diff --git a/physnum/rap6/Convergence.m~ b/physnum/rap6/Convergence.m~
deleted file mode 100644
index bbf181e..0000000
--- a/physnum/rap6/Convergence.m~
+++ /dev/null
@@ -1,80 +0,0 @@
-% Ce script Matlab automatise la production de resultats
-% lorsqu'on doit faire une serie de simulations en
-% variant un des parametres d'entree.
-% 
-% Il utilise les arguments du programme (voir ConfigFile.h)
-% pour remplacer la valeur d'un parametre du fichier d'input
-% par la valeur scannee.
-%
-
-%% Parametres %%
-%%%%%%%%%%%%%%%%
-
-repertoire = './'; % Chemin d'acces au code compile (NB: enlever le ./ sous Windows)
-executable = 'exercice6'; % Nom de l'executable (NB: ajouter .exe sous Windows)
-input = 'configuration.in'; % Nom du fichier d'entree de base
-
-nsimul = 501; % Nombre de simulations a faire
-
-N = linspace(10, nsimul+9, nsimul);
-
-p = 0.501;
-
-outfile = 'outs/conv.dat';
-outdir = 'm_outs/';
-
-%% Simulations %%
-%%%%%%%%%%%%%%%%%
-
-output = cell(1, nsimul); % Tableau de cellules contenant le nom des fichiers de sortie
-interval = cell(1, nsimul);
-for i = 1:nsimul
-    output{i} = [outdir, 'N=', num2str(N(i)), 'conv.out'];
-    interval{i} = sprintf('{{0.06, %d}, {0.12, %d}}', N(i), N(i));
-    % Execution du programme en lui envoyant la valeur a scanner en argument
-    cmd = sprintf('%s%s %s \"interval=%s\" output=%s trapezium=%f trivial=true', repertoire, executable, input, interval{i}, output{i}, p);
-    disp(cmd);
-    system(cmd);
-end
-
-%% Analyse %%
-%%%%%%%%%%%%%
-
-% Parcours des resultats de toutes les simulations
-
-phi_th = 0.0036;
-phi_err = zeros(1,nsimul);
-
-for i = 1:nsimul % Parcours des resultats de toutes les simulations
-    data = load(output{i});
-    phi_err(i) = abs(data(1,3) - phi_th);
-    fprintf('N = %d, phi(0) = %.15g, |phi_th - phi(0)| = %.15g\n', N(i), data(1,3), phi_err(i));
-end
-
-%% Linear regression %%
-%%%%%%%%%%%%%%%%%%%%%%%
-
-m = polyfit(log(N), log(phi_err), 1);
-linreg = sprintf('Loglog fit: %.8g * x**(%. + %.8g', m(1), m(2));
-disp(linreg);
-
-%% File writing %%
-%%%%%%%%%%%%%
-
-% print data on file
-out = fopen(outfile, 'w');
-fprintf(out, 'N  phi_err\n');
-for i = 1:nsimul
-    fprintf(out, '%d %.15g\n', N(i), phi_err(i));
-end
-fclose(out);
-
-%% Figures %%
-%%%%%%%%%%%%%
-    
-% make figure with data
-figure
-loglog(N, phi_err,'k+')
-xlabel('N')
-ylabel('phi_err')
-grid on
diff --git a/physnum/rap6/Polarization.m b/physnum/rap6/Polarization.m
index 415beb3..7f100f9 100644
--- a/physnum/rap6/Polarization.m
+++ b/physnum/rap6/Polarization.m
@@ -1,149 +1,161 @@
 % Ce script Matlab automatise la production de resultats
 % lorsqu'on doit faire une serie de simulations en
 % variant un des parametres d'entree.
 % 
 % Il utilise les arguments du programme (voir ConfigFile.h)
 % pour remplacer la valeur d'un parametre du fichier d'input
 % par la valeur scannee.
 %
 
 %% Parametres %%
 %%%%%%%%%%%%%%%%
 
 repertoire = './'; % Chemin d'acces au code compile (NB: enlever le ./ sous Windows)
 executable = 'exercice6'; % Nom de l'executable (NB: ajouter .exe sous Windows)
 input = 'configuration.in'; % Nom du fichier d'entree de base
 
 nsimul = 2001; % Nombre de simulations a faire
 coeff = 4.0/5;
 
 N_1 = linspace(10, nsimul+9, nsimul);
 N_2 = floor(coeff * N_1);
 
 outE = cell(1, nsimul);
 outdir = 'm_outs/';
 
 b = 0.02;
 R = 0.12;
 
 p = 0.5;
 
 eps0 = 8.854e-12;
 a0 = 1e4;
 
 L = 1;
 
 outfile = 'outs/polarization.dat';
 
 %% Simulations %%
 %%%%%%%%%%%%%%%%%
 
 output = cell(1, nsimul); % Tableau de cellules contenant le nom des fichiers de sortie
 interval = cell(1, nsimul);
 for i = 1:nsimul
     output{i} = [outdir, 'N=', num2str(N_1(i)), 'conv_polar.out'];
     interval{i} = sprintf('{{0.02, %d}, {0.12, %d}}', N_1(i), N_2(i));
     % Execution du programme en lui envoyant la valeur a scanner en argument
     cmd = sprintf('%s%s %s \"interval=%s\" output=%s trapezium=%f eps0=%g trivial=false', repertoire, executable, input, interval{i}, output{i}, p, eps0);
     disp(cmd);
     system(cmd);
 end
 
 
 %% Analyse %%
 %%%%%%%%%%%%%
 
 % Compute div(D) and rho
 rho_pol = zeros(nsimul, 1);
 q_pol = zeros(nsimul, 1);
 
 q_ref = - pi * L * eps0 * a0 * b^2 * 7/16;
 
 for i=1:nsimul
     data = load(output{i});
     [rquart, divD, divE, rmid, E, alph] = div_electric_D(data(:,1), data(:,3), b, R, eps0);
     Nb = N_1(i);
     rho_pol(i) = divE(Nb) * eps0 - divD(Nb);
     q_pol(i) = pi * L * rho_pol(i) * rquart(Nb) * (rquart(Nb+1) - rquart(Nb-1));
 end
 
 %q_lim = q_pol(nsimul);
 q_err = abs(q_pol - q_ref);
 
 %% File writing %%
 %%%%%%%%%%%%%
 
 % print data on file
 out = fopen(outfile, 'w');
 fprintf(out, 'N1  N2  rho_pol q_pol q_err\n');
 for i = 1:nsimul
     fprintf(out, '%d %d %.15g %.15g %.15g\n', N_1(i), N_2(i), rho_pol(i) / (a0*eps0), q_pol(i), q_err(i));
 end
 fclose(out);
 
+%% Linear fit %%
+%%%%%%%%%%%%%%%%
+
+m = polyfit(N_1', rho_pol/(a0*eps0), 1);
+fprintf('Rho pol fit: %.8g * N + %.8g\n', m(1), m(2));
+
+%N_1_cut = N_1(40:end);
+%q_err_cut = q_err(40:end);
+m = polyfit(log(N_1(40:end)'), log(q_err(40:end)), 1);
+fprintf('Q pol fit: N^%.8g * %.8g\n', m(1), exp(m(2)));
+
+
 %% Figures %%
 %%%%%%%%%%%%%
     
 % make figure with data
 figure
-plot(N_1, rho_pol/(a0*eps), 'k+')
+plot(N_1, rho_pol/(a0*eps0), 'k+')
 xlabel('N_1')
 ylabel('rho_{pol}')
 grid on
 
 % make figure with data
 figure
 loglog(N_1, q_err, 'k+')
 xlabel('N_1')
 ylabel('q_{pol}')
 grid on
 
 %% Helper %%
 %%%%%%%%%%%%
 
 function [rmid, E] = electric_field(r, phi)
     N = length(r) - 1;
     rmid = zeros(N, 1);
     E = zeros(N, 1);
     
     for i=1:N
         h = r(i+1) - r(i);
         rmid(i) = r(i) + h/2;
         E(i) = - (phi(i+1) - phi(i))/ h;
     end
 end
 
 function eps = epsilon(r, b, R)
     if (r <= (b*(1-1e-12)))
         eps = 1;
     else
         eps = 8 - 6 * (r-b)/(R-b);
     end
 end
 
 function a = alpha(r, b, R, E, eps0)
     a = r * eps0 * epsilon(r, b, R) * E;
 end
 
 function [rquart, divD, divE, rmid, E, alph] = div_electric_D(r, phi, b, R, eps0)
 
     [rmid, E] = electric_field(r, phi);
     N = length(rmid)-1;
     rquart = zeros(N, 1);
     divD = zeros(N, 1);
     divE = zeros(N, 1);
     alph = zeros(N+1, 1);
     
     %alph(1) = alpha(rmid(1), b, R, E(1), eps0);
     a_init = alpha(rmid(1), b, R, E(1), eps0);
     for i=1:N
         %a_init = alpha(rmid(i), b, R, E(i), eps0);
         a_next = alpha(rmid(i+1), b, R, E(i+1), eps0);
         alph(i+1) = a_next;
         h = rmid(i+1) - rmid(i);
         rquart(i) = rmid(i) + h/2;
         divD(i) = (a_next - a_init) / (h * rquart(i));
         divE(i) = (rmid(i+1) * E(i+1) - rmid(i) * E(i)) / (h * rquart(i));
         a_init = a_next;
     end
 end
diff --git a/physnum/rap6/Verify.m b/physnum/rap6/Verify.m
index 131f8d3..e448b61 100644
--- a/physnum/rap6/Verify.m
+++ b/physnum/rap6/Verify.m
@@ -1,166 +1,166 @@
 % Ce script Matlab automatise la production de resultats
 % lorsqu'on doit faire une serie de simulations en
 % variant un des parametres d'entree.
 % 
 % Il utilise les arguments du programme (voir ConfigFile.h)
 % pour remplacer la valeur d'un parametre du fichier d'input
 % par la valeur scannee.
 %
 
 %% Parametres %%
 %%%%%%%%%%%%%%%%
 
 repertoire = './'; % Chemin d'acces au code compile (NB: enlever le ./ sous Windows)
 executable = 'exercice6'; % Nom de l'executable (NB: ajouter .exe sous Windows)
 input = 'configuration.in'; % Nom du fichier d'entree de base
 
-N_1 = 500;
-N_2 = 400;
+N_1 = 100;
+N_2 = 80;
 b = 0.02;
 R = 0.12;
 
 p = 0.5;
 
 eps0 = 8.854e-12;
 a0 = 1e4;
 
 output = 'm_outs/verify.out';
 
 outfile = 'outs/verify.dat';
 
 %% Simulations %%
 %%%%%%%%%%%%%%%%%
 
 interval = sprintf('{{%.15g, %d}, {%.15g, %d}}', b, N_1, R, N_2);
 % Execution du programme en lui envoyant la valeur a scanner en argument
 cmd = sprintf('%s%s %s \"interval=%s\" output=%s eps0=%g trapezium=%f trivial=false', repertoire, executable, input, interval, output, eps0, p);
 disp(cmd);
 system(cmd);
 
 
 %% Analyse %%
 %%%%%%%%%%%%%
 
 % Compute div(D) and rho
 
  data = load(output);
  [rquart, divD, divE, rmid, E, alph] = div_electric_D(data(:,1), data(:,3), b, R, eps0);
  
  rho = density(rquart, a0, eps0, b);
  rho_pol = divE * eps0 - divD;
     
  %for i=1:length(rquart)
  %   fprintf('r = %.15g, div(D)(r) = %.15g, rho(r) = %.15g\n', rquart(i), divD(i)/(a0*eps0), rho(i)/(a0*eps0));
  %end
 
 %% File writing %%
 %%%%%%%%%%%%%
 
 % print data on file
 out = fopen(outfile, 'w');
 fprintf(out, 'r divD rho rho_pol err\n');
 for i = 1:length(rquart)
     fprintf(out, '%.15g %.15g %.15g %.15g %.15g\n', rquart(i), divD(i)/(a0*eps0), rho(i)/(a0*eps0), rho_pol(i)/(a0*eps0), abs(rho(i) - divD(i))/ (a0*eps0));
 end
 fclose(out);
 
 %% Figures %%
 %%%%%%%%%%%%%
     
 % make figure with data
 figure
 plot(rquart, divD/(a0*eps0))%,'k+')
 xlabel('r')
 ylabel('div(D)(r)/(a0*eps0)')
 grid on
 
 figure
 semilogy(rquart, abs(divD - rho)/(a0*eps0))%,'k+')
 xlabel('r')
 ylabel('err')
 grid on
 
 figure
 plot(rquart, rho_pol/(a0*eps0))%,'k+')
 xlabel('r')
 ylabel('eps_0 * div(E)(r)-div(D)(r)')
 grid on
 
 figure
 plot(data(:,1), data(:,3))%,'k+')
 xlabel('r')
 ylabel('phi(r)')
 grid on
 
 figure
 plot(rmid, E) %,'k+')
 xlabel('r')
 ylabel('E(r)')
 grid on
 
 figure
 plot(rmid, alph) %'k+')
 xlabel('r')
 ylabel('alpha(r)')
 grid on
 
 %% Helper %%
 %%%%%%%%%%%%
 
 function [rmid, E] = electric_field(r, phi)
     N = length(r) - 1;
     rmid = zeros(N, 1);
     E = zeros(N, 1);
     
     for i=1:N
         h = r(i+1) - r(i);
         rmid(i) = r(i) + h/2;
         E(i) = - (phi(i+1) - phi(i))/ h;
     end
 end
 
 function eps = epsilon(r, b, R)
     if (r <= (b*(1-1e-12)))
         eps = 1;
     else
         eps = 8 - 6 * (r-b)/(R-b);
     end
 end
 
 function a = alpha(r, b, R, E, eps0)
     a = r * eps0 * epsilon(r, b, R) * E;
 end
 
 function [rquart, divD, divE, rmid, E, alph] = div_electric_D(r, phi, b, R, eps0)
 
     [rmid, E] = electric_field(r, phi);
     N = length(rmid)-1;
     rquart = zeros(N, 1);
     divD = zeros(N, 1);
     divE = zeros(N, 1);
     alph = zeros(N+1, 1);
     
     %alph(1) = alpha(rmid(1), b, R, E(1), eps0);
     a_init = alpha(rmid(1), b, R, E(1), eps0);
     for i=1:N
         %a_init = alpha(rmid(i), b, R, E(i), eps0);
         a_next = alpha(rmid(i+1), b, R, E(i+1), eps0);
         alph(i+1) = a_next;
         h = rmid(i+1) - rmid(i);
         rquart(i) = rmid(i) + h/2;
         divD(i) = (a_next - a_init) / (h * rquart(i));
         divE(i) = (rmid(i+1) * E(i+1) - rmid(i) * E(i)) / (h * rquart(i));
         a_init = a_next;
     end
 end
 
 function rho = density(r, a0, eps0, b)
     rho = zeros(length(r), 1);
     for i=1:length(r)
         if r(i) < b
             rho(i) = eps0 * a0 * (1 - (r(i)/b)^2);
         else
             rho(i) = 0;
         end
     end
 end
diff --git a/physnum/rap6/Verify.m~ b/physnum/rap6/Verify.m~
deleted file mode 100644
index 5f4349b..0000000
--- a/physnum/rap6/Verify.m~
+++ /dev/null
@@ -1,166 +0,0 @@
-% Ce script Matlab automatise la production de resultats
-% lorsqu'on doit faire une serie de simulations en
-% variant un des parametres d'entree.
-% 
-% Il utilise les arguments du programme (voir ConfigFile.h)
-% pour remplacer la valeur d'un parametre du fichier d'input
-% par la valeur scannee.
-%
-
-%% Parametres %%
-%%%%%%%%%%%%%%%%
-
-repertoire = './'; % Chemin d'acces au code compile (NB: enlever le ./ sous Windows)
-executable = 'exercice6'; % Nom de l'executable (NB: ajouter .exe sous Windows)
-input = 'configuration.in'; % Nom du fichier d'entree de base
-
-N_1 = 500;
-N_2 = 400;
-b = 0.02;
-R = 0.12;
-
-p = 0.5;
-
-eps0 = 8.854e-12;
-a0 = 1e4;
-
-output = 'm_outs/verify.out';
-
-outfile = 'outs/verify.dat';
-
-%% Simulations %%
-%%%%%%%%%%%%%%%%%
-
-interval = sprintf('{{%.15g, %d}, {%.15g, %d}}', b, N_1, R, N_2);
-% Execution du programme en lui envoyant la valeur a scanner en argument
-cmd = sprintf('%s%s %s \"interval=%s\" output=%s eps0=%g trapezium=%f trivial=false', repertoire, executable, input, interval, output, eps0, p);
-disp(cmd);
-system(cmd);
-
-
-%% Analyse %%
-%%%%%%%%%%%%%
-
-% Compute div(D) and rho
-
- data = load(output);
- [rquart, divD, divE, rmid, E, alph] = div_electric_D(data(:,1), data(:,3), b, R, eps0);
- 
- rho = density(rquart, a0, eps0, b);
- rho_pol = divE * eps0 - divD;
-    
- %for i=1:length(rquart)
- %   fprintf('r = %.15g, div(D)(r) = %.15g, rho(r) = %.15g\n', rquart(i), divD(i)/(a0*eps0), rho(i)/(a0*eps0));
- %end
-
-%% File writing %%
-%%%%%%%%%%%%%
-
-% print data on file
-out = fopen(outfile, 'w');
-fprintf(out, 'r divD rho rho_pol err\n');
-for i = 1:length(rquart)
-    fprintf(out, '%.15g %.15g %.15g %.15g %.15g\n', rquart(i), divD(i)/(a0*eps0), rho(i)/(a0*eps0), rho_pol(i)/(a0*eps0), abs(rho(i) - divD(i))/ (a0*eps0));
-end
-fclose(out);
-
-%% Figures %%
-%%%%%%%%%%%%%
-    
-% make figure with data
-figure
-plot(rquart, divD/(a0*eps0))%,'k+')
-xlabel('r')
-ylabel('div(D)(r)/(a0*eps0)')
-grid on
-
-figure
-plot(rquart, rho_pol/(a0*eps0))%,'k+')
-xlabel('r')
-ylabel('eps_0 * div(E)(r)-div(D)(r)')
-grid on
-
-figure
-plot(rquart, rho_pol/(a0*eps0))%,'k+')
-xlabel('r')
-ylabel('eps_0 * div(E)(r)-div(D)(r)')
-grid on
-
-figure
-plot(data(:,1), data(:,3))%,'k+')
-xlabel('r')
-ylabel('phi(r)')
-grid on
-
-figure
-plot(rmid, E) %,'k+')
-xlabel('r')
-ylabel('E(r)')
-grid on
-
-figure
-plot(rmid, alph) %'k+')
-xlabel('r')
-ylabel('alpha(r)')
-grid on
-
-%% Helper %%
-%%%%%%%%%%%%
-
-function [rmid, E] = electric_field(r, phi)
-    N = length(r) - 1;
-    rmid = zeros(N, 1);
-    E = zeros(N, 1);
-    
-    for i=1:N
-        h = r(i+1) - r(i);
-        rmid(i) = r(i) + h/2;
-        E(i) = - (phi(i+1) - phi(i))/ h;
-    end
-end
-
-function eps = epsilon(r, b, R)
-    if (r <= (b*(1-1e-12)))
-        eps = 1;
-    else
-        eps = 8 - 6 * (r-b)/(R-b);
-    end
-end
-
-function a = alpha(r, b, R, E, eps0)
-    a = r * eps0 * epsilon(r, b, R) * E;
-end
-
-function [rquart, divD, divE, rmid, E, alph] = div_electric_D(r, phi, b, R, eps0)
-
-    [rmid, E] = electric_field(r, phi);
-    N = length(rmid)-1;
-    rquart = zeros(N, 1);
-    divD = zeros(N, 1);
-    divE = zeros(N, 1);
-    alph = zeros(N+1, 1);
-    
-    %alph(1) = alpha(rmid(1), b, R, E(1), eps0);
-    a_init = alpha(rmid(1), b, R, E(1), eps0);
-    for i=1:N
-        %a_init = alpha(rmid(i), b, R, E(i), eps0);
-        a_next = alpha(rmid(i+1), b, R, E(i+1), eps0);
-        alph(i+1) = a_next;
-        h = rmid(i+1) - rmid(i);
-        rquart(i) = rmid(i) + h/2;
-        divD(i) = (a_next - a_init) / (h * rquart(i));
-        divE(i) = (rmid(i+1) * E(i+1) - rmid(i) * E(i)) / (h * rquart(i));
-        a_init = a_next;
-    end
-end
-
-function rho = density(r, a0, eps0, b)
-    rho = zeros(length(r), 1);
-    for i=1:length(r)
-        if r(i) < b
-            rho(i) = eps0 * a0 * (1 - (r(i)/b)^2);
-        else
-            rho(i) = 0;
-        end
-    end
-end
diff --git a/physnum/rap6/graphs/conv_nontriv.tex b/physnum/rap6/graphs/conv_nontriv.tex
index b467c07..a94a754 100644
--- a/physnum/rap6/graphs/conv_nontriv.tex
+++ b/physnum/rap6/graphs/conv_nontriv.tex
@@ -1,51 +1,51 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 9cm,
     xlabel={$N_1 (= \frac{5}{4}N_2)$},
     xmin = 10, xmax = 510,
     legend style = 
     {
         at = {(0.95, 0.95)},
         anchor = north east
     },
     %cycle list name = color,
     grid style = dashed,
     ymajorgrids = true
 }
 
 \begin{loglogaxis} 
 [
     ylabel={$\abs*{\phi(b) - \phi_{lim}(b)}$ [\si{\volt}]},
     ymin = 5e-7, ymax = 5e-3
 ]
 
 \addplot+ 
     [   
         color = blue,
         mark = x,
         mark options = {
             draw = blue,
             fill = none
         },
         only marks
     ]
 gnuplot [raw gnuplot] {
     plot "outs/conv_nontrivial.dat" using 1:3 every 4;
 };
 
 \addplot
 [
     mark = none,
     color = black,
     domain = 1:500
 ] {0.244477989 * x^(-2.0835933)};
-\addlegendentry{Convérgence: $\mathcal{O}\left(\frac{1}{N_1^2}\right)$}
+\addlegendentry{\raisebox{4pt}[1.5em]{Convérgence: $\mathcal{O}\left(\frac{1}{N_1^2}\right)$}}
 
 \end{loglogaxis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/graphs/conv_triv.tex b/physnum/rap6/graphs/conv_triv.tex
index 0592ec5..2ec49a5 100644
--- a/physnum/rap6/graphs/conv_triv.tex
+++ b/physnum/rap6/graphs/conv_triv.tex
@@ -1,51 +1,51 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 9cm,
     xlabel={$N_1 (= N_2)$},
     xmin = 10, xmax = 510,
     legend style = 
     {
         at = {(0.95, 0.95)},
         anchor = north east
     },
     %cycle list name = color,
     grid style = dashed,
     ymajorgrids = true
 }
 
 \begin{loglogaxis} 
 [
     ylabel={$\abs*{\phi_0 - \phi(0)}$ [\si{\volt}]},
     ymin = 1e-11, ymax = 1e-7
 ]
 
 \addplot+ 
     [   
         color = blue,
         mark = x,
         mark options = {
             draw = blue,
             fill = none
         },
         only marks
     ]
 gnuplot [raw gnuplot] {
     plot "outs/conv.dat" using 1:2;
 };
 
 \addplot
 [
     mark = none,
     color = black,
     domain = 10:510
 ] {3.4568239e-06 * x^(-1.8637587)};
-\addlegendentry{Convérgence: $\mathcal{O}\left(\frac{1}{N^2}\right)$}
+\addlegendentry{\raisebox{4pt}[1.5em]{Convérgence: $\mathcal{O}\left(\frac{1}{N^2}\right)$}}
 
 \end{loglogaxis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/graphs/divD.tex b/physnum/rap6/graphs/divD.tex
index ec9a864..18401ee 100644
--- a/physnum/rap6/graphs/divD.tex
+++ b/physnum/rap6/graphs/divD.tex
@@ -1,44 +1,50 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 10cm,
     xlabel={$r$ [\si{\metre}]},
     xmin = 0, xmax = 0.12,
     legend style = 
     {
-        at = {(0.05, 0.95)},
-        anchor = north west
+        at = {(0.95, 0.95)},
+        anchor = north east
     },
     %cycle list name = color,
     grid style = dashed,
-    ymajorgrids = true
+    ymajorgrids = true,
+    x tick label style={
+       /pgf/number format/.cd,
+       fixed,
+       fixed zerofill,
+       precision=2
+    }
 }
 
 \begin{axis} 
 [
     ylabel={$\nabla \cdot D / (a_0 \cdot \epsilon_0)$},
     ymin = -0.1, ymax = 1.1
 ]
 
 \addplot+ 
     [   
         color = orange,
         mark = +,
         mark options = {
             draw = orange,
             fill = none
         },
         only marks
     ]
 gnuplot [raw gnuplot] {
     plot "outs/verifyNhigh.dat" using 1:2 every 5;
 };
 \addlegendentry{$N_1 = 500$}
 
 \end{axis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/graphs/divD_diff_rho.tex b/physnum/rap6/graphs/divD_diff_rho.tex
index fa7f9de..d1fd467 100644
--- a/physnum/rap6/graphs/divD_diff_rho.tex
+++ b/physnum/rap6/graphs/divD_diff_rho.tex
@@ -1,44 +1,50 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 10cm,
     xlabel={$r$ [\si{\metre}]},
     xmin = 0, xmax = 0.12,
     legend style = 
     {
-        at = {(0.05, 0.95)},
-        anchor = north west
+        at = {(0.95, 0.95)},
+        anchor = north east
     },
     %cycle list name = color,
     grid style = dashed,
-    ymajorgrids = true
+    ymajorgrids = true,
+    x tick label style={
+       /pgf/number format/.cd,
+       fixed,
+       fixed zerofill,
+       precision=2
+    }
 }
 
 \begin{semilogyaxis} 
 [
     ylabel={$\abs*{\rho_{lib} - \nabla \cdot D} / (a_0 \cdot \epsilon_0)$},
-    ymin = 1e-11, ymax = 1e-3
+    ymin = 5e-10, ymax = 1e-5
 ]
 
 \addplot+ 
     [   
         color = blue,
         mark = x,
         mark options = {
             draw = blue,
             fill = none
         },
         only marks
     ]
 gnuplot [raw gnuplot] {
     plot "outs/verifyNhigh.dat" using 1:5 every 5;
 };
 \addlegendentry{$N_1 = 500$}
 
 \end{semilogyaxis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/graphs/qpol_conv.tex b/physnum/rap6/graphs/qpol_conv.tex
index 65ef4eb..5894f84 100644
--- a/physnum/rap6/graphs/qpol_conv.tex
+++ b/physnum/rap6/graphs/qpol_conv.tex
@@ -1,51 +1,52 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 10cm,
     xlabel={$N_1 (= \frac{5}{4}N_2)$},
-    xmin = 10, xmax = 1.1e3,
+    xmin = 10, xmax = 5e2,
     legend style = 
     {
-        at = {(0.05, 0.95)},
-        anchor = north west
+        at = {(0.05, 0.05)},
+        anchor = south west
     },
     %cycle list name = color,
     grid style = dashed,
     ymajorgrids = true
 }
 
 \begin{loglogaxis} 
 [
     ylabel={$\abs*{Q_{{pol}_{num}} - Q_{{pol}_{ref}}}$ [\si{\coulomb}]},
-    ymin = 1e-15, ymax = 1.1e-13
+    ymin = 8e-15, ymax = 1.1e-13
 ]
 
 \addplot+ 
     [   
         color = orange,
         mark = x,
         mark options = {
             draw = orange,
             fill = none
         },
         only marks
     ]
 gnuplot [raw gnuplot] {
-    plot "outs/polar.dat" using 1:5 every 5;
+    plot "outs/polarization.dat" using 1:5 every 5;
 };
+\addlegendentry{\raisebox{4pt}[1.5em]{Convérgence: $\mathcal{O}\left(\frac{1}{N_1}\right)$}}
 
-%\addplot
-%[
-%    mark = none,
-%    color = black,
-%    domain = 10:1.1e3
-%] {};
-\addlegendentry{Convérgence: $O(frac{1}{N_1})$}
+\addplot
+[
+    mark = none,
+    color = black,
+    domain = 10:1.1e3
+] {2.8663369e-12 * x^(-0.98257081)};
+\addlegendentry{Fit: $y \approx 2.8 \cdot 10^{-12} \; x^{-1}$}
 
 \end{loglogaxis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/graphs/rhopol.tex b/physnum/rap6/graphs/rhopol.tex
index 631a2a9..ba9d1bb 100644
--- a/physnum/rap6/graphs/rhopol.tex
+++ b/physnum/rap6/graphs/rhopol.tex
@@ -1,51 +1,69 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 10cm,
     xlabel={$r$ [\si{\metre}]},
-    xmin = 0, xmax = 0.12,
+    xmin = 0, xmax = 0.05,
     legend style = 
     {
-        at = {(0.05, 0.95)},
-        anchor = north west
+        at = {(0.95, 0.05)},
+        anchor = south east
     },
     %cycle list name = color,
     grid style = dashed,
     ymajorgrids = true
+    %x tick label style={
+    %   /pgf/number format/.cd,
+    %   fixed,
+    %   fixed zerofill,
+    %   precision=2
+    %}
 }
 
 \begin{axis} 
 [
     ylabel={$\rho_{pol} / (a_0 \cdot \epsilon_0)$},
-    ymin = 1e-8, ymax = -35
+    ymin = -8, ymax = 2
 ]
 
 \addplot+ 
     [   
         color = blue,
-        smooth,
         mark = none
     ]
 gnuplot [raw gnuplot] {
     plot "outs/verifyNlow.dat" using 1:4;
 };
 \addlegendentry{$N_1 = 50$}
 
+\node[label={0:{$(-2.937)$}},
+             black,
+             circle,
+             fill,
+             inner sep=1pt,
+             on layer = foreground] at (axis cs:0.02052, -2.937) {};
+
 \addplot+ 
     [   
-        color = orange,
-        smooth,
+        color = red,
         mark = none
     ]
 gnuplot [raw gnuplot] {
-    plot "outs/verifyNhigh.dat" using 1:4 every 4;
+    plot "outs/verifyNhigher.dat" using 1:4 every 2;
 };
-\addlegendentry{$N_1 = 500$}
+\addlegendentry{$N_1 = 100$}
+
+\node[label={0:{$(-5.953)$}},
+             black,
+             circle,
+             fill,
+             inner sep=1pt,
+             on layer = foreground] at (axis cs:0.02026, -5.953) {};
 
 \end{axis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/graphs/rhopol_augm.tex b/physnum/rap6/graphs/rhopol_augm.tex
index 908c337..0f703d9 100644
--- a/physnum/rap6/graphs/rhopol_augm.tex
+++ b/physnum/rap6/graphs/rhopol_augm.tex
@@ -1,43 +1,54 @@
 \begin{tikzpicture}
 
 \pgfplotsset
 {
     %scale only axis,
     %scaled x ticks = base 10:2,
     %height = 18cm,
     width = 10cm,
     xlabel={$N_1 (= \frac{5}{4}N_2)$},
     xmin = 0, xmax = 2100,
     legend style = 
     {
-        at = {(0.05, 0.95)},
-        anchor = north west
+        at = {(0.95, 0.95)},
+        anchor = north east
     },
     %cycle list name = color,
     grid style = dashed,
     ymajorgrids = true
 }
 
 \begin{axis} 
 [
     ylabel={$\rho_{pol} / (a_0 \cdot \epsilon_0)$},
-    ymin = -5e6, ymax = 0
+    ymin = -130, ymax = 0
 ]
 
-\addplot+ 
+\addplot 
     [   
         color = red,
-        mark = x,
+        mark = asterisk,
         mark options = {
             draw = red,
             fill = none
         },
-        only marks
+        only marks,
+        on layer = main
     ]
 gnuplot [raw gnuplot] {
-    plot "outs/polarization.dat" using 1:3 every 5;
+    plot "outs/polarization.dat" using 1:3 every 20;
 };
+\addlegendentry{Pics de densité}
+
+\addplot+
+[
+    mark = none,
+    color = black,
+    domain = 0:2100,
+    on layer = foreground
+] {0.10837105 - 0.060344277 * x};
+\addlegendentry{Fit: $y \approx 0.11 - 0.06 \cdot x$}
 
 \end{axis}
 
 \end{tikzpicture}
diff --git a/physnum/rap6/m_outs/verify.out b/physnum/rap6/m_outs/verify.out
index 3b2b47f..c939516 100644
--- a/physnum/rap6/m_outs/verify.out
+++ b/physnum/rap6/m_outs/verify.out
@@ -1,901 +1,181 @@
-0 1.999998e-06 1.09543994641903 
-4e-05 1.5999912e-05 1.09543594642303 
-8e-05 3.199944e-05 1.09542394648303 
-0.00012 4.79982e-05 1.09540394674303 
-0.00016 6.3995808e-05 1.09537594744303 
-0.0002 7.999188e-05 1.09533994891903 
-0.00024 9.5986032e-05 1.09529595160303 
-0.00028 0.00011197788 1.09524395602303 
-0.00032 0.00012796704 1.09518396280303 
-0.00036 0.000143953128 1.09511597266303 
-0.0004 0.00015993576 1.09503998641903 
-0.00044 0.000175914552 1.09495600498303 
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-0.00076 0.000303560568 1.09399646770303 
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-0.00084 0.000335406792 1.09367672434303 
-0.00088 0.000351318 1.09350488344303 
-0.00092 0.000367220759999999 1.09332506578303 
-0.00096 0.000383114688 1.09313727352303 
-0.001 0.000398999400000001 1.09294150891903 
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 0.12 0 0 
diff --git a/physnum/rap6/outs/verify.dat b/physnum/rap6/outs/verify.dat
new file mode 100644
index 0000000..f1669e6
--- /dev/null
+++ b/physnum/rap6/outs/verify.dat
@@ -0,0 +1,180 @@
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diff --git a/physnum/rap6/outs/verifyNhigher.dat b/physnum/rap6/outs/verifyNhigher.dat
new file mode 100644
index 0000000..f1669e6
--- /dev/null
+++ b/physnum/rap6/outs/verifyNhigher.dat
@@ -0,0 +1,180 @@
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+0.11125 5.54168031593092e-07 0 0.00846085697942084 5.54168031593092e-07
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+0.11375 6.28359366475147e-07 0 0.00935345370311995 6.28359366475147e-07
+0.115 6.70263933191203e-07 0 0.00986517779935097 6.70263933191203e-07
+0.11625 7.15979232533122e-07 0 0.0104283128271382 7.15979232533122e-07
+0.1175 7.66056356137954e-07 0 0.0110499877996142 7.66056356137954e-07
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diff --git a/physnum/rap6/rapport.pgf-plot.gnuplot b/physnum/rap6/rapport.pgf-plot.gnuplot
index c12baa8..35a67ab 100644
--- a/physnum/rap6/rapport.pgf-plot.gnuplot
+++ b/physnum/rap6/rapport.pgf-plot.gnuplot
@@ -1,2 +1,2 @@
 set table "rapport.pgf-plot.table"; set format "%.5f"
-set format "%.7e";; plot "outs/conv_nontrivial.dat" using 1:3 every 4; 
+set format "%.7e";; plot "outs/polarization.dat" using 1:5 every 5; 
diff --git a/physnum/rap6/rapport.pgf-plot.table b/physnum/rap6/rapport.pgf-plot.table
index f23b656..e907a26 100644
--- a/physnum/rap6/rapport.pgf-plot.table
+++ b/physnum/rap6/rapport.pgf-plot.table
@@ -1,130 +1,405 @@
 
-# Curve 0 of 1, 125 points
-# Curve title: ""outs/conv_nontrivial.dat" using 1:3 every 4"
+# Curve 0 of 1, 400 points
+# Curve title: ""outs/polarization.dat" using 1:5 every 5"
 # x y type
-1.3000000e+01 1.0160991e-03  i
-1.7000000e+01 6.0710267e-04  i
-2.1000000e+01 4.0255407e-04  i
-2.5000000e+01 2.7193371e-04  i
-2.9000000e+01 2.0429694e-04  i
-3.3000000e+01 1.5905215e-04  i
-3.7000000e+01 1.2730893e-04  i
-4.1000000e+01 1.0418684e-04  i
-4.5000000e+01 8.4412031e-05  i
-4.9000000e+01 7.1564738e-05  i
-5.3000000e+01 6.1437273e-05  i
-5.7000000e+01 5.3312633e-05  i
-6.1000000e+01 4.6695454e-05  i
-6.5000000e+01 4.0438537e-05  i
-6.9000000e+01 3.6003351e-05  i
-7.3000000e+01 3.2257353e-05  i
-7.7000000e+01 2.9064746e-05  i
-8.1000000e+01 2.6321619e-05  i
-8.5000000e+01 2.3592904e-05  i
-8.9000000e+01 2.1567466e-05  i
-9.3000000e+01 1.9790544e-05  i
-9.7000000e+01 1.8223059e-05  i
-1.0100000e+02 1.6833329e-05  i
-1.0500000e+02 1.5407963e-05  i
-1.0900000e+02 1.4319383e-05  i
-1.1300000e+02 1.3341136e-05  i
-1.1700000e+02 1.2458792e-05  i
-1.2100000e+02 1.1660204e-05  i
-1.2500000e+02 1.0824211e-05  i
-1.2900000e+02 1.0173267e-05  i
-1.3300000e+02 9.5785206e-06  i
-1.3700000e+02 9.0336803e-06  i
-1.4100000e+02 8.5333105e-06  i
-1.4500000e+02 8.0017786e-06  i
-1.4900000e+02 7.5820626e-06  i
-1.5300000e+02 7.1939038e-06  i
-1.5700000e+02 6.8342136e-06  i
-1.6100000e+02 6.5002725e-06  i
-1.6500000e+02 6.1416110e-06  i
-1.6900000e+02 5.8553790e-06  i
-1.7300000e+02 5.5882066e-06  i
-1.7700000e+02 5.3384371e-06  i
-1.8100000e+02 5.1045901e-06  i
-1.8500000e+02 4.8512722e-06  i
-1.8900000e+02 4.6474229e-06  i
-1.9300000e+02 4.4557542e-06  i
-1.9700000e+02 4.2753141e-06  i
-2.0100000e+02 4.1052415e-06  i
-2.0500000e+02 3.9197404e-06  i
-2.0900000e+02 3.7694626e-06  i
-2.1300000e+02 3.6273285e-06  i
-2.1700000e+02 3.4927598e-06  i
-2.2100000e+02 3.3652283e-06  i
-2.2500000e+02 3.2253444e-06  i
-2.2900000e+02 3.1113971e-06  i
-2.3300000e+02 3.0030988e-06  i
-2.3700000e+02 2.9000820e-06  i
-2.4100000e+02 2.8020083e-06  i
-2.4500000e+02 2.6939307e-06  i
-2.4900000e+02 2.6054864e-06  i
-2.5300000e+02 2.5210827e-06  i
-2.5700000e+02 2.4404769e-06  i
-2.6100000e+02 2.3634444e-06  i
-2.6500000e+02 2.2782171e-06  i
-2.6900000e+02 2.2081993e-06  i
-2.7300000e+02 2.1411474e-06  i
-2.7700000e+02 2.0768963e-06  i
-2.8100000e+02 2.0152917e-06  i
-2.8500000e+02 1.9469016e-06  i
-2.8900000e+02 1.8905274e-06  i
-2.9300000e+02 1.8363792e-06  i
-2.9700000e+02 1.7843413e-06  i
-3.0100000e+02 1.7343052e-06  i
-3.0500000e+02 1.6785939e-06  i
-3.0900000e+02 1.6325368e-06  i
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diff --git a/physnum/rap6/rapport.tex b/physnum/rap6/rapport.tex
index 4317d85..c7d7fec 100644
--- a/physnum/rap6/rapport.tex
+++ b/physnum/rap6/rapport.tex
@@ -1,620 +1,674 @@
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 % ======= Le document commence ici ======
 
 \begin{document}
 % Le titre, l'auteur et la date
 \title{Physique Numérique I - II - Exercice 6}
 \date{\today}
 \author{Ancarola Raffaele, Cincotti Armando\\{\small \mail{raffaele.ancarola@epfl.ch}}, {\small \mail{armando.cincotti@epfl.ch}}}
 \maketitle
 \tableofcontents % Table des matieres
 
 % Quelques options pour les espacements entre lignes, l'identation 
 % des nouveaux paragraphes, et l'espacement entre paragraphes
 \baselineskip=16pt
 \parindent=15pt
 \parskip=5pt
 
 \section{Introduction} %-----------------------------------------
 En physique, les systèmes étudiés les plus intéressant, souvent ne possèdent
 pas de solution détérminable analityquement. Pour ceci il est nécéssaire d'utiliser
 des schémas numériques et de discrétiser le problème pour pouvoir l'étudier correctement.
 Dans le rapport qui suit, un système éléctrodynamique sera étudié 
 en utilisant la méthode des éléments finis pour la discrétisation du problème.
 Cette méthode sera donc expliquée et illustrée, ensuite testée et en fin employée
 sur un problème non trivial pour pouvoir en déduire différentes caractéristiques 
 intéressantes.
 
 \section{Analyse du problème}
 
 \paragraph{Potentiel éléctrique.}
 
 Soit un diéléctrique dans un volume $\Omega$ avec bord $\partial\Omega$, 
 il est cherché une fonction décrivant le potentiel eléctrique $\phi(\vec{x})$
 tel qu'il est à $V_0$ sur $\partial\Omega$. Le milieu est de constante diéléctrique $\epsilon_r(\vec{x})$
 et avec densité de charges libres $\rho_{lib}(\vec{x})$. Une des équations  de Maxwell pour l'éléctrodynamique
 veut :
 
 \begin{equation}\label{maxwell}
 \nabla \cdot \vec{D}(\vec{x}) = \rho_{lib}(\vec{x})
 \end{equation}
 
 où $\vec{D}(\vec{x}) =- \epsilon_0\epsilon_r(\vec{x})\nabla\phi(\vec{x})$ est le champs d'induction éléctrique, aussi appelé champs de déplacement.
 
 \subsection{Forme variationelle du problème.}
 En choisissant une fonction $\eta(\vec{x})$ quelconque de régularité $C^1$ et 0 sur $\partial\Omega$,
 et en utilisant les propritée véctorielle de $\nabla \cdot$, de l'équation (\ref{maxwell}) il est possible de déduire
 les rélations suivantes :
 
 \begin{align*}
 \nabla \cdot (\eta(\vec{x})\epsilon_r(\vec{x})\nabla\phi(\vec{x}))
 -\nabla\eta(\vec{x}) \cdot \epsilon_r\nabla \phi(\vec{x}) 
 =
  \eta(\vec{x}) \nabla \cdot (\epsilon_r(\vec{x})\nabla\phi(\vec{x}))\\
 \Rightarrow\quad \nabla\cdot( \eta(\vec{x})\epsilon_r(\vec{x}) \nabla\phi(\vec{x}))
 -\nabla\eta(\vec{x})\cdot \epsilon_r(\vec{x})\nabla\phi(\vec{x}) 
  = - \eta(\vec{x})\frac{\rho_{lib}(\vec{x})}{\epsilon_0} \Leftrightarrow \\
  \iiint \limits_{\Omega} \nabla\cdot(\eta(\vec{x})\epsilon_r(\vec{x}) \nabla\phi(\vec{x}))  
   -\nabla\eta(\vec{x})\cdot \epsilon_r(\vec{x})\nabla\phi(\vec{x}))dV 
   = -\iiint \limits_{\Omega} \eta(\vec{x})\frac{\rho_{lib}(\vec{x})}{\epsilon_0} dV\\
 \Leftrightarrow \iint \limits_{\partial\Omega} \eta(\vec{x})\epsilon_r(\vec{x}) \nabla\phi(\vec{x}) \cdot d\vec{\sigma} 
 -  \iiint \limits_{\Omega} \nabla\eta(\vec{x})\cdot \epsilon_r(\vec{x})\nabla\phi(\vec{x})) dV 
   = -\iiint \limits_{\Omega} \eta(\vec{x})\frac{\rho_{lib}(\vec{x})}{\epsilon_0} dV\\
 \end{align*}
 
 Pour la dérnière équivalence il a été employé le théorème de la divergence.
 Comme par définition $\eta(\vec{x})$ est 0 sur $\delta\Omega$ on en déduit en fin :
 
 \begin{equation}\label{general}
 \iiint \limits_{\Omega} \nabla\eta(\vec{x})\cdot \epsilon_r(\vec{x})\nabla\phi(\vec{x})) dV 
   = \iiint \limits_{\Omega} \eta(\vec{x})\frac{\rho_{lib}(\vec{x})}{\epsilon_0} dV
 \end{equation} 
 
 \paragraph{Description du système.}
 
 Le système étudié ici voit $\Omega$ comme étant un domaine cylindrique de rayon $R$ et longeur $L_z$
 , et par les symmétries cylindriques du système,
 les fonctions définies sur ce domaine dépendent uniquement de $r$, et on choisit $\eta = \eta(r)$.
 Donc en passant par un changement en cordonnées cylindriques, en sachant que la jacobiènne d'un tel
 changement a pour détérminant $r$, l'équation (\ref{general}) devient :
 
 \begin{equation*}
 \iiint \limits_{\Omega(r,\theta,z)} \epsilon_r(r) \frac{\partial\eta(r)}{\partial r}\frac{\partial\phi(r)}{\delta r}r drd\theta dz 
 = \iiint  \limits_{\Omega(r,\theta,z)} \eta(r)\frac{\rho_{lib}(r)}{\epsilon_0}r drd\theta dz 
 \end{equation*}
 
 Comme les fonctions à l'intérieur de l'intégrale dépendent uniquement de r, cette équation peut être simplifiée pour obtenir :
 
 \begin{equation}\label{cylindriques}
 \int \limits_{0}^{R} \epsilon_r(r) \frac{\partial\eta(r)}{\partial r}\frac{\partial\phi(r)}{\delta r}r dr
 = \int \limits_{0}^{R} \eta(r)\frac{\rho_{lib}(r)}{\epsilon_0}r dr
 \end{equation}
 
 
 \subsection{Discretisation par la méthode des éléments finis.}
 
 Pour discrétiser le problème, l'on commence en partageant l'intervalle $[0,R]$ par des plus petits intavalles 
 $[r_i,r_{i+1}]$, pour $i = 1,2,...n$. En suite les fonctions $\eta(r)$ et $\phi(r)$ sont approximées
 par une somme discrète de fonctions chapeau $\Lambda_i$ ayant valeur 1 sur $r_i$, linéaires dans l'intervalle $[r_{i-1}, r_{i+1}]$
 et 0 ailleur. 
 
 \begin{equation}\label{discr}
 \phi(r) = \sum_{k=1}^{n}\phi_k\Lambda_k(r) \; \quad \quad \; \eta(r) = \sum_{k=1}^n \eta_k\Lambda_k(r)
 \end{equation}
 
 Ainsi faisant, l'équation (\ref{cylindriques})
 peut s'écrire comme suit :
 
 \begin{equation*}
 \sum_{i=1}^n \eta_i  \sum_{j=1}^n (\int \limits_{0}^R \epsilon_r(r)r\frac{\partial \Lambda_i(r)}{\partial r}\frac{\partial \Lambda_j(r)}{\partial r} dr)\phi_j =
 \sum_{i=1}^n \eta_i (\int \limits_{0}^{R} \Lambda_i(r)\frac{\rho(r)}{\epsilon_0}r dr)  \quad \quad \forall\;\; \eta(r) = \sum_{k=1}^n \eta_k\Lambda_k(r)
 \end{equation*}
 
 Comme cette équation est valide pour toute forme de $\eta(r)$, elle peut être simplifiée pour obtenir :
 
 \begin{equation}\label{eqdiscrete}
 \sum_{j=1}^n (\int \limits_{0}^R \epsilon_r(r)r\frac{\partial \Lambda_i(r)}{\partial r}\frac{\partial \Lambda_j(r)}{\partial r} dr)\phi_j =
  \int \limits_{0}^{R} \Lambda_i(r)\frac{\rho(r)}{\epsilon_0}r dr \quad \quad \forall i
 \end{equation}
 
 Il est intéressant en fin de rémarquer que l'équation (\ref{eqdiscrete}), en considérant i de 1 à $n+1$, est équivalente à l'équation
 vectorielle $A\vec{\phi}=\vec{b}$ où $A$ est une matrice carrée symmétrique dont les coéfficients sont :
 
 \begin{equation}\label{Aij}
 A_{i,j} = \int \limits_{0}^R \epsilon_r(r)r\frac{\partial \Lambda_i(r)}{\partial r}\frac{\partial \Lambda_j(r)}{\partial r} dr
 \end{equation}
  
 où par convetion l'indice i est indice de ligne et j de colonne.
 Les coéfficient de $\vec{b}$ sont donnés par :
 
 \begin{equation}\label{bi}
 b_i =  \int \limits_{0}^{R} \Lambda_i(r)\frac{\rho(r)}{\epsilon_0}r dr
 \end{equation}
 
 \paragraph{Détermination de $A_{i.j}$ et $b_i$.}
 
 \begin{equation*}
 \end{equation*}
 
 L'équation (\ref{Aij}) peut s'écrire comme
 $A_{i,j} = \sum_{k=1}^n \int \limits_{r_k}^{r_{k+1}} \epsilon_r(r)r\frac{\partial \Lambda_i(r)}{\partial r}\frac{\partial \Lambda_j(r)}{\partial r} dr$.
 De cette forme, par la définition des fonctions $\Lambda_j$ il peut être 
 facilement déduit que les seules coéfficients non nuls de la matrice A
 sont ceux pour lequels $j=i=k$, $j=i=k+1$, $j=k, i=k+1$ et $j=k+1 ,i=k$, 
 car les dérivées de ces fonctions ne sont jamais non-nulles au même temps
 dans d'autres cas. De ça il est possible de déduire que la matrice $A$ est tridiagonale.
 Pour faciliter le calcul des intègrales, la fonctions $f(r) = \epsilon_r(r)r$ est approximée 
 par une ``valeur moyenne''  $\epsilon_{r,k}^{*}$ sur l'intérval $[r_k, r_{k+1}]$, qui
 corréspond à la valeur par laquelle la fonction $f(r)$ serait approximée à l'intérieur de l'intégrale
 lorsque l'on applique la formule du trapèze mixte à celle du point milieu
 (voir équation (\ref{trapezio2})).
 Il ne reste plus que détérminer la valeur de l'intégrale pour les quatres cas non nuls. 
 Soit $h_k = r_{k+1} - r_k$ la longuer de l'intérval $[r_k, r_{k+1}]$, alors $\Lambda_k = -\frac{1}{h_k}$ et
  $\Lambda_{k+1} =\frac{1}{h_k} $ sur cet interval, il en suit directement que $\int \limits_{r_k}^{r_{k+1}} \frac{\partial \Lambda_k(r)}{\partial r}\frac{\partial \Lambda_k(r)}{\partial r} dr = \frac{1}{h_k}$ et $\int \limits_{r_k}^{r_{k+1}} \frac{\partial \Lambda_k(r)}{\partial r}\frac{\partial \Lambda_{k+1}(r)}{\partial r} dr = -\frac{1}{h_k}$. 
  Ceci implique donc les équations suivantes:
  
 \begin{align*}
 & A_{i,i} = \frac{\epsilon_{r,i-1}^{*}}{h_{i-1}} + \frac{\epsilon_{r,i}^{*}}{h_{i}} \;\; i=2,... n; & & A_{1,1} = \frac{\epsilon_{r,1}^{*}}{h_{1}}
 & A_{n+1,n+1} = \frac{\epsilon_{r,n}^{*}}{h_n} \\
 & A_{i,i+1}= -\frac{\epsilon_{r,i}^{*}}{h_{i}}\; \; i = 1,... n; & & A_{i,i-1} = -\frac{\epsilon_{r,i-1}^{*}}{h_{i-1}}\;\; i = 2,... n+1;
 \end{align*}
 \begin{equation}\label{coefficienti}
 A_{i,j} = 0 \quad \forall j \notin \{i-1,i,i+1\}
 \end{equation}
 
 Pour la détérmintation des coéfficient $b_i$ le même raisonnement est fait, et l'on rémarque que l'intégrale 
 à l'équation (\ref{bi}) est non-nulle seulement sur les interavalles $[r_{i-1},r_{i}]$ et $[r_i,r_{i+1}]$ pour $i = 2,... n$
 , de même sur l'intérval $[r_1,r_2]$ pour $i=1$ et $[r_{n},r_{n+1}]$ pour $i=n+1$.
 L'intégrale dans ce cas est simplement approximée par la formule du trapèze mixte à celle du point milieu (\ref{trapezio}) 
 donnant les coéfficient qui suivent :
 
 \begin{align*}
 b_i = h_{i-1}[p\frac{\rho_{lib}(r_i)r_i}{2\epsilon_0} + (1-p)\frac{\rho_{lib}(r_{i-1/2})r_{i-1/2}}{2\epsilon_0}]
 \\
 +\;\;h_{i}[p\frac{\rho_{lib}(r_i)r_i}{2\epsilon_0} + (1-p)\frac{\rho_{lib}(r_{i+1/2})r_{i+1/2}}{2\epsilon_0}]
 \quad \; i = 2,... n. 
 \end{align*}
 \begin{equation*}
 b_1 = h_1[p\frac{\rho_{lib}(r_1)r_1}{2\epsilon_0} + (1-p)\frac{\rho_{lib}(r_{1+1/2})r_{1+1/2}}{2\epsilon_0}]
 \end{equation*}
 \begin{equation}\label{bcoeff}
 b_{n+1} = h_{n}[p\frac{\rho_{lib}(r_{n+1})r_{n+1}}{2\epsilon_0} + (1-p)\frac{\rho_{lib}(r_{n+1/2})r_{n+1/2}}{2\epsilon_0}]
 \end{equation}
 
 où $r_{i +1/2} = \frac{r_{i+1} + r_i}{2}$ et $r_{i -1/2} = \frac{r_{i} + r_{i-1}}{2}$, et où $p \in [0,1]$ est choisi en configuration
 avec les autres paramètres constants du système.
 
 
 En fin, pour imposer la condition limite $V(R) = V_0$ dans l'équation véctoriel
 on pose en dérnière ligne $A_{n+1,n} = 0$, $A_{n+1,n+1} = 1$ et $b_{n+1} = V_0$.
 L'algoritme permettant en fin de détérminer la valeur de $\vec{\phi}$ dans l'équation
 $A \vec{\phi} = \vec{b}$ est décrit à page 87 du polycopier de cours \cite{ref2}. Par l'application
 de cet algorithme, il est possible d'obtenir le vecteur de composantes $\phi_i = \phi(r_i)$, pour avoir
 une {\it mappe} des valeurs de $\phi$ sur les points discrétisées du système.
 
 \subsection{Champs éléctrique}
 
 Lorsque le champs scalaire $\phi(r)$ à été détérminé par discrétisation en un vecteur $\vec{\phi}$, 
 il est intéréssant de détérminer aussi les champs véctoriel dérivant de ce potentiel. Le but ici 
 est de détérminer la méthode numérique permettant d'obtenir le champs de déplacement 
 $D(r) = \epsilon_0\epsilon_r(r)E(r)$ avec $E(r) = -\nabla\phi = \frac{\partial \phi}{\partial r}(r)$.
 Par la méthode des éléments finis, comme décrit à la ligne (\ref{discr}) on obtient $E(r) =- \sum_{j} \phi_j \frac{\partial\Lambda_j}{\partial r}(r)$.
 Lorsque l'on cherche d'évaluer $E(r^{'})$ pour $r^{'} \in ]r_k,r_{k+1}[$ l'on observe que seulement deux fonctions de base $\Lambda_j$
 contribuent à la somme ( $\Lambda_k$ et  $\Lambda_{k+1}$), ce qui implique :
 
 \begin{equation}\label{D}
 D(r^{'})= \epsilon_0\epsilon_r(r^{'})E(r^{'})\;\;\;\; \text{ où } \;E(r^{'}) = \frac{\phi_{k+1} - \phi_k}{h_k} \quad\quad \forall \; \; r^{'} \in {]r_k, r_{k+1}[}
 \end{equation}
 
 Il est possible d'observer donc que le champs éléctrique est approximé par une formules des différences finies entre $r_k$ et $r_{k+1}$.
 
 \paragraph{Verification de la loi de Maxwell (\ref{maxwell}).}
 Maintenant que le champs de déplacement $D(r)$ peut être détérminé, 
 il est intéressant de tester l'algorithme en vérifiant la loi de maxwell pour l'éléctrodynamique
 du système. Par l'équation (\ref{maxwell}) et par la forme cylindrique de 
 l'omperateur de divergence ($\nabla \cdot$) il est possible d'obtenir la
 rélation suivante :
 
 \begin{equation}\label{maxdiscr}
 \frac{1}{r}\frac{\partial}{\partial r}(r D(r))=\rho_{lib}(r)
 \end{equation}
 
 Soit $\alpha(r) = r D(r)$, pour vérifier numériquement l'équation (\ref{maxdiscr}) il est nécessaire d'évaluer numériquement
 $\frac{\partial \alpha(r)}{\partial r}$. Pour faire ainsi, sont d'abord évaluée les valeur $\alpha(r_{k+1/2})$ pour
 $k=1,... n$, car la foction est discontinue sur tout $r_k$ par construction de $\frac{\partial \phi}{\partial r}$ (Rappel:
 $r_{k+1/2} = \frac{r_{k+1} +r_k}{2}$). Ainsi, on approxime la dérivée partielle de $\alpha$ évaluée dans le point
 $r_{mid,k} = \frac{r_{k+1/2} + r_{k+3/2}}{2}$, lequel corréspond au point intémédaire entre les points pour lesquels
 on a calculé $\alpha(r)$, par $\frac{\partial\alpha(r_{mid,k})}{\partial r} = \frac{\alpha(r_{k+3/2}) - \alpha(r_{k+1/2})}{r_{k+3/2} - r_{k+1/2}}$.
 
 Ainsi faisant, numériquement il ne reste qu'à vérifier la rélation suivante :
 
 \begin{equation}\label{div}
 \frac{1}{r_{mid,k}} \frac{\partial\alpha(r_{mid,k})}{\partial r} = \rho_{lib}(r_{mid,k})
 \end{equation}
 
 \section{Simulations Numériques}
 
 Pour discrétiser le problème on partage l'intérval $[0,R]$ en $N_1$ intervalles entre $r = 0$ et $r = b$, et en $N_2$ intervalles
 entre $r = b$ et $r = R$. La condition de bord est $\phi(R) = V_0$ avec $V_0$ donné.
 
 \subsection{Cas trivial}
 
 Pour tester l'algorithme est simulé le cas trivial où $\rho_{lib}(r)= \epsilon_0$ et $\epsilon_{r}(r) = 1$, avec $V_0 = 0$, 
 $b = 0.06$ \si{\metre}, $R = 0.12$ \si{\metre} et $N_1 = N_2$.
 
 \paragraph{Solution analytique.}
 
 De l'équation (\ref{maxdiscr}) l'on déduit :
 
 \begin{equation}\label{solving}
 -\frac{1}{r}\frac{\partial}{\partial r}(r \epsilon_0\epsilon_r(r)\frac{\partial \phi(r)}{\partial r})=\rho_{lib}(r)
 \end{equation}
 
 En appliquant les hypothèse pour le cas trivial plus la condition de bord, il est possible de résoudre l'équation différentielle
 pour obtenir la solution analytique suivante :
 
 \begin{equation}\label{solution}
 \phi(r) = V_0 + \frac{1}{4}(R^2 - r^2)
 \end{equation}
 
 L'on observe de l'équation (\ref{solution}) que pour ce cas trivial $\phi(0) = \frac{R^2}{4}$. 
 
 \begin{minipage}{\textwidth}
 \hspace{-0.05\textwidth}
 \begin{minipage}{0.45\textwidth}
 \paragraph{Étude de convergence}
 Ayant obtenu la valeur réelle du potentiel en $r=0$, 
 il est facile d'étudier la convergence du schéma numérique en ce point, en évaluant l'erreur 
 $|\phi_0 - \phi(0)|$ en fonction de la discretisation faite, c'est à dire en fonction du nombre de points
 $N$ par lequel l'interval $[0,R]$ est discrétisé.
 Le graphe en Figure \ref{} montre que la valeur simulée $\phi_0$ converge effectivement vers $\phi(0)$.
 L'algorithme donc converge à l'ordre 2, car étant le graphe en log log cet ordre correspond à la pente de la courbe.
 \end{minipage}
 \hspace{0.02\textwidth}
 \begin{minipage}{0.6\textwidth}
 \resizebox{\textwidth}{!}{
     \input{graphs/conv_triv.tex}
 }
 \captionof{figure}{Test de convérgence en $\phi(r = 0)$}
 \label{conv_triv_fig}
 \end{minipage}
 \end{minipage}
 
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 %\begin{figure}[h]
 %\hspace{-0.1\textwidth}
 %\begin{subfigure}{0.55\textwidth}
 %\resizebox{\textwidth}{!}{
 %    \input{graphs/phi_nontriv.tex}
 %}
 %\end{subfigure}
 %\hspace{0.02\textwidth}
 %\begin{subfigure}{0.55\textwidth}
 %\resizebox{\textwidth}{!}{
 %    \input{graphs/E_nontriv.tex}
 %}
 %\end{subfigure}
 %\end{figure}
 
 
 \subsection{Cas non trivial}
 
 Maintenant l'on a  $V_0 = 0$, $b = 0.02$ \si{\metre}, $R = 0.12$ \si{\metre}, et 
 
 \begin{equation}\label{epsir}
 \epsilon_r(r) = \begin{cases} 1 & (0\leq r <b)\\ 8 - 6\frac{r - b}{R - b} & (b \leq r \leq R) \end{cases}
 \end{equation}
 
 \begin{equation}\label{rhoo}
 \rho_{lib}(r) = \begin{cases} \epsilon_0 a_0 (1 - (\frac{r}{b})^2) & (0\leq r <b)\\ 0 & (b \leq r \leq R) \end{cases}
 \end{equation}
 
 avec $a_0 = 10^4$ \si{\volt\per\metre^2}
 
 \paragraph{Solutions numérique.}
 
 Sur les graphes en Figure \ref{} et \ref{} il est possible d'observer respectivement les soultions
 numériques pour le potentiel $\phi(r)$ et le champs éléctriques $E(r)$ dans le cylindre.
 Le graphe \ref{} illustre le potentiel en fonction de r, et il est possible d'y observer que ceci
 prends deux formes distinctes. Ceci est du naturellement à la discontinuité du milieu
 entre le vide et le diéléctrique et cette discontinuité peut s'observer plus clairement sur le graphe
  \ref{} du champs éléctrique. Dans le premier interval $\phi(r)$ subit une descente approximativement
 parabolique. En passant par une estimation de l'équation (\ref{solving}) lorsque l'on y injecte l'équation
 (\ref{rhoo}), il est en effet possible de determiner que $\phi(r)$ descend à l'ordre 4 sur $r$. En effet la situation est
 similaire au cas trivial si l'on considère le domaine représentant le cylindre vide de rayon $b$. 
 
 \begin{figure}[h]
 \hspace{-0.15\textwidth}
 \begin{subfigure}{0.65\textwidth}
 \resizebox{\textwidth}{!}{
     \input{graphs/phi_nontriv.tex}
 }
 \caption{$\phi(r)$}
 \label{phi_subfig}
 \end{subfigure}
 \hspace{0.02\textwidth}
 \begin{subfigure}{0.65\textwidth}
 \resizebox{\textwidth}{!}{
     \input{graphs/E_nontriv.tex}
 }
 \caption{$E(r)$}
 \label{E_subfig}
 \end{subfigure}
 \caption{Graphe de $\phi(r)$ et $E(r)$ pour différents $N_1$}
 \label{phiE_fig}
 \end{figure}
 
 Sur l'intérval $[b,R]$, $\phi(r)$ descend plus lentement et sa dérivée $E(r)$ tend donc à se minimiser et elle est discontinue en $r=b$ comme anticipé.
 Ce comportement peut s'expliquer par le fait que le diéléctrique s'oppose à l'action du champs éléctrique par l'éffet d'une polarisation
 qui en contient donc les effets.
 
 \begin{minipage}{\textwidth}
 \hspace{-0.05\textwidth}
 \begin{minipage}{0.45\textwidth}
 En confrontant les deux graphes il est possible de vérifier en fin le bon comportement et cohérence physique de $E(r)$ 
 qui approxime bien le gradient de $\phi(r)$.
 Pour finir, pour une discretisation assez grande une valeur de $\phi(r=b)$ est cherchée pour ensuite étudier la convergence
 en $N_2 \propto N_1$  du potentiel en cette valeur pour $r=b$. Le graphe en Figure \ref{} illustre qu'en effet l'algorithme converge
 aussi et de nouveau à l'ordre 2 comme dans le cas trivial.
 \end{minipage}
 \hspace{0.02\textwidth}
 \begin{minipage}{0.6\textwidth}
 \resizebox{\textwidth}{!}{
     \input{graphs/conv_nontriv.tex}    
 }
 \captionof{figure}{Test de convérgence de $\phi(r = b)$}
 \label{conv_nontriv_fig}
 \end{minipage}
 \end{minipage}
 
-%TODO inserire grafico test di convergenza
-
 \subsubsection{Verifier (\ref{div}).}
 
 En utilisant les différences finies pour l'operateur $d/dr$ il est possible de calculer la divergence du champ 
 de déplacement $\nabla \cdot D$ et vérifier ainsi numériquement l'équation (\ref{div}).
 Le graphe en Figure \ref{} illustre en effet la solution numérique de $\nabla \cdot D$
 et l'image de $\rho(r)$ comme défini en (\ref{rhoo}).
 
 %TODO, inserire grafico Div D e rholib
+\begin{figure}[h]
+\hspace{-0.15\textwidth}
+\begin{subfigure}{0.65\textwidth}
+\resizebox{\textwidth}{!}{
+    \input{graphs/divD.tex}
+}
+\caption{$\nabla \cdot D$}
+\label{divD_subfig}
+\end{subfigure}
+\hspace{0.02\textwidth}
+\begin{subfigure}{0.65\textwidth}
+\resizebox{\textwidth}{!}{
+    \input{graphs/divD_diff_rho.tex}
+}
+\caption{Différence entre $\nabla \cdot D$ et $\rho_{lib}$}
+\label{divDrho_subfig}
+\end{subfigure}
+\caption{Verification de l'équation du système (\ref{div})}
+\label{divDrho_fig}
+\end{figure}
 
 %TODO Correggere
 Sur ce graphe il est possible d'observer que la rélation (\ref{div}) est vérifiée car les deux 
 courbes coïncident, or la loi de Maxwell est vérifée aussi numériquement.
 
-\paragraph{Charges de polarisation.}
+\subsubsection{Charges de polarisation.}
 Lorsque un diélèctrique est traversé par un champs éléctrique $E(r)$, il se polarise
 pour contraster l'effet de ce dernier. Elles aparaissent donc des charges de polarisation 
 sur l'intèrface vide-diélèctrique, dont la charge totale est $Q_{pol}$ et il est possible d'en évaluer numériquement la densitée $\rho_{pol}(r)$ 
 par la rélation $\rho_{pol}(r) = \epsilon_0 \nabla \cdot E (r) - \nabla \cdot D (r)$.  
 
 \paragraph{Valeurs de référence pour les charges de polarisation.}
-À partir de l'équation différentielle du système (\ref{}), on peut trouver une
+À partir de l'équation différentielle du système (\ref{maxdiscr}), on peut trouver une
 éxpréssion analytique da $\alpha$ en sachant que $\alpha(0) = 0$ et intégrant
 sur le domaine $[0, r]$:
 
 \begin{equation}\label{alpha}
 \alpha(r) = \int_{0}^{r} s \; \rho_{lib}(s) ds =
 \begin{cases}
 \frac{1}{2} a_0 \; \epsilon_0 \; r^2 \; (1 - \frac{r^2}{2b^2}) & \text{si  } 0 \leq r < b \\
 \frac{1}{4} a_0 \; \epsilon_0 \; b^2 & \text{si  } b \leq r \leq R
 \end{cases}
 \end{equation}
 
 En considerant maintenant la grandeur $\rho_{pol}(r)$, si on l'intègre sur un
 domaine volumique $\Omega$ à forme de tube mince autour de rayon moyen $b$ et d'épaisseur $h_- + h_+$
 (trés petits), on obtien en utilisant le \textit{théorème de la divérgence}:
 
-\begin{equation}\label{qpol_ref_vec}
+\begin{equation*}
 Q_{pol} := \iiint_\Omega \nabla \cdot (\epsilon_0 \vec{E} - \vec{D}) dV = \epsilon_0 \iint_{\delta \Omega} \vec{E} (1 - \epsilon_r) \; d\vec{\sigma}
-\end{equation}
+\end{equation*}
 
 Puisque $\vec{E}$ est projecté selon la diréction radiale par rapport au centre
 du cilindre, alors il est toujours pérpendiculaire à la surface $\delta \Omega$.
 On peut donc enlever la notation véctorielle, ce qui amène en coordonnées
 cilindriques:
 
-\begin{equation}
+\begin{equation*}
 Q_{pol} = L \epsilon_0 \int_{0}^{2\pi} \left. r \; E(r) (1 - \epsilon_r(r)) \right\rvert_{r = b - h_-}^{r = b + h_+} \; d\theta
-\end{equation}
+\end{equation*}
 
 où $L$ est l'hauteur du cilindre.
 Puisque $\epsilon_r(r) = 1$ si $r < b$, la valeur évaluée à $b-h_-$ s'annulle et
 si on considère que $E(r) = \frac{\alpha(r)}{\epsilon_0 \epsilon_r(r) r}$, on
 obtien l'éxpréssion suivante en faisant tendre $h_+$ vers $0$:
 
 \begin{equation}\label{qpol_ref}
 Q_{pol} = 2 \pi \; L \; \alpha(b+h_+) \left(\frac{1}{\epsilon_r(b+h_+)} - 1\right)
 = - \frac{7}{16} \pi \; L \; \epsilon_0 \; a_0 \; b^2
 \end{equation}
 
 D'autre côté, au niveau de discrétisation numérique, en applicant la règle du
 trapèze pour l'intégrant $r \rho_{pol}$ évalué sur le même intérval qu'en avance
 et on supposant que $\rho_{pol}(r) \approx 0 \quad \forall r \neq b$, l'approximation
 numérique qui en dérive est la suivante:
 
 \begin{equation}\label{qpol_num}
 Q_{pol} = \iiint_{\Omega} \rho_{pol} dV = 2\pi \; L \int_{b-h_-}^{b+h_+} r \; \rho_{pol} dr \approx
 \pi \; L \; r_{N_b} \; \rho_{pol}(r_{N_b}) \cdot (r_{N_b+1} - r_{N_b-1})
 \end{equation}
 
 où $N_b$ est tel que $\rho_{pol}(r_{N_b}) \neq 0$ donne la singularité charactèristique des
 charges de polarisation.
 
 \paragraph{Solution numérique}
-Le graphe en Figure \ref{} illustre $\rho_{pol}(r)$ pour différentes discretisations.
+Le graphe en Figure \ref{rhopol_pic_subfig} illustre $\rho_{pol}(r)$ pour différentes discretisations.
 Il est possible d'observer que la densitée est non nulle seleument en proximité de l'intérface, et
 que le pic de $\rho_{pol}$ est négatif en impliquant que les charges de polarisation ont un signe négatif.
-En effet $E(r)$ a un sens positif (voir graphe \ref{}), donc par opposition à cela le diéléctrique cumule des charges négatives 
+En effet $E(r)$ a un sens positif (voir graphe \ref{E_subfig}), donc par opposition à cela le diéléctrique cumule des charges négatives 
 sur son intérface.
-En suite il est possible d'observer que le pic augmente en valeur absolu lorsque des $N$
+En suite il est possible d'observer que le pic augmente lineairement en valeur absolu lorsque des $N$
 plus grands sont pris pour la discretisation du système. Ceci est du au fait que physiquement
 les charges de polarisation se cumulent uniquement sur l'intérface du diéléctrique en $r=b$, mais
 par discretisation il est impossible de representer cela. Pourtant la charge totale de polarisation $Q_{pol}$,
-laquelle est donnée par l'intégrale sur $r$ de la courbe $\rho_{pol}(r)$, se conserve numériquement.
+laquelle est donnée par l'intégrale sur $r$ de la courbe $r \; \rho_{pol}(r)$, se conserve numériquement.
+
+\begin{figure}[h]
+\hspace{-0.15\textwidth}
+\begin{subfigure}{0.65\textwidth}
+\resizebox{\textwidth}{!}{
+    \input{graphs/rhopol.tex}
+}
+\caption{$\rho_{pol}$}
+\label{rhopol_subfig}
+\end{subfigure}
+\hspace{0.02\textwidth}
+\begin{subfigure}{0.65\textwidth}
+\resizebox{\textwidth}{!}{
+    \input{graphs/rhopol_augm.tex}
+}
+\caption{Pic de dénsité de charges en $r = b$ en fonction de $N_1$}
+\label{rhopol_pic_subfig}
+\end{subfigure}
+\caption{Verification de l'équation du système (\ref{div})}
+\label{rhopol_fig}
+\end{figure}
+
 En effet, pour des discretisation plus {\it fines} les charges de polarisation restent dans l'intervalle 
 approchant $r=b$, et donc pour un interval plus mince le pic de densité en ce point augmente
 en valeur absolue en conservant $Q_{pol}$.  
 
-%TODO inserire grafico dove si vede che rho_pol aumenta quando gli N sono più grandi
-
+\begin{minipage}{\textwidth}
+\hspace{-0.1\textwidth}
+\begin{minipage}{0.4\textwidth}
 En fin, en intégrant donc sous la courbe $\rho_{pol}(r)$ pour différentes discretisations de $N_1$ et $N_2$, il
-est possible de chercher une convergence sur la valeur totale de $Q_{pol}$.
-%TODO inserisci valore di Qpol
+est possible de chercher une convergence sur la valeur totale de $Q_{pol}$. Puisque
+la charge dépend aussi de la hauteur $L$ du cilindre, cette valeur à été fixé à $1$ \si{\metre}. 
+Ce qu'on observe par la figure (\ref{q_pol_fig}) c'est que les valeurs numériques convergent vers la valeur de
+référence $Q_{pol_{ref}}$ calculée en précedence par l'éxpréssion (\ref{qpol_ref}), qui vaut:
+\begin{equation*}
+Q_{pol} = -4.8677 \cdot 10^{-11} \; \text{\si{\coulomb}}
+\end{equation*}
+\end{minipage}
+\hspace{0.02\textwidth}
+\begin{minipage}{0.7\textwidth}
+\resizebox{\textwidth}{!}{
+    \input{graphs/qpol_conv.tex}
+}
+\captionof{figure}{Test de convérgence di $Q_{pol}$ vers la valeur de référence $Q_{pol_{ref}}$}
+\label{q_pol_fig}
+\end{minipage}
+\end{minipage}
 
 \section{Conclusions}
 
 Grace à la méthode des éléments finis, il a été possible de construire un algorithme
 pour la détérmination numérique d'une solution à un problème à condition de bords
 non trivial. La façon la plus immédiate de tester un tel algorithme est celle de tester sa cohérence physique 
 par exemple en vérifiant une des lois de Maxwell de l'éléctrodynamique, ou même de le tester
 sur un problème dont la solution analytique est connue explicitement.
 De même une fois verifiée la convergence de l'algorithme, il est possible de l'utiliser
 pour obtenir numériquement la solution à un problème non trivial pour pouvoir en déduire
 différentes observations, et par exemple en tirer des infos intéresantes comme la charge de
 polarisation totale du dièlèctrique dans le système étudié.
 Celle ci est une procédure d'étude très récurrente est typique en physique, lorsque le système
 étudié ne possède pas de solutions pouvant être détérminé analytiquement.
 
 
 \section{Annexes}
 
 \subsection{Formule du trapèze et du point du milieu}
 
 Pour approximer l'intégrale unidimensionnelle d'une fonction $f(x)$ dans l'intérval $[x_k,x_{k+1}]$ de longuer
 $h_k = x_{k+1} - x_k$ il est possible d'employer la formule du trapèze mixte à celle du point du milieu, pour un facteur $p \in [0,1]$:
  
 \begin{equation}\label{trapezio}
 \int \limits_{x_k}^{x_{k+1}} f(x) dx \approx  h_k [p\frac{f(x_k)+f(x_{k+1})}{2} + (1-p)f(\frac{x_k+x_{k+1}}{2})]
 \end{equation}
 
 Pour obtenir cette formule la fonction $f(x)$ est approximée à une {\it valeur moyenne} qui prend la fonction dans l'intérval d'intégration :
 \begin{equation}\label{trapezio2}
 f_{moy}(x) = p\frac{f(x_k)+f(x_{k+1})}{2} + (1-p)f(\frac{x_k+x_{k+1}}{2})
 \end{equation}
 
 Ainsi faisant en intégrant sur cette valeur constante il en résulte le facteur $h_k$ apparaissant dans l'équation (\ref{trapezio}).
 
 \begin{thebibliography}{99}
 
 
 \bibitem{ref2}
 L. VILLARD, PHYSIQUE NUMERIQUE I - II ,Swiss Plasma Center, Faculté des Sciences de Base, Section de Physique, Ecole Polytecnique Fédérale de Lausanne.
 \bibitem{ref1} 
 ECOLE POLYTECNIQUE FEDERALE DE LAUSANNE, Semestre de printemps 2019, Physique Numérique I - II - Exercice 5.
 \bibitem{ref3}
 L. VILLARD, Cours de Physique Numérique pour physiciens, Faculté des Sciences de Base, Section de Physique, Ecole Polytecnique Fédérale de Lausanne.
 \end{thebibliography}
 
 \end{document} %%%% THE END %%%%
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+
diff --git a/physnum/rap6/tikzext/rapport-figure5.dep b/physnum/rap6/tikzext/rapport-figure5.dep
new file mode 100644
index 0000000..875d298
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure5.dep
@@ -0,0 +1 @@
+tikzext/rapport-figure5.pdf: tikzext/rapport-figure5.pgf-plot.table
diff --git a/physnum/rap6/tikzext/rapport-figure5.dpth b/physnum/rap6/tikzext/rapport-figure5.dpth
new file mode 100644
index 0000000..e69de29
diff --git a/physnum/rap6/tikzext/rapport-figure5.md5 b/physnum/rap6/tikzext/rapport-figure5.md5
new file mode 100644
index 0000000..73982a2
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure5.md5
@@ -0,0 +1 @@
+\def \tikzexternallastkey {9C20D6C68881923E66F579C98DBF8895}%
diff --git a/physnum/rap6/tikzext/rapport-figure5.pdf b/physnum/rap6/tikzext/rapport-figure5.pdf
new file mode 100644
index 0000000..8678e70
Binary files /dev/null and b/physnum/rap6/tikzext/rapport-figure5.pdf differ
diff --git a/physnum/rap6/tikzext/rapport-figure5.pgf-plot.gnuplot b/physnum/rap6/tikzext/rapport-figure5.pgf-plot.gnuplot
new file mode 100644
index 0000000..336e133
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure5.pgf-plot.gnuplot
@@ -0,0 +1,2 @@
+set table "tikzext/rapport-figure5.pgf-plot.table"; set format "%.5f"
+set format "%.7e";; plot "outs/verifyNhigh.dat" using 1:5 every 5; 
diff --git a/physnum/rap6/tikzext/rapport-figure5.pgf-plot.table b/physnum/rap6/tikzext/rapport-figure5.pgf-plot.table
new file mode 100644
index 0000000..d025431
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure5.pgf-plot.table
@@ -0,0 +1,184 @@
+
+# Curve 0 of 1, 179 points
+# Curve title: ""outs/verifyNhigh.dat" using 1:5 every 5"
+# x y type
+2.0000000e-04 1.4999949e-06  i
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+
diff --git a/physnum/rap6/tikzext/rapport-figure6.dep b/physnum/rap6/tikzext/rapport-figure6.dep
new file mode 100644
index 0000000..760a7dd
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure6.dep
@@ -0,0 +1,2 @@
+tikzext/rapport-figure6.pdf: tikzext/rapport-figure6.pgf-plot.table
+tikzext/rapport-figure6.pdf: tikzext/rapport-figure6.pgf-plot.table
diff --git a/physnum/rap6/tikzext/rapport-figure6.dpth b/physnum/rap6/tikzext/rapport-figure6.dpth
new file mode 100644
index 0000000..e69de29
diff --git a/physnum/rap6/tikzext/rapport-figure6.md5 b/physnum/rap6/tikzext/rapport-figure6.md5
new file mode 100644
index 0000000..9ca6f48
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure6.md5
@@ -0,0 +1 @@
+\def \tikzexternallastkey {660AA2D549787249346EF43777B61C02}%
diff --git a/physnum/rap6/tikzext/rapport-figure6.pdf b/physnum/rap6/tikzext/rapport-figure6.pdf
new file mode 100644
index 0000000..77250fa
Binary files /dev/null and b/physnum/rap6/tikzext/rapport-figure6.pdf differ
diff --git a/physnum/rap6/tikzext/rapport-figure6.pgf-plot.gnuplot b/physnum/rap6/tikzext/rapport-figure6.pgf-plot.gnuplot
new file mode 100644
index 0000000..b6461c0
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure6.pgf-plot.gnuplot
@@ -0,0 +1,2 @@
+set table "tikzext/rapport-figure6.pgf-plot.table"; set format "%.5f"
+set format "%.7e";; plot "outs/verifyNhigher.dat" using 1:4 every 2; 
diff --git a/physnum/rap6/tikzext/rapport-figure6.pgf-plot.table b/physnum/rap6/tikzext/rapport-figure6.pgf-plot.table
new file mode 100644
index 0000000..024ea93
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure6.pgf-plot.table
@@ -0,0 +1,94 @@
+
+# Curve 0 of 1, 89 points
+# Curve title: ""outs/verifyNhigher.dat" using 1:4 every 2"
+# x y type
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+8.0000000e-04 0.0000000e+00  i
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+
diff --git a/physnum/rap6/tikzext/rapport-figure7.dep b/physnum/rap6/tikzext/rapport-figure7.dep
new file mode 100644
index 0000000..3bfa1f6
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure7.dep
@@ -0,0 +1 @@
+tikzext/rapport-figure7.pdf: tikzext/rapport-figure7.pgf-plot.table
diff --git a/physnum/rap6/tikzext/rapport-figure7.dpth b/physnum/rap6/tikzext/rapport-figure7.dpth
new file mode 100644
index 0000000..e69de29
diff --git a/physnum/rap6/tikzext/rapport-figure7.md5 b/physnum/rap6/tikzext/rapport-figure7.md5
new file mode 100644
index 0000000..a1b061a
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure7.md5
@@ -0,0 +1 @@
+\def \tikzexternallastkey {E8163C6B2061F977BEA2B7DBC93F9D91}%
diff --git a/physnum/rap6/tikzext/rapport-figure7.pdf b/physnum/rap6/tikzext/rapport-figure7.pdf
new file mode 100644
index 0000000..4734ac7
Binary files /dev/null and b/physnum/rap6/tikzext/rapport-figure7.pdf differ
diff --git a/physnum/rap6/tikzext/rapport-figure7.pgf-plot.gnuplot b/physnum/rap6/tikzext/rapport-figure7.pgf-plot.gnuplot
new file mode 100644
index 0000000..d62c596
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure7.pgf-plot.gnuplot
@@ -0,0 +1,2 @@
+set table "tikzext/rapport-figure7.pgf-plot.table"; set format "%.5f"
+set format "%.7e";; plot "outs/polarization.dat" using 1:3 every 20; 
diff --git a/physnum/rap6/tikzext/rapport-figure7.pgf-plot.table b/physnum/rap6/tikzext/rapport-figure7.pgf-plot.table
new file mode 100644
index 0000000..93b58d1
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure7.pgf-plot.table
@@ -0,0 +1,105 @@
+
+# Curve 0 of 1, 100 points
+# Curve title: ""outs/polarization.dat" using 1:3 every 20"
+# x y type
+2.9000000e+01 -1.6579467e+00  i
+4.9000000e+01 -2.8631919e+00  i
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+
diff --git a/physnum/rap6/tikzext/rapport-figure8.dep b/physnum/rap6/tikzext/rapport-figure8.dep
new file mode 100644
index 0000000..f4eb2d3
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure8.dep
@@ -0,0 +1 @@
+tikzext/rapport-figure8.pdf: tikzext/rapport-figure8.pgf-plot.table
diff --git a/physnum/rap6/tikzext/rapport-figure8.dpth b/physnum/rap6/tikzext/rapport-figure8.dpth
new file mode 100644
index 0000000..e69de29
diff --git a/physnum/rap6/tikzext/rapport-figure8.md5 b/physnum/rap6/tikzext/rapport-figure8.md5
new file mode 100644
index 0000000..969314a
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure8.md5
@@ -0,0 +1 @@
+\def \tikzexternallastkey {F49A38487DF9823BBFBD31BBCF46FD69}%
diff --git a/physnum/rap6/tikzext/rapport-figure8.pdf b/physnum/rap6/tikzext/rapport-figure8.pdf
new file mode 100644
index 0000000..988a352
Binary files /dev/null and b/physnum/rap6/tikzext/rapport-figure8.pdf differ
diff --git a/physnum/rap6/tikzext/rapport-figure8.pgf-plot.gnuplot b/physnum/rap6/tikzext/rapport-figure8.pgf-plot.gnuplot
new file mode 100644
index 0000000..eb484fa
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure8.pgf-plot.gnuplot
@@ -0,0 +1,2 @@
+set table "tikzext/rapport-figure8.pgf-plot.table"; set format "%.5f"
+set format "%.7e";; plot "outs/polarization.dat" using 1:5 every 5; 
diff --git a/physnum/rap6/tikzext/rapport-figure8.pgf-plot.table b/physnum/rap6/tikzext/rapport-figure8.pgf-plot.table
new file mode 100644
index 0000000..e907a26
--- /dev/null
+++ b/physnum/rap6/tikzext/rapport-figure8.pgf-plot.table
@@ -0,0 +1,405 @@
+
+# Curve 0 of 1, 400 points
+# Curve title: ""outs/polarization.dat" using 1:5 every 5"
+# x y type
+1.4000000e+01 1.4861697e-13  i
+1.9000000e+01 1.2296461e-13  i
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