Maxwell

Layer Potentials

\[\begin{split} \begin{array}{rcl} \mathrm{SL}\big(\boldsymbol j\big)(\boldsymbol x) &=& \kappa \, \displaystyle {\int\limits_\Gamma \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|\boldsymbol x-\boldsymbol y\|}}{\| \boldsymbol x-\boldsymbol y\|} \, \boldsymbol j(\boldsymbol y)\, \mathrm{d}\sigma_y + \frac{1}{\kappa} \nabla \int\limits_\Gamma \frac{1}{4\,\pi}\, \frac{e^{i\,\kappa\,\|\boldsymbol x-\boldsymbol y\|}}{\| \boldsymbol x-\boldsymbol y\|} \, \mathrm{div}_\Gamma \boldsymbol j(\boldsymbol y)\, \mathrm{d}\sigma_y } \\ \mathrm{DL}\big(\boldsymbol n \times \boldsymbol m\big)(\boldsymbol x) &=& \nabla \times \displaystyle {\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|\boldsymbol x-\boldsymbol y\|}}{\| \boldsymbol x-\boldsymbol y\|} } \, \boldsymbol n(\boldsymbol y) \times \boldsymbol{m}(\boldsymbol y)\, \mathrm{d}\sigma_y }\end{array} \end{split}\]

Maxwell Equations

Let \(\Omega \in \mathbb R^3\) denote a perfect electric conductor and \(\gamma_R \boldsymbol E^i = -\boldsymbol m\) the given Dirichlet trace of an incoming time-harmonic electromagnetic signal \(\boldsymbol E^i\). The scattered electromagnetic field with components \(\boldsymbol E\) and \(\boldsymbol H\) solves the following equations in the exterior domain \(\Omega^c\):

\[\begin{split} \left\{ \begin{array}{ccl} \mathbf{curl} \, \boldsymbol H &=& -i\omega\varepsilon \boldsymbol E\,, \\ \mathbf{curl} \, \boldsymbol E &=& i\omega\mu \boldsymbol H\,, \\ \gamma_R\, \boldsymbol E &=& \boldsymbol m \end{array} \right. \end{split}\]

From here we can derive two second order equations: one for the electric field \(\boldsymbol E\) and one for the magnetic field \(\boldsymbol H\).

Maxwell Dirichlet BVP

The electric field \(\boldsymbol E\) solves the second order equation with Dirichlet boundary conditions:

\[\begin{split}\begin{array}{rcl l} \mathbf{curl} \, \mathbf{curl}\, \boldsymbol E - \kappa^2 \, \boldsymbol E &=& \boldsymbol 0, \quad &\textnormal{in } \Omega^c \subset \mathbb R^3\,,\\ \gamma_R \,\boldsymbol E &=& \boldsymbol m, \quad & \textnormal{on }\Gamma\,, \\ \left\| \mathbf{curl} \, \boldsymbol E( x) - i\,\omega\,\epsilon \, \boldsymbol E( x)\right\| &=& \mathcal O\left( \displaystyle \frac{1}{\| x\|^2}\right), &\textnormal{for} \; \|x\| \to \infty\,.\end{array}\end{split}\]

1. Direct Ansatz

\[\begin{split} \begin{array}{r rcl } \textnormal{direct ansatz } & \boldsymbol E(x) &=& \mathrm{SL}\left( \gamma_N \, \boldsymbol E\right)(x) + \mathrm{DL}\left( \gamma_D \,\boldsymbol E\right)(x) \\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{div}_\Gamma, \Gamma): \; \big\langle \gamma_R \, \mathrm{SL} (\boldsymbol j), \boldsymbol v \big\rangle_{-\frac12} &=& \big\langle \boldsymbol m, \boldsymbol v\big\rangle_{-\frac12} - \big\langle \gamma_R\,\mathrm{DL}(\boldsymbol n \times \boldsymbol{m}), \boldsymbol v\big\rangle_{-\frac12} \\ \textnormal{discretisation} & \mathrm{V} \, \mathbf{j} &=& \left( \dfrac12 \mathrm{M} - \mathrm{K}\right) \,\mathbf{m} \end{array}\end{split}\]

2. Indirect Ansatz

\[\begin{split} \begin{array}{r rcl } \textnormal{indirect ansatz} & \boldsymbol E(x) &=& \mathrm{SL}\left(\boldsymbol j^t\right)(x)\\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{div}_\Gamma, \Gamma): \; \big\langle \gamma_R \, \mathrm{SL} (\boldsymbol j^t), \boldsymbol v \big\rangle_{-\frac12} &=& \big\langle \boldsymbol m, \boldsymbol v\big\rangle_{-\frac12} \\ \textnormal{discretisation} & \mathrm{V} \, \mathbf{j^t} &=& \mathrm{M} \,\mathbf{m} \end{array}\end{split}\]

Notes:

• \(\boldsymbol j^t\) is the Neumann trace of the total electric field \(\boldsymbol E^t = \boldsymbol E + \boldsymbol E^i\).

Maxwell Neumann BVP

The magnetic field \(\boldsymbol H\) solves the second order equation with Neumann boundary conditions:

\[\begin{split} \left\{ \begin{array}{rcl l} \mathbf{curl} \, \mathbf{curl}\, \boldsymbol H - \kappa^2 \, \boldsymbol H &=& \boldsymbol 0, \quad &\textnormal{in } \Omega^c \subset \mathbb R^3\,,\\ \gamma_N \,\boldsymbol H &=& -\dfrac{i\omega\varepsilon}{\kappa} \, \boldsymbol n\times \boldsymbol m, \quad & \textnormal{on }\Gamma\,, \\[1ex] \left\| \mathbf{curl} \, \boldsymbol H( x) + i\,\omega\,\mu \, \boldsymbol H( x)\right\| &=& \mathcal O\left( \displaystyle \frac{1}{\| x\|^2}\right), &\textnormal{for} \; \|x\| \to \infty\end{array} \right. \end{split}\]

1. Direct Ansatz

Representation formula:

\[\begin{split}\begin{array}{rcl} \boldsymbol H(x) &=& \mathrm{SL}\left( \gamma_N \, \boldsymbol H\right)(x) +\mathrm{DL}\left( \gamma_D\,\boldsymbol H\right)(x) \\[1ex] &=& -\dfrac{i\omega\varepsilon}{\kappa} \, \Big(\underbrace{ \kappa\, \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol n(y)\times \boldsymbol m(y)\, \mathrm{d}\sigma_y + \frac{1}{\kappa} \nabla \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi}\, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \mathrm{curl}_\Gamma \,\boldsymbol m(y)\, \mathrm{d}\sigma_y }_{\displaystyle{\mathrm{SL}(\boldsymbol n \times \boldsymbol m)} } \Big) \\ && + \dfrac{ \kappa }{ i\omega\mu} \underbrace{ \nabla \times \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol{j}(y) \, \mathrm{d}\sigma_y }_{\displaystyle{ \mathrm{DL} (\boldsymbol{j}) } }\end{array}\end{split}\]

Apply rotated Dirichlet trace:

\[\begin{split}\begin{array}{c rcl} & \gamma_D \,\boldsymbol H &=& -\dfrac{i\omega\varepsilon}{\kappa} \gamma_D \,\mathrm{SL}(\boldsymbol n\times \boldsymbol m) + \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j) \\[1ex] \Rightarrow & \dfrac{\kappa}{i\omega\mu}\boldsymbol j &=& -\dfrac{i\omega\varepsilon}{\kappa} \gamma_D \, \mathrm{SL}(\boldsymbol n\times \boldsymbol m) + \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j) \\ \Rightarrow & \boldsymbol j &=& \gamma_D \,\mathrm{SL}( \boldsymbol n\times \boldsymbol m) + \gamma_D\, \mathrm{DL}(\boldsymbol j) \end{array}\end{split}\]

Thus,

\[\begin{split} \begin{array}{r rcl } \textnormal{direct ansatz} & \boldsymbol H(x) &=& -\dfrac{i\omega\varepsilon}{\kappa} \mathrm{SL}(\boldsymbol n \times \boldsymbol m) + \dfrac{\kappa}{i\omega\mu} \mathrm{DL}(\boldsymbol j) \\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{curl}_\Gamma, \Gamma): \; \big\langle \boldsymbol v, \boldsymbol j\big\rangle_{-\frac12} - \big\langle \boldsymbol v, \gamma_D \,\mathrm{DL} (\boldsymbol j) \big\rangle_{-\frac12} &=& \big\langle \boldsymbol v, \gamma_D\, \mathrm{SL}(\boldsymbol n \times \boldsymbol{m}) \big\rangle_{-\frac12} \\ \textnormal{discretisation} & \left( \dfrac12 \mathrm{M}^\intercal + \mathrm{K}^\intercal\right) \,\mathbf{j} &=& -\mathrm D \, \mathbf m \end{array}\end{split}\]

2. Indiret Ansatz

Representation formula:

\[\begin{split} \begin{array}{rcl} \boldsymbol H(x) &=& \mathrm{DL}\left( \gamma_R\,\boldsymbol H^t\right)(x) \\ &=& \dfrac{ \kappa }{ i\omega\mu} \underbrace{ \nabla \times \displaystyle\int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \boldsymbol{j}^t(y) \, \mathrm{d}\sigma_y }_{\displaystyle{ \mathrm{DL} (\boldsymbol{j}^t) } } \end{array}\end{split}\]

Apply rotated Dirichlet trace and use \(\boldsymbol j = \boldsymbol j^t - \boldsymbol j^i\):

\[\begin{split}\begin{array}{ l c rcl} && \gamma_D \,\boldsymbol H &=& \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j^t) \\ &\Rightarrow & \dfrac{\kappa}{i\omega\mu}\boldsymbol j &=& \dfrac{\kappa}{i\omega\mu} \gamma_D \,\mathrm{DL}( \boldsymbol j^t) \\ &\Rightarrow & \boldsymbol j^t &=& \gamma_D \,\mathrm{DL}(\boldsymbol j^t) + \boldsymbol j^i \end{array}\end{split}\]

Thus,

\[\begin{split} \begin{array}{r rcl } \textnormal{indirect ansatz } & \boldsymbol H(x) &=& \dfrac{\kappa}{i\omega\mu} \mathrm{DL}\left(\boldsymbol j^t\right) \\ \textnormal{variational formulation } & \forall \, \boldsymbol v\in H^{-\frac12}(\mathrm{curl}_\Gamma, \Gamma): \; \left\langle \boldsymbol v, \boldsymbol j^t\right\rangle_{-\frac12} - \left\langle \boldsymbol v, \gamma_D \,\mathrm{DL} (\boldsymbol j^t) \right\rangle_{-\frac12} &=& \left\langle \boldsymbol v, \boldsymbol{j}^i \right\rangle_{-\frac12}\\ \textnormal{discretisation} & \left( \dfrac12 \mathrm{M}^\intercal + \mathrm{K}^\intercal\right) \,\mathbf{j}^t &=& \mathrm M \, \mathbf j^i \end{array}\end{split}\]

Notes:

• The indirect ansatz for Dirichlet problem is often called EFIE (electric field integral equation).

• The indirect ansatz for Neumann problem is often called MFIE (magnetic field integral equation).

• MFIE equation is only valid on closed boundaries whereas EFIE holds in a generalized form on open screens

• Also the magnetic field leads to boundary integral equations for \(\boldsymbol j\) and \(\boldsymbol j^t\). The boundary integral equations are integral equations of second type.

• For low frequencie problems it is necessary to introduce explicitly the electrostatic potential. This leads to an additional equation which is a weak form of the continuity equation relating traces on the boundary holds. Here the normal trace, i.e. the Neuman trace of the electrostatic potential pops up.

• Scattering at arbitrary dielectrics and pec bodies leads to a system of equations which is coupled by corresponding interface conditions.

• In case the wave number \(\kappa\) corresponds to an interior eigenvalue of \(\mathbf{curl}\mathbf{curl}\) the BIE is not uniquely solvable. Instead of EFIE one consideres the combined electric field integral equation (CFIE).

• consider \(H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\) conforming finite elements for test and trial space. Here is the hypersingular operator entry \(lk\)

\[ \mathrm{D}_{lk} = \int\limits_\Gamma \int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \langle \boldsymbol n(y)\times \boldsymbol v_l(y), \boldsymbol n(x) \times \boldsymbol \varphi_k(x)\rangle\, \mathrm{d}\sigma_y \, \mathrm{d}\sigma_x - \int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi}\, \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \, \mathrm{curl}_\Gamma \,\boldsymbol v_l(y)\, \mathrm{curl}_\Gamma\,\boldsymbol \varphi_k(x) \mathrm{d}\sigma_y \mathrm{d}\sigma_x \]

• consider a trial function \(\boldsymbol \psi_k \in H^{-\frac12}(\mathrm{div}_\Gamma,\Gamma)\) and a test function \(\boldsymbol v_l \in H^{-\frac12}(\mathrm{curl}_\Gamma,\Gamma)\). Here is the adjoint double layer potential operator entry \(lk\)

\[ \mathrm{K}'_{lk} = \int\limits_\Gamma \int\limits_\Gamma \displaystyle{ \frac{1}{4\,\pi} \, \big\langle \boldsymbol n(y)\times \boldsymbol v_l(y), \nabla_{x} \frac{e^{i\,\kappa\,\|x-y\|}}{\| x-y\|} } \times \boldsymbol \psi_k(y) \big\rangle\, \mathrm{d}\sigma_y \, \mathrm{d}\sigma_x \]