# 2D formulation for Electrostatic Problems

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− | :<math>\left[ \frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial}{\partial y} \right) \right] | + | :<math>\left[ \frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial}{\partial y} \right) \right]V(x,y) = f(x,y)</math> |

− | ::<math> A(V) = \ | + | ::<math> A(V) = \left[ \frac{\partial}{\partial x}\cdot \left( \varepsilon_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(\varepsilon_{y} \cdot \frac{\partial}{\partial y} \right) \right]V(x,y) + \rho_S = 0 ~~ in ~ \Omega </math> |

## Revision as of 14:48, 30 October 2009

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).

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.