FEM In Brief David Garmire, Course: EE693I UH Dept. of Electrical Engineering 4/20/2008

Partial (Spatial) Differential Equation We want the PDE to hold over our domain – subject to boundary conditions (BC). We want to approximate u(x): How do we choose “best” coeffs. s.t. PDE holds? - coefficients- basis function

Idea: use the variational approach Variational form: – v i (x) - ith basis function (smooth, satisfying BCs) – We can write an equation that must hold, as well – Apply integration by parts:

Then: write f(x) and u(x) in terms of v i (x) Now we can solve a matrix system Objective: solve for each a term

Solution Now we have the solution of the PDE! Note: f(x) can also be represented in our basis – Then our matrix equation becomes