47. The function space \(H(\operatorname{div})\)#

\(\DeclareMathOperator{\opdiv}{div}\) We will

  • give a clear definition of the space \(H(\operatorname{div})\)

  • understand its boundary values, i.e. its trace

  • understand its continuity properties

As for the Sobolev space \(H^1\), we need a weak definition of the differential operator \(\operatorname{div}\). We call

\[ d = \operatorname{div} \sigma \]

the weak divergence iff

\[ \int_\Omega d \, \varphi = - \int_\Omega \sigma \nabla \varphi \]

holds for all smooth testfunctions with compact support \(\varphi \in C_0^\infty(\Omega)\).

We define the function space

\[ H(\operatorname{div}) = \{ \sigma \in [L_2]^2 : \operatorname{div} \sigma \in L_2 \} \]

equipped with the norm

\[ \| \sigma \|_{H(\operatorname{div})} = \left( \| \sigma \|_{L_2}^2 + \| \operatorname{div} \sigma \|_{L_2}^2 \right)^{1/2}. \]

Smooth functions are dense in \(H(\operatorname{div})\), i.e. \(\forall \, \sigma \in H(\operatorname{div})\) and \(\forall \varepsilon > 0\) there exists \(\sigma^\varepsilon \in C^\infty\) such that

\[ \| \sigma - \sigma^\varepsilon \|_{H(\operatorname{div})} < \varepsilon \]

(proven by commuting mollifiers).

For smooth functions \(\sigma\) and \(\varphi\) there holds the integration by parts formula

\[ \int_\Omega \operatorname{div} \sigma \varphi = -\int_\Omega \sigma \nabla \varphi + \int_{\partial \Omega} \sigma_n \varphi \]

47.1. Normal-trace of functions in \(H(\operatorname{div})\)#

From the integration by parts formula applied to smooth functions \(\sigma\) we get

\[\begin{eqnarray*} \left| \int_{\partial \Omega} \sigma_n \varphi \right| & = & \left| \int_\Omega \operatorname{div} \sigma \, \varphi + \int_\Omega \sigma \nabla \varphi \right| \\ & \leq & \| \operatorname{div} \sigma \|_{L_2} \| \varphi \|_{L_2} + \| \sigma \|_{L_2} \| \nabla v \|_{L_2} \\ & \leq & \| \sigma \|_{H(\operatorname{div})} \| \varphi \|_{H^1} \end{eqnarray*}\]

and thus

\[\begin{eqnarray*} \| \sigma_n \|_{H^{-1/2}} & = & \sup_{v \in H^{1/2}} \frac{\left< v, \sigma_n \right>}{\| v \|_{H^{1/2}}} \\ & \approx & \sup_{\varphi \in H^1(\Omega)}  \frac{\left< \varphi_{|\partial \Omega}, \sigma_n \right>}{\| \varphi \|_{H^1(\Omega)}} \leq \| \sigma \|_{H(\operatorname{div})} \end{eqnarray*}\]

We have shown that

\[ \| \sigma_n \|_{H^{-1/2}} \prec \| \sigma \|_{H(\operatorname{div})} \quad \text{ for smooth $\sigma$ } \]

Now we use density of smooth functions in \(H(\operatorname{div})\) to uniquely extend the trace operator \(\operatorname{tr}_n\) to the whole space \(H(\operatorname{div})\).

Having well-defined normal boundary values, we get now the integration by parts formula for arbitrary functions \(\sigma \in H(\operatorname{div})\).

Exercise: Prove that for all \(\sigma_n \in H^{-1/2}(\partial \Omega)\) there exists a \(\sigma \in H(\operatorname{div})\) such that \(\operatorname{tr}_n \sigma = \sigma_n\).

47.2. \(H(\operatorname{div})\) on sub-domains#

Assume that we have a non-overlapping domain decomposition

\[ \Omega = \Omega_1 \cup \Omega_2 \ldots \Omega_n \]

with interfaces

\[ \gamma_{ij} = \overline{\Omega_i} \cap \overline{\Omega_j} \]

Theorem: for \(\sigma \in [L_2(\Omega)]^n\) such that

  • \(\sigma_{|\Omega_i} \in H(\operatorname{div}, \Omega_i)\) with \(d_i = \operatorname{div}_{\Omega_i} \sigma_{|\Omega_i}\)

  • \(\sigma_{\Omega_i} n_i = - \sigma_{\Omega_j} n_j\) on \(\gamma_{ij}\)

then there is \(\sigma \in H(\operatorname{div}, \Omega)\) with

\[ (\operatorname{div} \sigma)_{|\Omega_i} = d_i \]

Proof: We apply the integration by parts formula on sub-domains, and observe that boundary terms cancel out:

\[\begin{eqnarray*} \int_\Omega d \, \varphi & = & \sum_{\Omega_i} \int_{\Omega_i} d_i \, \varphi = \sum_{\Omega_i} \int_{\Omega_i} \operatorname{div}_{\Omega_i} \sigma_{|\Omega_i} \varphi \\ & = & \sum_{\Omega_i} -\int_{\Omega_i} \sigma_{|\Omega_i} \nabla \varphi + \int_{\partial \Omega_i} n_i \sigma_{|\Omega_i} \varphi \\ & = & \int_\Omega \sigma \, \nabla \varphi \end{eqnarray*}\]