I am trying to use poisson equation to plot the electric scalar potential in close 2D space. The details in in this video(https://youtu.be/oPuF8v92lnc?t=854
and https://youtu.be/bLiaz-IHX_Y?t=723)

The following in written in matlab for quick prototype.

% solve matrix
clearvars
% number of rows ans columns
m = 21
n = 41

% m = 3
% n = 3
% initialise arrays
U = zeros(m*n,m*n);

dx = 0.01;
dy = 0.01;

for i = 1:m
    for j = 1:n
        index = (i-1)*n+j;
        U(index, index) = -2/dx^2 -2/dy^2;
        if j-1 > 0
            U(index,index-1) = 1/dx^2;
        end

        if j+1 <= n 
            U(index,index+1) = 1/dx^2;
        end

        if i-1 > 0
            U(index,index-3) = 1/dy^2;
        end

        if i+1 <= m
            U(index,index+3) = 1/dy^2;
        end
    end
end

% U

V0 = 0.5
% U*results =B
B = zeros(m*n,1);
B((m-1)*n+1:m*n, 1) = -V0/dy^2;


% Solve Linear Equations in Matrix Form
results = linsolve(U,B);

final =zeros(m,n);

for k = 1:m
    final(k,:) = results((k-1)*n+1:k*n);
end    

% plot final results
f1 = figure('Name','Voltage surface plot','NumberTitle','off');
[X,Y] = meshgrid(0:0.1:4, 0:0.1:2);
s = surf(X,Y,final,'FaceAlpha',0.5)
title("Potential V(x,y) for finite conducting box, " + "V_0=" + V0 + " volts")

f2 = figure('Name','imagesc','NumberTitle','off');
    
clims = [4 18];
imagesc(X(1,:), Y(:,1), final)
title("Potential V(x,y) for finite conducting box, " + "V_0=" + V0 + " volts")

Image form author.

Image from me.