# 2D formulation for Electrostatic Problems

The 2D Electrostatic Poisson's equation given by the governing PDE and its boundary conditions:

We will apply the residual formulation based on the Weighted Residual Method (WRM).

with:

- and the weighting functions.

Where is the numerical approach of the unknown :

This is:

with the infinit condition factor and the field produced whe is fixed by .

The weak form of this expression can be obtained using the integration by parts. In addition, if :

Remembering that:

is the gradient potential with:

and:

The electric field and electric displacement field can be written as follows:

We will now use the Galerkin Method . So, finally, the integral expression ready to create the matricial system of equations is:

Note that **K** is a coefficients matrix that depends on the geometrical and physical properties of the problem, **a** is the vector with the * n* unknowns to be obtained and

**f**is a vector that depends on the source values and boundary conditions.