Notation

Throughout these notebooks a consistent notation shall be used as far as possible.

Symbol(s)	Description
\(\Omega\)	domain
\(\partial\Omega\)	domain boundary
\(\partial\Omega_i\subset\partial\Omega\)	subset of the domain boundary
\(\text{d}\Omega\)	integration measure over the cells
\(\text{d}\Gamma\)	integration measure over the cell facets
\(\textbf{e}_x, \textbf{e}_y, \textbf{e}_z\)	unit vectors
\(\textbf{x}=(x, y, z) = x\textbf{e}_x + y\textbf{e}_y + z\textbf{e}_z\)	spatial coordinates
\(t\)	time
\(\Delta t\)	timestep
\(\mathcal{D}(\cdot)\)	finite difference operator
\(h(\textbf{x})\)	local cell size
\(\mathcal{F}\)	set of cell facets
\(\left[\!\left[ \cdot \right]\!\right]\)	cell facet jump operator
\(\{\cdot\}\)	cell facet average operator
\(u, v, \dots\)	scalar quantity
\(\textbf{u}, \textbf{v}, \dots\)	vector quantity
\(\textbf{u} = u_x\textbf{e}_x + u_y\textbf{e}_y + u_z\textbf{e}_z\)	vector quantity components
\(\mathsf{U}, \mathsf{V}, \dots\)	tensor quantity
\(\mathsf{U} = ((U_{xx}, U_{xy}), (U_{yx}, U_{yy})) \)	tensor quantity components
\(V_u\)	function space to which \(u\) belongs
\(\mathbb{BVP}\)	boundary value problem
\(\mathbb{IBVP}\)	initial boundary value problem
\(\mathbb{IVP}\)	initial value problem
\(\mathbb{EVP}\)	eigenvalue problem
\(\mathbb{S}\)	specification
\(\mathbb{F}\)	linearized or time-discretized weak forms sequence
\(u_0\)	initial condition on \(u\)
\(u_\text{D}\)	Dirichlet boundary condition on \(u\)
\(u_\text{N}\)	Neumann or natural boundary condition on \(u\)
\(u_\text{E}\)	essential boundary condition on \(u\)
\(\mathbb{R}\)	the set of real numbers
\(\mathbb{C}\)	the set of complex numbers