# Finite Element Method Formulation for the Poisson's Equation

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

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: $D = \begin{bmatrix} K_x & 0 \\ 0 & K_y \end{bmatrix}$ $\partial \Omega = \partial x \partial y$ $x=[x,y]^T = \sum N_i[x_i,y_i]^T$

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: $B = \begin{bmatrix} B_1... & B_n \end{bmatrix}$ $B_i = \begin{bmatrix} \frac{\partial N_i}{\partial x} \\ \frac{\partial N_i}{\partial y} \end{bmatrix}$

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).