Numerical calculation of electric field in dielectric medium

I want to numerically calculate the electric field between two parallel plates with finite length. Where parallel plates are connected to source of constant potential difference:

I have boundary conditions on the edges of numerical area (potential $V = 0$) and "inside-boundary" conditions on parallel plates (for instance, potential on left plate $V = V_1$, on right plate $V = 0$).

Then I use the FDM method and solve system for Laplace's equations:

$$\nabla^2 \varphi=0$$

For every point within the numerical area. Then, from a value of potential in every point in numerical area, I calculate the electric field.

In theory, always it is OK, but my question is - where must I consider relative permittivity of dielectric medium? In the Laplace equation there is no relative permittivity and I think that my algorithm is contradictory with real physics.

I'd appreciate any help.

In actuality, you're solving Poisson's equation, $\nabla^2\phi = -\rho/\epsilon$, where $\rho$ is the charge density and $\epsilon$ the permittivity.
If you had a particular charge distribution then the permittivity would affect the magnitude of the resulting potential. However, by imposing boundary conditions on the potential, you're implicitly accounting for the permittivity. So, for example, if you wanted to double the permittivity, you would simply halve $V_1$.