Notation#
Throughout these notebooks a consistent notation shall be used as far as possible.
Symbol(s) |
Description |
|---|---|
Space |
|
\(\Omega\) |
domain |
\(\partial\Omega\) |
domain boundary |
\(\partial\Omega_{\text{B}}\subset\partial\Omega\) |
subset of the domain boundary for a boundary type \(\text{B}\) |
\(\text{d}\Omega\) |
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 |
\(\textbf{x}=\begin{pmatrix} x \\ y \\ z\end{pmatrix} = x\textbf{e}_x + y\textbf{e}_y + z\textbf{e}_z\) |
spatial coordinates |
\(~\) |
|
Time |
|
\(t\) |
time |
\(\Delta t\) |
time-step |
\(t^n\) |
time at the \(n^{\text{th}}\) time-level |
\(\mathcal{D}\) |
finite difference operator |
\(\mathcal{D}_{u,w,\dots}=\mathcal{D}_u\circ\mathcal{D}_w\dots\) |
argument-wise finite difference operator |
\(\mathscr{D}^{\text{IM}}_u\) |
the set of all finite difference operators that are explicit with respect to \(u\) |
\(\mathscr{D}^{\text{EX}}_u\) |
the set of all finite difference operators that are explicit with respect to \(u\) |
\(\text{FE}\) |
forward Euler finite difference operator |
\(\text{BE}\) |
backward Euler finite difference operator |
\(\text{CN}\) |
Crank-Nicolson finite difference operator |
\(\text{AB}_n\) |
order \(n\) Adams-Bashforth finite difference operator |
\(\text{AM}_n\) |
order \(n\) Adams-Moulton finite difference operator |
\(~\) |
|
Partial differential equations |
|
\(\mathscr{L}_{\textbf{x}}\) |
spatial differential operator |
\(\mathscr{L}_{\textbf{x},t}\) |
spatial and temporal differential operator |
\(\mathbb{BVP}_u\) |
boundary value problem to solve for \(u\) |
\(\mathbb{IBVP}_u\) |
initial boundary value problem to solve for \(u\) |
\(\mathbb{IVP}_u\) |
initial value problem to solve for \(u\) |
\(\mathbb{EVP}_u\) |
eigenvalue problem to solve for \(u\) |
\(\mathbb{S}_u\) |
specification of a problem solving for \(u\) |
\(u_0\) |
initial condition on \(u\) |
\(u_\text{D}\) |
Dirichlet boundary condition on \(u\) |
\(u_\text{E}\) |
essential boundary condition on \(u\) |
\(u_\text{N}\) |
Neumann or natural boundary condition on \(u\) |
\(u_\text{S}\) |
strong boundary condition on \(u\) |
\(u_\text{W}\) |
weak boundary condition on \(u\) |
\(u_\text{I}\) |
inflow boundary condition on \(u\) |
\(u_\text{R}\) |
Robin boundary condition on \(u\) |
\(~\) |
|
Finite element method |
|
\(\mathscr{T}\) |
tesselation of the domain |
\(\bigcup_{\mathcal{K}} \mathcal{K}\) |
union of cells forming the mesh |
\(h\) |
local cell size |
\(\mathcal{F}\) |
set of cell facets |
\(\mathcal{V}\) |
set of cell vertices |
\(V_u\) |
trial function space to which \(u\) belongs |
\(\hat{V}_v\) |
test function space to which \(v\) belongs |
\(\xi_j(\mathbf{x})\) |
finite element basis functions |
\(\sum_jU_j\xi_j\) |
finite element approximation |
\(\{U_j\}_j\leftrightarrow\textbf{U}\) |
degrees of freedom vector |
\(L^2(\Omega)\) |
Lebesgue function space on the domain \(\Omega\) |
\(H^1(\Omega)\) |
Sobolev function space on the domain \(\Omega\) |
\(C^0(\Omega)\) |
set of continuous functions on the domain \(\Omega\) |
\(\mathrm{P}_k\) |
continuous Lagrange element of degree \(k\) |
\(\mathrm{DP}_k\) |
discontinuous Lagrange element of degree \(k\) |
\(\mathrm{BDM}_k\) |
Brezzi–Douglas–Marini element of degree \(k\) |
\(\mathcal{R}\) |
residual of the equation |
\(\left[\!\left[ \cdot \right]\!\right]\) |
cell facet jump operator |
\(\{\cdot\}\) |
cell facet average operator |
\(\tau_{\text{SUPG}}\) |
SUPG stabilization parameter |
\(\alpha_{\text{DG}}\) |
DG penalty parameter |
\(\alpha_{\text{W}}\) |
weak enforcement penalty parameter |
\(\mathcal{E}\) |
the error in the numerical solution |
\(\mathbb{F}_{u,w\dots}\) |
sequence of weak forms solving for \(u,w\dots\) |
\(~\) |
|
General mathematics |
|
\(u, v, \dots\) |
scalar quantity |
\(\textbf{u}, \textbf{v}, \dots\) |
vector quantity |
\(\textbf{u} = \begin{pmatrix} u_x \\ u_y \\ u_z\end{pmatrix}=u_x\textbf{e}_x + u_y\textbf{e}_y + u_z\textbf{e}_z\) |
vector quantity components |
\(\mathsf{U}, \mathsf{V}, \dots\) |
tensor quantity |
\(\mathsf{U} = \begin{pmatrix} U_{xx} & U_{xy} & U_{xz} \\ U_{yx} & U_{yy} & U_{yz} \\ U_{zx} & U_{zy} & U_{zz} \end{pmatrix} \) |
tensor quantity components |
\(\dfrac{\mathrm{d}}{\mathrm{d}x}\) |
ordinary derivative operator |
\(\partial_x = \dfrac{\partial}{\partial x}\) |
partial derivative operator |
\(\nabla = (\partial_x, \partial_y, \partial_z)\) |
gradient operator |
\(\mathbf{n}\) |
outward unit normal vector |
\(\mathrm{H}\) |
Heaviside step function |
\(\mathsf{I}\) |
identity tensor |
\(\det\) |
matrix determinant |
\(\min_{\mathbf{x} \in \Omega}\) |
minimum over the domain \(\Omega\) |
\(\max_{\mathbf{x} \in \Omega}\) |
maximum over the domain \(\Omega\) |
\(\mathrm{vol}(\Omega)\) |
volume of domain \(\Omega\) |
\(\langle \cdot \rangle_{\Omega}\) |
space-averaging over the domain \(\Omega\) |
\(\overline{\;\cdot\;}^{[t,t']}\) |
time-averaging over interval \([t,t']\) |
\(\mathbb{R}\) |
the set of real numbers |
\(\mathbb{C}\) |
the set of complex numbers |
\(~\) |
|
Fluid mechanics |
|
\(\textbf{u}\) |
velocity |
\(p\) |
pressure |
\(\psi\) |
streamfunction |
\(\boldsymbol{\omega}\) |
vorticity |
\(\rho\) |
fluid density |
\(\mu\) |
fluid viscosity |
\(\textbf{f}\) |
body force |
\(\tau\) |
deviatoric stress |
\(\phi\) |
porosity |
\(\mathsf{K}\) |
permeability |
\(c\) |
solute concentration |
\(\theta\) |
temperature |
\(\mathsf{D}\) |
solutal dispersion |
\(\mathsf{G}\) |
thermal dispersion |
\(g\) |
gravity constant |
\(\,{\textbf{e}}_g\) |
gravity unit vector |
\(~\) |
|
Acronyms |
|
PDE |
partial differential equation |
ODE |
ordinary differential equation |
DNS |
direct numerical simulation |
CG |
continuous Galerkin |
DG |
discontinuous Galerkin |
SUPG |
streamline-upwind Petrov–Galerkin |
CFL |
Courant–Friedrichs–Lewy |