Laplace

Layer Potentials

\[\begin{split} \begin{array}{r rcl} \mathrm{SL}\left( j \right) (\boldsymbol x) &=& \displaystyle{ \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \frac{1}{\| \boldsymbol x- \boldsymbol y\|} } \, j(\boldsymbol y)\, \mathrm{d}\sigma_y } \\ \mathrm{DL}\left(m \right)(\boldsymbol x) &=& \displaystyle{ \int\limits_\Gamma \displaystyle{\frac{1}{4\,\pi}\, \frac{ \langle \boldsymbol n_y, \boldsymbol x- \boldsymbol y \rangle }{\| \boldsymbol x- \boldsymbol y\|^3} } \, m(\boldsymbol y)\, \mathrm{d}\sigma_y } \end{array}\end{split}\]

Laplace Dirichlet BVP

Let \(u\) denote the electrostatic potential that arises under given Dirichlet boundary condition inside a source-free domain \(\Omega \in \mathbb R^3\). Thus, \(u\) solves the interior boundary value problem

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

From here we can choose an direct or an indirect ansatz.

1. Direct Ansatz

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

2. Indirect Ansatz

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

Laplace Neumann BVP

Let \(u\) denote the electrostatic potential that arises under given Neumann boundary condition inside a source-free domain \(\Omega \in \mathbb R^3\). Thus, \(u\) solves the boundary value problem

\[\begin{split} \left\{ \begin{array}{rcl l} \Delta u &=& 0\,, \quad & \Omega \subset \mathbb R^3\,, \\ \gamma_1 u &=& j\,, \quad & \Gamma = \partial \Omega\,. \end{array} \right. \end{split}\]

From here we can choose an direct or an indirect ansatz.

1. Direct Ansatz

\[\begin{split} \begin{array}{r rcl } \textnormal{ansatz} & u &=& \mathrm{SL}(j) - \mathrm{DL}(m) \\ \textnormal{variational formulation } & \forall v\in H^{\frac12}(\Gamma): \, \big\langle v, \gamma_1 \left(\mathrm{DL}(m)\right) \big\rangle_{-\frac12} &=& \big\langle v, j\big\rangle_{-\frac12} - \big\langle v, \gamma_1 \left(\mathrm{SL}(j)\right) \big\rangle_{-\frac12} \\ \textnormal{discretisation} & \left( \mathrm{D} + \mathrm{S}\right) \mathrm{m} &=& \left( \frac12 \mathrm{M} - \mathrm{K}' \right) \, \mathrm{j} \end{array} \end{split}\]

2. Indirect Ansatz

\[\begin{split} \begin{array}{r rcl } \textnormal{ansatz} & u &=& \mathrm{DL}(m) \\ \textnormal{variational formulation } & \forall \quad v\in H^{\frac12}(\Gamma):\, \big\langle v, \gamma_1 \left(\mathrm{DL}(m)\right) \big\rangle_{-\frac12} &=& -\big\langle v, j\big\rangle_{-\frac12} \\ \textnormal{discretisation} & \left( \mathrm{D} + \mathrm S\right) \, \mathrm{m} &=& -\mathrm{M}\,\mathrm{j} \end{array} \end{split}\]