874 x y - 485436 x z + 334124 x + 233437 y z + 258749 z^{2} - 153562 z + 2250\right)}{2250}\\\displaystyle 115 x y z - \tfrac{95 x z^{2}}{2} + 40 y^{2} z - 37 y z^{2} - 44 y z - y - \tfrac{45 z^{3}}{2} + 28 z^{2} + 2 z\\\displaystyle 20 x^{2} y + 20 x y^{2} - 20 x y + \tfrac{213749 x z^{2}}{2250} + \tfrac{120937 y z^{2}}{2250} + y + \tfrac{94781 z^{3}}{1125} - \tfrac{58781 z^{2}}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 343124 x^{2} - 479812 x y - 427498 x z + 334124 x + 413437 y^{2} + 78749 y z - 153562 y + 2250\right)}{2250}\\\displaystyle \tfrac{y \left(135 x y - 4 y^{2} - 35 y z - 32 y + 2\right)}{2}\\\displaystyle \tfrac{y \left(45000 x^{2} - 123749 x y + 382498 x z - 45000 x - 435937 y^{2} + 493312 y z + 144562 y + 253124 z^{2} - 262124 z + 2250\right)}{2250}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(- 221062 x^{2} - 188437 x y - 348749 x z + 180562 x + 2250\right)}{2250}\\\displaystyle \tfrac{65 x^{3}}{2} - 7 x^{2} y - \tfrac{205 x^{2} z}{2} + 28 x^{2} + 40 x y^{2} + 85 x y z - 44 x y + 2 x - y\\\displaystyle - \tfrac{528749 x^{3}}{2250} - \tfrac{30937 x^{2} y}{2250} + \tfrac{113156 x^{2} z}{375} + \tfrac{72281 x^{2}}{1125} + 20 x y^{2} + \tfrac{98437 x y z}{1125} - 20 x y + \tfrac{126562 x z^{2}}{1125} - \tfrac{131062 x z}{1125} + y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{17531 x^{3}}{375} + \tfrac{342559 x^{2} y}{2250} + \tfrac{38699 x^{2} z}{450} - \tfrac{87467 x^{2}}{1125} + \tfrac{29081 x y^{2}}{225} + \tfrac{138934 x y z}{1125} - \tfrac{33412 x y}{225} - \tfrac{719 x z^{2}}{125} - \tfrac{15184 x z}{1125} + \tfrac{4106 x}{225} + \tfrac{53437 y^{3}}{2250} + \tfrac{5623 y^{2} z}{2250} - \tfrac{13406 y^{2}}{375} - \tfrac{29813 y z^{2}}{450} + \tfrac{74252 y z}{1125} - \tfrac{563 y}{750} - \tfrac{101251 z^{3}}{2250} + \tfrac{22894 z^{2}}{225} - \tfrac{156377 z}{2250} + \tfrac{14344}{1125}\\\displaystyle - \tfrac{255 x^{3}}{2} - 572 x^{2} y - \tfrac{875 x^{2} z}{2} + 363 x^{2} - \tfrac{1373 x y^{2}}{2} - 1224 x y z + 1034 x y - \tfrac{985 x z^{2}}{2} + 836 x z - \tfrac{691 x}{2} - 242 y^{3} - \tfrac{1423 y^{2} z}{2} + 596 y^{2} - 652 y z^{2} + 1114 y z - 465 y - \tfrac{365 z^{3}}{2} + 473 z^{2} - \tfrac{801 z}{2} + 110\\\displaystyle \tfrac{281251 x^{3}}{2250} + \tfrac{205313 x^{2} y}{450} + \tfrac{53969 x^{2} z}{125} - \tfrac{82294 x^{2}}{225} + \tfrac{1119377 x y^{2}}{2250} + \tfrac{1202066 x y z}{1125} - \tfrac{960752 x y}{1125} + \tfrac{236701 x z^{2}}{450} - \tfrac{988316 x z}{1125} + \tfrac{802127 x}{2250} + \tfrac{374063 y^{3}}{2250} + \tfrac{134269 y^{2} z}{225} - \tfrac{168094 y^{2}}{375} + \tfrac{1461941 y z^{2}}{2250} - \tfrac{234788 y z}{225} + \tfrac{299063 y}{750} + \tfrac{82219 z^{3}}{375} - \tfrac{619033 z^{2}}{1125} + \tfrac{100519 z}{225} - \tfrac{130219}{1125}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 3 of the reference element.

\(\displaystyle l_{12}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{0}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{0}\)
where \(f_{0}\) is the 0th face;
and \(\hat{\boldsymbol{n}}_{0}\) is the normal to facet 0.

\(\displaystyle \boldsymbol{\phi}_{12} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 183750 x^{2} - 143936 x y - 222439 x z + 174750 x + 221968 y z + 428657 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \tfrac{z \left(- 163564 x y + 231782 x z - 183750 y^{2} - 202811 y z + 174750 y + 418843 z^{2} - 208875 z + 2250\right)}{375}\\\displaystyle \tfrac{2 z \left(132766 x z + 127859 y z + 70875 z^{2} - 87375 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 183750 x^{2} - 109061 x y - 257314 x z + 174750 x + 371968 y^{2} + 278657 y z - 208875 y + 2250\right)}{375}\\\displaystyle \tfrac{y \left(68218 x y + 1125 y^{2} + 31157 y z - 34125 y + 2250\right)}{375}\\\displaystyle \tfrac{2 y \left(17234 x y + 115532 x z + 97141 y^{2} + 52843 y z - 34125 y + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(1125 x^{2} + 78032 x y + 21343 x z - 34125 x + 2250\right)}{375}\\\displaystyle \tfrac{x \left(381782 x^{2} - 128689 x y + 268843 x z - 208875 x - 183750 y^{2} - 237686 y z + 174750 y + 2250\right)}{375}\\\displaystyle \tfrac{2 x \left(92234 x^{2} + 22141 x y + 62657 x z - 34125 x + 105718 y z + 48750 z^{2} - 53250 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 263 x^{3} + \tfrac{91468 x^{2} y}{125} + \tfrac{148157 x^{2} z}{125} - 699 x^{2} + \tfrac{277747 x y^{2}}{375} + 2166 x y z - \tfrac{515122 x y}{375} + \tfrac{504503 x z^{2}}{375} - \tfrac{741878 x z}{375} + 627 x + \tfrac{101968 y^{3}}{375} + \tfrac{377593 y^{2} z}{375} - \tfrac{277811 y^{2}}{375} + \tfrac{434282 y z^{2}}{375} - 1824 y z + \tfrac{247468 y}{375} + \tfrac{158657 z^{3}}{375} - \tfrac{391189 z^{2}}{375} + \tfrac{304157 z}{375} - 191\\\displaystyle \tfrac{111782 x^{3}}{375} + \tfrac{317003 x^{2} y}{375} + \tfrac{387407 x^{2} z}{375} - \tfrac{297439 x^{2}}{375} + \tfrac{101282 x y^{2}}{125} + 2166 x y z - \tfrac{554378 x y}{375} + \tfrac{424468 x z^{2}}{375} - 1824 x z + \tfrac{257282 x}{375} + 263 y^{3} + \tfrac{138343 y^{2} z}{125} - 699 y^{2} + \tfrac{465247 y z^{2}}{375} - \tfrac{702622 y z}{375} + 627 y + \tfrac{148843 z^{3}}{375} - \tfrac{371561 z^{2}}{375} + \tfrac{294343 z}{375} - 191\\\displaystyle - \tfrac{85532 x^{3}}{375} - \tfrac{231782 x^{2} y}{375} - \tfrac{612878 x^{2} z}{375} + \tfrac{237814 x^{2}}{375} - \tfrac{221968 x y^{2}}{375} - 3084 x y z + 1176 x y - \tfrac{283532 x z^{2}}{125} + \tfrac{1078628 x z}{375} - \tfrac{221282 x}{375} - \tfrac{75718 y^{3}}{375} - \tfrac{573622 y^{2} z}{375} + \tfrac{218186 y^{2}}{375} - \tfrac{273718 y z^{2}}{125} + \tfrac{1039372 y z}{375} - \tfrac{211468 y}{375} - 862 z^{3} + 1926 z^{2} - 1248 z + 184\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{13}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{1}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{1}\)
where \(f_{1}\) is the 1st face;
and \(\hat{\boldsymbol{n}}_{1}\) is the normal to facet 1.

\(\displaystyle \boldsymbol{\phi}_{13} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(- 20250 x^{2} - 59434 x y - 39791 x z + 22050 x - 9000 y^{2} + 16217 y z + 9000 y + 50158 z^{2} - 13725 z - 450\right)}{75}\\\displaystyle \tfrac{z \left(68434 x y - 7217 x z + 38250 y^{2} + 37091 y z - 36450 y - 45658 z^{2} + 15525 z - 450\right)}{75}\\\displaystyle \tfrac{z \left(- 13967 x z - 8533 y z + 900 z^{2} + 7200 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(- 20250 x^{2} - 133909 x y + 34684 x z + 22050 x + 97217 y^{2} - 30842 y z - 13725 y - 9000 z^{2} + 9000 z - 450\right)}{75}\\\displaystyle \tfrac{y \left(61217 x y + 29025 y^{2} + 658 y z - 20925 y - 450\right)}{75}\\\displaystyle \tfrac{y \left(67967 x y - 81934 x z - 36467 y^{2} + 46834 y z - 12600 y - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 303 x^{3} - \tfrac{34217 x^{2} y}{75} + \tfrac{12842 x^{2} z}{75} + 111 x^{2} - 6 x - 120 y^{2} z - 120 y z^{2} + 120 y z\\\displaystyle \tfrac{x \left(- 43217 x^{2} + 95209 x y - 9658 x z + 15525 x + 38250 y^{2} + 10316 y z - 36450 y - 450\right)}{75}\\\displaystyle \tfrac{x \left(31967 x^{2} - 467 x y - 27034 x z - 12600 x - 8066 y z - 18000 z^{2} + 19800 z - 450\right)}{75}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1923 x^{3} - \tfrac{107683 x^{2} y}{25} - \tfrac{154742 x^{2} z}{25} + 4719 x^{2} - \tfrac{243607 x y^{2}}{75} - 9246 x y z + \tfrac{520582 x y}{75} - \tfrac{431843 x z^{2}}{75} + \tfrac{708818 x z}{75} - 3687 x - \tfrac{64783 y^{3}}{75} - \tfrac{259408 y^{2} z}{75} + \tfrac{196841 y^{2}}{75} - \tfrac{306467 y z^{2}}{75} + 6744 y z - \tfrac{198883 y}{75} - \tfrac{111842 z^{3}}{75} + \tfrac{290959 z^{2}}{75} - \tfrac{245942 z}{75} + 891\\\displaystyle \tfrac{28783 x^{3}}{75} + \tfrac{51907 x^{2} y}{75} + \tfrac{83908 x^{2} z}{75} - \tfrac{78041 x^{2}}{75} + \tfrac{15883 x y^{2}}{25} + 2094 x y z - \tfrac{119182 x y}{75} + \tfrac{81467 x z^{2}}{75} - 2016 x z + \tfrac{70183 x}{75} + 327 y^{3} + \tfrac{34442 y^{2} z}{25} - 951 y^{2} + \tfrac{105143 y z^{2}}{75} - \tfrac{172418 y z}{75} + 903 y + \tfrac{26342 z^{3}}{75} - \tfrac{73159 z^{2}}{75} + \tfrac{67742 z}{75} - 279\\\displaystyle \tfrac{103967 x^{3}}{75} + \tfrac{243467 x^{2} y}{75} + \tfrac{380768 x^{2} z}{75} - \tfrac{256534 x^{2}}{75} + \tfrac{175033 x y^{2}}{75} + 7344 x y z - 5016 x y + \tfrac{128567 x z^{2}}{25} - \tfrac{588668 x z}{75} + \tfrac{201617 x}{75} + \tfrac{35533 y^{3}}{75} + \tfrac{170032 y^{2} z}{75} - \tfrac{119666 y^{2}}{75} + \tfrac{81133 y z^{2}}{25} - \tfrac{377932 y z}{75} + \tfrac{133183 y}{75} + 1452 z^{3} - 3576 z^{2} + 2778 z - 654\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{14}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{2}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{2}\)
where \(f_{2}\) is the 2nd face;
and \(\hat{\boldsymbol{n}}_{2}\) is the normal to facet 2.

\(\displaystyle \boldsymbol{\phi}_{14} = \begin{cases} \left(\begin{array}{c}\displaystyle 6 z \left(40 x^{2} + 72 x y + 6 x z - 44 x - 96 y z + 26 z^{2} + 28 z + 1\right)\\\displaystyle \tfrac{2 z \left(22500 x^{2} - 24262 x y + 45881 x z - 22500 x - 10762 y^{2} + 118857 y z + 6262 y - 119738 z^{2} + 3619 z + 1125\right)}{375}\\\displaystyle \tfrac{2 z \left(- 32869 x z - 29738 y z - 41869 z^{2} + 21619 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 y \left(40 x^{2} - 50 x y + 128 x z - 44 x - 16 y^{2} - 54 y z + 28 y + 1\right)\\\displaystyle 120 x^{2} z - \tfrac{1762 x y^{2}}{375} + 120 x z^{2} - 120 x z + \tfrac{52238 y^{3}}{375} - \tfrac{75524 y^{2} z}{375} + \tfrac{19762 y^{2}}{375} + 6 y\\\displaystyle \tfrac{2 y \left(55369 x y - 88238 x z - 15262 y^{2} - 22107 y z - 8119 y - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 6 x \left(- 2 x^{2} - 24 x y + 74 x z - 16 x + 1\right)\\\displaystyle \tfrac{2 x \left(- 156619 x^{2} + 192619 x y + 82762 x z + 3619 x - 10762 y^{2} - 98024 y z + 6262 y + 22500 z^{2} - 22500 z + 1125\right)}{375}\\\displaystyle \tfrac{2 x \left(145369 x^{2} - 105262 x y - 185869 x z - 8119 x + 75524 y z - 34238 z^{2} + 29738 z + 1125\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 1452 x^{3} - 4152 x^{2} y - 4236 x^{2} z + 3576 x^{2} - 3756 x y^{2} - 7344 x y z + 6528 x y - 3588 x z^{2} + 6360 x z - 2778 x - 1056 y^{3} - 2916 y^{2} z + 2760 y^{2} - 2664 y z^{2} + 5016 y z - 2358 y - 804 z^{3} + 2256 z^{2} - 2106 z + 654\\\displaystyle \tfrac{496762 x^{3}}{375} + \tfrac{394162 x^{2} y}{75} + \tfrac{1654048 x^{2} z}{375} - \tfrac{463762 x^{2}}{125} + \tfrac{2379334 x y^{2}}{375} + \tfrac{4326668 x y z}{375} - \tfrac{732524 x y}{75} + \tfrac{345562 x z^{2}}{75} - \tfrac{3020096 x z}{375} + \tfrac{430012 x}{125} + \tfrac{301762 y^{3}}{125} + \tfrac{520124 y^{2} z}{75} - \tfrac{2199334 y^{2}}{375} + \tfrac{755286 y z^{2}}{125} - \tfrac{3957668 y z}{375} + \tfrac{337912 y}{75} + \tfrac{570524 z^{3}}{375} - \tfrac{307762 z^{2}}{75} + \tfrac{1363798 z}{375} - \tfrac{395512}{375}\\\displaystyle - \tfrac{69262 x^{3}}{375} - \tfrac{529048 x^{2} y}{375} - \tfrac{67462 x^{2} z}{75} + \tfrac{100762 x^{2}}{125} - \tfrac{170062 x y^{2}}{75} - \tfrac{1644668 x y z}{375} + \tfrac{1247096 x y}{375} - \tfrac{574834 x z^{2}}{375} + \tfrac{196124 x z}{75} - \tfrac{133012 x}{125} - \tfrac{390524 y^{3}}{375} - \tfrac{435786 y^{2} z}{125} + \tfrac{188962 y^{2}}{75} - \tfrac{244724 y z^{2}}{75} + \tfrac{1950668 y z}{375} - \tfrac{720298 y}{375} - \tfrac{102262 z^{3}}{125} + \tfrac{786334 z^{2}}{375} - \tfrac{129112 z}{75} + \tfrac{166012}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 2 of the reference element.

\(\displaystyle l_{15}:\boldsymbol{v}\mapsto\displaystyle\int_{f_{3}}\boldsymbol{v}\cdot(1)\hat{\boldsymbol{n}}_{3}\)
where \(f_{3}\) is the 3rd face;
and \(\hat{\boldsymbol{n}}_{3}\) is the normal to facet 3.

\(\displaystyle \boldsymbol{\phi}_{15} = \begin{cases} \left(\begin{array}{c}\displaystyle \tfrac{z \left(343124 x^{2} + 421874 x y + 485436 x z - 334124 x - 233437 y z - 258749 z^{2} + 153562 z - 2250\right)}{375}\\\displaystyle 3 z \left(- 230 x y + 95 x z - 80 y^{2} + 74 y z + 88 y + 45 z^{2} - 56 z - 2\right)\\\displaystyle - 120 x^{2} y - 120 x y^{2} + 120 x y - \tfrac{213749 x z^{2}}{375} - \tfrac{120937 y z^{2}}{375} - \tfrac{189562 z^{3}}{375} + \tfrac{117562 z^{2}}{375} - 6 z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{y \left(343124 x^{2} + 479812 x y + 427498 x z - 334124 x - 413437 y^{2} - 78749 y z + 153562 y - 2250\right)}{375}\\\displaystyle 3 y \left(- 135 x y + 4 y^{2} + 35 y z + 32 y - 2\right)\\\displaystyle \tfrac{y \left(- 45000 x^{2} + 123749 x y - 382498 x z + 45000 x + 435937 y^{2} - 493312 y z - 144562 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle \tfrac{x \left(221062 x^{2} + 188437 x y + 348749 x z - 180562 x - 2250\right)}{375}\\\displaystyle 3 x \left(- 65 x^{2} + 14 x y + 205 x z - 56 x - 80 y^{2} - 170 y z + 88 y - 2\right)\\\displaystyle \tfrac{x \left(528749 x^{2} + 30937 x y - 678936 x z - 144562 x - 45000 y^{2} - 196874 y z + 45000 y - 253124 z^{2} + 262124 z - 2250\right)}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - \tfrac{35062 x^{3}}{125} - \tfrac{342559 x^{2} y}{375} - \tfrac{38699 x^{2} z}{75} + \tfrac{174934 x^{2}}{375} - \tfrac{58162 x y^{2}}{75} - \tfrac{277868 x y z}{375} + \tfrac{66824 x y}{75} + \tfrac{4314 x z^{2}}{125} + \tfrac{30368 x z}{375} - \tfrac{8212 x}{75} - \tfrac{53437 y^{3}}{375} - \tfrac{5623 y^{2} z}{375} + \tfrac{26812 y^{2}}{125} + \tfrac{29813 y z^{2}}{75} - \tfrac{148504 y z}{375} + \tfrac{563 y}{125} + \tfrac{101251 z^{3}}{375} - \tfrac{45788 z^{2}}{75} + \tfrac{156377 z}{375} - \tfrac{28688}{375}\\\displaystyle 765 x^{3} + 3432 x^{2} y + 2625 x^{2} z - 2178 x^{2} + 4119 x y^{2} + 7344 x y z - 6204 x y + 2955 x z^{2} - 5016 x z + 2067 x + 1452 y^{3} + 4269 y^{2} z - 3576 y^{2} + 3912 y z^{2} - 6684 y z + 2778 y + 1095 z^{3} - 2838 z^{2} + 2397 z - 654\\\displaystyle - \tfrac{281251 x^{3}}{375} - \tfrac{205313 x^{2} y}{75} - \tfrac{323814 x^{2} z}{125} + \tfrac{164588 x^{2}}{75} - \tfrac{1119377 x y^{2}}{375} - \tfrac{2404132 x y z}{375} + \tfrac{1921504 x y}{375} - \tfrac{236701 x z^{2}}{75} + \tfrac{1976632 x z}{375} - \tfrac{799877 x}{375} - \tfrac{374063 y^{3}}{375} - \tfrac{268538 y^{2} z}{75} + \tfrac{336188 y^{2}}{125} - \tfrac{1461941 y z^{2}}{375} + \tfrac{469576 y z}{75} - \tfrac{297563 y}{125} - \tfrac{164438 z^{3}}{125} + \tfrac{1238066 z^{2}}{375} - \tfrac{200588 z}{75} + \tfrac{258188}{375}\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

This DOF is associated with face 3 of the reference element.

\(\displaystyle l_{16}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}1\\0\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{16} = \begin{cases} \left(\begin{array}{c}\displaystyle 4 z\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 y\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 4 x\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle - 4 x - 4 y - 4 z + 4\\\displaystyle 0\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{17}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\1\\0\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{17} = \begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 z\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 y\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 4 x\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle - 4 x - 4 y - 4 z + 4\\\displaystyle 0\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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

\(\displaystyle l_{18}:\boldsymbol{v}\mapsto\boldsymbol{v}(\tfrac{1}{4},\tfrac{1}{4},\tfrac{1}{4})\cdot\left(\begin{array}{c}0\\0\\1\end{array}\right)\)

\(\displaystyle \boldsymbol{\phi}_{18} = \begin{cases} \left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 z\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 1, 0), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 y\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (1, 0, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle 4 x\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((0, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\\\left(\begin{array}{c}\displaystyle 0\\\displaystyle 0\\\displaystyle - 4 x - 4 y - 4 z + 4\end{array}\right)&\text{in }\operatorname{Tetrahedron}(((1, 0, 0), (0, 1, 0), (0, 0, 1), (1/4, 1/4, 1/4)))\end{cases}\)

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