Subdomains

In this section, we consider a domain \(\Omega\) which a union of multiple subdomain, i.e.

\[ \Omega := \bigcup_{i=1}^n \Omega_i, \quad \Omega_i \bigcap \Omega_j = \emptyset, ~ i \neq j \]

We shall denote by \(\mathcal{I}\) the set of all internal interfaces of \(\Omega\).

We shall also need the following operators

• The jump of the function \(u\), defined as \([\![ u ]\!] := u|\_{\Omega_{i_1}} - u|\_{\Omega_{i_2}}\) for two adjacent subdomains \(\Omega_{i_1}\) and \(\Omega_{i_2}\)

• The average of the function \(u\), defined as \(\{u\} := \frac{1}{2} \left( u|\_{\Omega_{i_1}} + u|\_{\Omega_{i_2}} \right)\) for two adjacent subdomains \(\Omega_{i_1}\) and \(\Omega_{i_2}\)

In general, these operators are defined based on the considered variational formulation. We shall see in the following examples how to define them.