Finite Element Method Formulation for the Poisson's Equation

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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:

\left[ \frac{\partial}{\partial x}\cdot \left( K_{x} \cdot \frac{\partial}{\partial x}\right) + \frac{\partial}{\partial y}\cdot \left(K_{y} \cdot \frac{\partial}{\partial y} \right) \right]\varphi(x,y) = f(x,y)

for a domain Ω, equivalent to write:

\nabla \cdot (K \nabla \varphi )= f

where K is the material property, \varphi 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 (\varphi_{0}) on a boundary ({\Gamma_{\varphi}}) of the domain:

\varphi(x,y)\mid_{\Gamma_{\varphi}}  = \varphi_{0}

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:

K \nabla \varphi(x,y)\mid_{\Gamma_{q}}  = q_{0}

Finite Element Method Formulation

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

    \int_{\Omega} W [\nabla^T D \nabla \varphi + \rho_v]\partial \Omega 
    + \oint_{\Gamma_q}\overline{W}[n^T D \nabla \varphi +  \overline{q}]\partial \Gamma_q=0

with W, \overline{W} the weights for each differential equation. If \overline{W}=W and using the integration by parts, this equation becomes:

    \int_{\Omega} \nabla^T W^T D \nabla \varphi \partial \Omega =
    \int_{\Omega} W^T \rho_v \partial \Omega 
    - \oint_{\Gamma_q} W^T \overline{q_n} \partial \Gamma_q
    - \oint_{\Gamma_{\varphi}} W^T q_n \partial \Gamma_{\varphi}

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

D = 
     K_x & 0      \\
     0   & K_y
\partial \Omega = \partial x \partial y
x=[x,y]^T = \sum N_i[x_i,y_i]^T

with Ni the shape forms of the element.

\varphi = \sum N_i \varphi_i = N a^{(e)}
q=- D \nabla \varphi = - D \nabla N a^{(e)} = - D B a^{(e)}

with B defined as:

B = 
      B_1...  & B_n
B_i =
     \frac{\partial N_i}{\partial x}      \\
     \frac{\partial N_i}{\partial y}

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

K \cdot a = f

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.

K^{(e)} =
    \int_{\Omega^{(e)}} B^T D B \partial \Omega^{(e)}
f^{(e)} =
    \int_{\Omega^{(e)}} N^T \rho_v \partial \Omega^{(e)}
    - \oint_{\Gamma_q^{(e)}} N^T \overline{q} \partial \Gamma_q^{(e)}
    - \oint_{\Gamma_{\varphi}^{(e)}} n^T N^T q_n \partial \Gamma_{\varphi}^({(e)}

A very symple example (to be prepared).

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