This document introduces the finite element method for solving partial differential equations. It discusses using a "master element" to perform calculations that then get transformed to individual mesh elements. The method is described for a general diffusion equation, integrating it by parts and discretizing it using basis functions defined on mesh elements. This leads to a system of equations relating the unknown values at different nodes in the mesh at each time step. Transforming between the master element coordinates and the actual mesh coordinates completes the description of how the finite element method sets up and solves the discrete system of equations approximating the original PDE.

1. Intro to Finite Elements Method
Martin Jones
It is possible to do calculations on an arbitrary grid of finite elements given just one master element.
The idea is useful because performing calculations on arbitrary elements in curvilinear coordinates proves
awkward, and the character of these calculations such as limits of integration, etc..., changes from element to
element. We call the Master element, ˆΩ, and calculations for the mesh we are interested in are performed for
each individual element on it, then ‘transformed’ back to the original mesh coordinates, with their respective
material properties, or solution, in place.
∂φ
∂t
= · (D(x) φ) (1)
Where φ and D are concentration and diffusivity, repectively, scalar functions that depend on space and
time. And we have φ(x, 0) = φ0(x) and φ · n = 0 on the surface for our boundary conditions.
Now multiply both sides by an arbitrary test function V (x) and integrate over a volume Ω. Also,
integrating by parts on the RHS, the equation evolves to the following form, (making use first of the following
theorem):
Ω
V · (D φ)dΩ =
S
V (D φ · n)dS −
Ω
D( V · φ)dΩ
Ω
∂φ
∂t
V (x)dΩ = −
Ω
D(x) V · φdΩ (2)
Now we will discretize and write with basis functions in N dimensional space (using Einstein notation) where
N is on the order of the number of elements in the mesh,
φ(x, tk) = φk
i ψi (3)
and
V (x) = Vjψj (4)
Using the finite difference formula for the time derivative and plugging in we get,
Ω
φk+1
i − φk
i
∆t
· ψi(Vj · ψj)dΩ = −
Ω
D(x) (Vjψj) · (φk+1
i ψi)dΩ (5)
Now we may write N equations with N unknowns due to the arbitrariness of V (1 on the ith
node and zero
every where else). One gets,
1
∆t
M + K φk+1
=
1
∆t
Mφk
(6)
Where M is given by,
Mi,j =
Ω
ψiψjdΩ (7)
and K is given by,
Ki,j =
Ω
D(x) ψi · ψjdΩ (8)

2. These equations describe the physics on the local grid. We want to describe them on the global grid,
or our final mesh. Remembering the jacobian relating the differentials between the two coordinate systems:
(x, y), and (ξ, η),
dx =
∂x
∂ξ
dξ +
∂x
∂η
dη (9)
and,
dy =
∂y
∂ξ
dξ +
∂y
∂η
dη (10)
We can obtain our final equations through these equations ultimately. They are.
ke
ij =
Ωe
k
∂ψe
i
∂x
∂ψe
j
∂x
+ bψe
i ψe
j dxdy (11)
fe
i =
Ωe
fψe
i dxdy (12)
Pe
ij =
δΩe
pψe
i ψe
j ds (13)
γe
i =
δΩe
γψe
i ds (14)
2