Helmholtz

Layer Potentials

\[\begin{split} \begin{array}{rcl} \mathrm{SL}\left( j \right) (\boldsymbol x) &=& \displaystyle{ \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \frac{e^{i\, \kappa \, |\boldsymbol x-\boldsymbol y|} }{\| \boldsymbol x-\boldsymbol y\|} } \, j(\boldsymbol y)\, \mathrm{d}\sigma_y } \\ \mathrm{DL}\left(m \right)(\boldsymbol x) &=& \displaystyle{ \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \displaystyle{ \boldsymbol n_y \cdot \nabla_y }\displaystyle{ \frac{ e^{i\,\kappa\,\|\boldsymbol x-\boldsymbol y\|}}{\| \boldsymbol x-\boldsymbol y\|}} } \, m(\boldsymbol y)\, \mathrm{d}\sigma_y } \end{array}\end{split}\]

Helmholtz Dirichlet BVP

Let \(u\) denote the acoustic potential which is caused by Dirichlet boundary condition on \(\Gamma\) and which propagates in \(\Omega^c \in \mathbb R^3\). Thus, \(u\) solves the exterior boundary value problem

\[\begin{split} \left\{ \begin{array}{rcl l} \Delta u + \kappa^2 u &=& 0\,, \quad & \Omega^c \subset \mathbb R^3\,, \\ \gamma_0 u &=& m\,, \quad & \Gamma = \partial \Omega\,. \end{array} \right. \end{split}\]

To stabilize interior eigenvalues, one consideres a combined field integral equation. The combinded field integral equation combines single and double layer integral operators, one option is the Brakhage-Werner formulation:

\[\begin{split} \begin{array}{ll rcl } &\textnormal{ansatz} & u(x) &=& i \, \kappa \, \mathrm{ SL}(j) - \mathrm{DL}(j) \\ &\textnormal{variational formulation } & \forall v\in H^{-\frac12}(\Gamma): \; \left\langle \gamma_0 u , v \right\rangle_{-\frac12} &=& i \, \kappa \, \left\langle \gamma_0 \left(\mathrm{SL}(j)\right), v \right\rangle_{-\frac12} - \left\langle \gamma_0 \left(\mathrm{DL}(j)\right), v\right\rangle_{-\frac12} \\ & \textnormal{discretisation} & \left( \frac12 \, \mathrm M + i \, \kappa \,\mathrm{V} + \mathrm K\right) \, \mathrm{j} &=& \mathrm{M} \, \mathrm{m} \end{array} \end{split}\]