1. Procedures in deriving element equations (General field problems)
Governing equation
Identify boundary conditions
Discretize the solution domain, Ω
For variational approach
Define the functional, I(Ø)
Minimize the functional. Or
For method of weighted residual (MWR)
Integrate by parts (Reduce order of integral and introduce BC) .

2. To derive an element equation for 2-D heat transfer problem.
Example
To derive an element equation for 2-D heat transfer problem.
Governing differential equation (steady state)
Boundary conditions
Specified T=T(x,y) on s1
*(T-T) convective heat loss
q heat loss on boundary due to conduction
Discretize the Ωdomain into M number of elements. For each element:
For variational approach
The functional has only 1st order derivatives, thus T is continuous across the interface and T is constant within element. ň s1 s2 x y Ω

3. Minimize the functional
Terms to be evaluated: .

4. yield: By method of weight residual (Galerkin’s Method)
Integrate by parts
Let u=Ni
Results:
Apply B.C : = -q-h(T(e) –Tα) on s2
yield: