General formulation for Electrostatic Problems
The 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).
- and the weighting functions.
Where is the numerical approach of the unknown :
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 :
is the gradient potential with:
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.