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Preface

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Conventions used in this book

The following typographical conventions are used in the book

Italic indicates new terms

typewriter is used on program listings as well as when referring to programming elements, e.g. functions, variables, statements, data types, environment variables or keywords.

\$ typewriter or > typewriter displays commands that the user types literally without the \$ or >.

this is a general note.
this is a general warning.
be cautious

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}\$

Spaces

  • \$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\$.