Finite Element Method Formulation for the Poisson's Equation
- 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.
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 (q0) is to take on the boundary (Γq) 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).