On page 8 of "A gentle introduction to the Finite Element Method" by Francisco–Javier Sayas, a space is defined on a domain omega with an overbar.
Previously he had used the omega domain to represent the physical domain. Does anyone have any ideas as to what the overbar means?
On
Mathematically, it is the closure of $\Omega$: \begin{equation} \bar\Omega = \Omega \cup \Gamma \end{equation} in which $\Omega$ is the domain (a connected open set) and $\Gamma$ is the boundary of the problem.
The domain should generally be bounded, i.e. it can be enclosed by a circle with finite radius. That basically means that it is possible to "close" the open set by it's boundary.
As an example in $\mathbb{R}^2$, consider the following case of a double barrier basket option: \begin{equation} \Omega = \{S \in \mathbb{R}^2 : a_1S_1 + a_2S_2>a_0 \ and \ b_1S_1 + b_2S_2<b_0 \} \\ \Gamma = \{S \in \mathbb{R}^2 : a_1S_1 + a_2S_2=a_0 \ or \ b_1S_1 + b_2S_2=b_0 \} \\ \bar\Omega = \Omega \cup \Gamma \end{equation} It is necessary to specify the boundary and the domain separately as the value of the option is only (strictly) positive in the domain and becomes zero on the boundary.
The overbar usually means 'closure', in some appropriate contextual sense. It's $\Omega \cup \{\text{something else}\}$ so that extra properties are satisfied. Very likely in this case it's for completentess, i.e. so that Cauchy sequences in $\overline{\Omega}$ converge in $\overline{\Omega}$ (though don't take my word for it).