Variable

Meaning

n

A natural number that is either 1 or 2

m

$m\in \mathbb{N}$ is a natural number

N

$N\in \mathbb{N}$ is a natural number

$k_w$

$k_w\in ℝ$ is a constant

x

$x\in ℝ$ is a single variable

$x$

$x=\left(x_1\mathrm{,}\cdots \mathrm{,}{x}_n\right)\in {ℝ}^n$ , $n\ge 2$

t

$t\ge 0$ is a non-negative real variable

T

$T>0$ is a positive real number

L

$L>0$ is a positive real number

a

$a\in ℝ$ is a real number

b

$b\in ℝ$ is a real number

k

$k\in \mathbb{N}$ is a natural number

$a\left(\cdot \mathrm{,}\cdot \right)$

A bilinear form

$\beta$

$\beta =\left({\beta }_1\mathrm{,}\cdots \mathrm{,}{\beta }_n\right)\in {\mathbb{N}}^N$ is a multi-index

${\beta }_k$

${\beta }_k=\frac{k\pi }{L}{k}_w$ , $k\in \mathbb{N}$ , $k\ge 1$

$\alpha$

$\alpha \in ℝ$ is a real number

$t_i,i=\mathrm{0,1,2}$

$t_i\ge 0$ is a real non-negative number

Ω

$\Omega \subset {ℝ}^n$ is a domain

$\partial \Omega$

the boundary of the domain $\Omega \subset {ℝ}^n$

V

A Hilbert space V

X

A vector space X

u

A scalar function $u\equiv u\left(x\mathrm{,}t\right)$

v

A scalar function $v\equiv v\left(x\mathrm{,}t\right)$

$\phi$

A scalar function $\phi \equiv \phi \left(x\mathrm{,}t\right)$

$\mathcal{N}$

A linear operator $\mathcal{N}$

$\mathcal{C}$

A letter used to design a space of continuous functions

${\mathcal{T}}_h$

A triangulation with diameter h>

$X_h$

A vector space of piecewise linear polynomials

$X_{1h}$

A vector space of functions of two variables

$u_h$

A function in $X_h$

C

$C>0$ is an arbitrary constant

$w_h$

A function in $X_h$

${\Pi }_h$

The projection of X to $X_h$

${\gamma }_k$

A function ${\gamma }_k\equiv {\gamma }_k\left(t\right)$ , $k\in \mathbb{N}$ , $k\ge 1$

$s_k$

A function $s_k\left(x\right)\sin \left(k\pi x/L\right)$

$\nu$

The convergence rate

${\nu }_{exp}$

The expected convergence rate