Mathematical Notations

Geometry and Meshes

  • \$d=1,2,3\$ geometrical dimension

  • \$\Omega \subset \mathbb{R}^d\$

  • \$K\$ a cell or element of a mesh

  • \$h\$ characteristic mesh size

  • \$k_{\mathrm{geo}}\$ polynomial order of the geometrical transformation

  • \$\delta=(h,k_{\mathrm{geo}})\$ discretization parameter pair for the geometrical transformation, default value \$k_{\mathrm{geo}}=1\$ (straight cells or elements)

  • \$\varphi^K_\delta: \hat{K} \rightarrow K\$, geometrical transformation

  • \$\mathcal{T}_{\delta}\$ a triangulation, \$\mathcal{T}_\delta = \{ K\; | \; K=\varphi^K_\delta (\hat{K}) \} \$

  • \$\Omega_h \equiv \cup_K {K}\$


  • \$P^k_{c,h} = \{ v_h \in C^0(\bar{\Omega}); \forall K \in \mathcal{T}_h,\ v_h \circ T_K \in \mathbb{P}^k\}\$ Space of continuous piecewise polynomial of total degree \$\leq k\$.