By default, I know that Mathematica makes the normal derivative of the potential function in Mathematica continuous across the boundary defined in my problem. My problem is that I need the normal derivative of the potential function times different dielectric constants to be continuous across the boundary. So in a more concrete sense I currently have the values (Potential[0.5, 1.5] - Potential[0.5, 1.5 - 10^-4])/10^-4 and (Potential[0.5, 1.5 + 10^-4] - Potential[0.5, 1.5])/10^-4 are very close.

What I need is:

43.6 *(Potential[0.5, 1.5] - Potential[0.5, 1.5 - 10^-4])/10^-4 and 3.9 *(Potential[0.5, 1.5 + 10^-4] - Potential[0.5, 1.5])/10^-4 to be very close.

Is there any way that I can do this?

To clarify a bit more: The function I need to make continuous across the boundaries is n*\[Epsilon]*grad(phi), where n is a normal unit vector, phi is the potential function, and [Epsilon] is a 2x2 diagonal matrix containing the dielectric constants that changes based on the region. There are two regions and for what I have written above I am considering the boundary at the top of the core of my shape which is located at d=1.5 and x= 0.5

Here is the full setup of geometry and boundary conditions for what I am working with:

ElectricIntensity[a1_, a2_, a3_, a4_, b1_, b2_, h_, c1_, c2_, d_, 
  H_] := Module[{}, \[CapitalOmega] = 
   RegionDifference[
    RegionDifference[Rectangle[{-a4 - R, -R}, {a4 + R, b2 + R}], 
     Polygon[{{a1, b2}, {a2, b2}, {a3, b1}, {a4, b1}, {a4, 
        b1 + h}, {a3, b1 + h}, {a2, b2 + h}, {a1, b2 + h}}]], 
    Polygon[{{-a1, b2}, {-a2, b2}, {-a3, b1}, {-a4, b1}, {-a4, 
       b1 + h}, {-a3, b1 + h}, {-a2, b2 + h}, {-a1, b2 + h}}]];
  \[CapitalOmega]Core = 
   Polygon[{{-a4, b1 - H}, {a4, b1 - H}, {a4, b1}, {c2, b1}, {c1, 
      d}, {-c1, d}, {-c2, b1}, {-a4, b1}}];
  \[Epsilon][x_, y_] := If[{x, y} \[Element] \[CapitalOmega]Core, ( {
      {29.16, 0},
      {0, 43.6}
     } ), ( {
      {3.9, 0},
      {0, 3.9}
     } )];
  Potential = 
   NDSolveValue[{Inactive[
        Div][{\[Epsilon][x, y].Inactive[Grad][u[x, y], {x, y}]}, {x, 
        y}] == 0, 
     DirichletCondition[u[x, y] == 0.5, 
      a1 <= x <= a2 && y == b2 || 
       a2 <= x <= a3 && y == (b1 - b2)/(a3 - a2) (x - a2) + b2 || 
       a3 <= x <= a4 && y == b1 || a1 <= x <= a2 && y == b2 + h || 
       a2 <= x <= a3 && y == (b1 - b2)/(a3 - a2) (x - a2) + b2 + h || 
       a3 <= x <= a4 && y == b1 + h || x == a1 && b2 <= y <= b2 + h ||
        x == a4 && b1 <= y <= b1 + h],
     DirichletCondition[
      u[x, y] == -0.5, -a2 <= x <= -a1 && y == b2 || -a3 <= x <= -a2 &&
         y == (b2 - b1)/(a3 - a2) (x + a3) + b1 || -a4 <= x <= 
         a4 - 3 && y == b1 || -a2 <= x <= -a1 && 
        y == b2 + h || -a3 <= x <= -a2 && 
        y == (b2 - b1)/(a3 - a2) (x + a3) + b1 + h || -a4 <= 
         x <= -a3 && y == b1 + h || x == -a1 && b2 <= y <= b2 + h || 
       x == -a4 && b1 <= y <= b1 + h]}, 
    u, {x, y} \[Element] \[CapitalOmega]]; 
  Derivative[1, 0][Potential][0.5, 1.5]]

Example of values used to run this code:

R = 1; AbsoluteTiming[
 ElectricIntensity[1, 2, 3, 5, 1, 2, 0.1, 1, 2, 1.5, 0.2]]