# Finite Element Method Formulation for the Poisson's Equation

## Contents |

### Suggested Prerequisites

- Linear algebra
- Partial differential equations

### Our Partial Differential Equation: Poisson's equation

As our first example, we will use the Poisson's equation in two-dimensional Cartesian coordinates, that it takes the form:

for a domain Ω, equivalent to write:

where **K** is the material property, is the unknown to obtain and **f** is a source term.

### Boundary Conditions

Two different boundary conditions will be considered:

#### Dirichlet boundary condition

The **first type** boundary condition specifies the values the solution needs to take on a boundary of the domain:

#### Neumann boundary condition

The **second type** boundary condition specifies the values that the derivative of a solution (**q _{0}**) is to take on the boundary (Γ

_{q}) of the domain:

### Finite Element Method Formulation

The Weak formulation using the Weighted Residual Form (Galerkin Method) of Poisson equation can be written as follows:

with the weights for each differential equation. If and using the integration by parts, this equation becomes:

for the two-dimensional Cartesian coordinates, each term can be written:

with **N _{i}** the shape forms of the element.

with **B** defined as:

These system of equations can be written as the following matricial form:

with **K** the stiffness matrix of the system, **a** the vector of nodal values to obatin (unknown vector), and **f** the source vector (independent forces or charges).

The superscript (e) refers to the equation applied for each element.

A very symple example (to be prepared).