Update examples/maxwell/weak_FEM_BEM_coupling/README.md

examples/maxwell/weak_FEM_BEM_coupling/README.md

 
-#The electromagnetic transmission-scattering problem
+## The electromagnetic transmission-scattering problem
 
 Dependencies:
 * GmshFEM, commit: bd0fb7ebb6b76f0541e25212b9cbcc42e7a0d0b7
 * Gmsh, commit: 2db112186d43fab77fca9d2b546c8f76b3ea6169
-* GmshDdm, commit: 1e539b2a25293011fd80a8b0600a2a55c27fdfbf
+* GmshDdm, commit: 2a2746ca938924279ad9efbe15e4dc000370ed7f
 * Bempp-cl, commit: 293135531f385f39ba29cf1c52cc2dba84614e6b
 
+
+## Problem description
+Let $` \Omega_{-} `$ be a unit sphere (the scatterer) and $`\Omega_{+} `$ the exterior domain. The boundary of $` \Omega_{-} `$  is denoted by $`\Gamma`$.
+We consider an incident electromagnetic plane wave propagating in $`\Omega_{+}`$. The total wave field $` \mathbf{E}`$ verifies the following  three-dimensional electromagnetic transmission-scattering problem:
+
+```math
+\mathbf{curl}((k_{-}\mathcal{Z}_{-})^{-1}\mathbf{curl }~\mathbf{E})- k_{-}\mathcal{Z}_{-}^{-1}\mathbf{E} =  \textbf{0} \text{ ~in~ } \Omega_{-}, \\
+\mathbf{curl}\mathbf{curl }~\mathbf{E}- k_{+}^{2}\mathbf{E} =  \textbf{0} \text{ ~in~ } \Omega_{+},\\ 
+```
+with $` k_\pm`$ and $`\mathcal{Z}_\pm`$ the wavenumbers and the impedances associated with  $` \Omega_\pm`$ respectively.  The total electric field satisfied the transmission conditions on $`\Gamma`$. For the exterior problem to be well posed and physically admissible, we assume that the scattered field verifies   the Silver-Muller radiation condition at infinity.
+
 ## Method overview:
 
 The method is an efficient weak coupling formulation between the boundary element method
@@ -15,22 +26,6 @@ domain decomposition method involving quasi-optimal transmission operators. The
 transmission boundary conditions are constructed through a localization process based on complex
 rational Padé approximants of the nonlocal Magnetic-to-Electric operators (see I. Badia, B. Caudron, X. Antoine, C. Geuzaine. "A well-conditioned weak coupling of boundary element and high-order finite element methods for time-harmonic electromagnetic scattering by inhomogeneous objects". SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2022)
 
-## Problem description
-Let ```math \Omega_{-} ``` be a unit sphere (the scatter) and ```math \Omega_{+} ``` the exterior domain.
-We consider an incident electromagnetic plane wave propagating in ```math \Omega_{+} ```. The total wave field ```math \mathbf{E}``` verifies the following  three-dimensional electromagnetic transmission-scattering problem:
-```math
-\begin{subequations}
-\begin{align}
-\curl \left(\frac{1}{k_{-}\mathcal{Z}_{-}}\curl\E\right) -  k_{-}\mathcal{Z}_{-}^{-1}\E &=  \textbf{0}, \text{ ~in~ } \Omega_{-},  \\
-\curl\curl\E - k_{+}^{2}\E & =  \textbf{0},  \text{ ~in~ } \Omega_{+}, \\
-\frac{1}{\iota k_{+}}\curl \textbf{E}_{\textbf{sc}} \times \frac{\textbf{x}}{\left\lVert\textbf{x} \right\rVert}-\textbf{E}_{\textbf{sc}} &=   \underset{r \to +\infty}{\mathcal{O}}(r^{-2}), \\
-\boldsymbol{\gamma}^{-}_{t}\E &=\boldsymbol{\gamma}^{+}_ {t}\E, \text{ ~on~ } \Gamma, \\
-\boldsymbol{\gamma}^{-}_{t} \left(\frac{1}{k_{-}\mathcal{Z}_{-}}\curl\E\right) &=\frac{1}{k_{+}\mathcal{Z}_{+}}\boldsymbol{\gamma}^{+}_ {t}\left(\curl\E\right), \text{ ~on~ } \Gamma,
-\end{align}
-\end{subequations}
-```
-with ```math k_\pm``` and ```math \mathcal{Z}_\pm``` the wavenumbers and the impedances associated with  ```math \Omega_\pm``` respectively.
-
 ## Installation
 
 Run
@@ -39,23 +34,24 @@ Run
   mkdir build && cd build
   cmake ..
   make
+  cd ..
 ```
 
 
 ## Usage
 
-Simply run:
+Run:
 ```
-  ./example [PARAM]
+  ./build/example [PARAM]
 ```
 with `[PARAM]`:
 * `-pade [x]`  (with `[x]` equal to 1 or 0)  is a parameter to indicate to use Padé or not
 * `-pointsByWl [x]` where `[x]` is the mesh size.
-* `-FEorderAlpha [x]` is the order of the finite element hierarchical basis of the interior electric field.
-* `-FEorderAlpha [x]` is the order of the finite element hierarchical basis of some fields.
+* `-FEorderAlpha [x]` is the order of the finite element hierarchical basis associated with the interior electric field.
+* `-FEorderAlpha [x]` is the order of the finite element hierarchical basis associated with certain fields.
 * `-help` to display all other available parameters
 
 The folder contains
 * the main script (`main.cpp`)
 * A python script for the BEM part (`bem.py`)
-* The pybind11 folder for information exchange between python and C++
+* A folder for information exchange between python and C++ (`pybind11`)