Unlike the traditional Taylor's series expansion method, the Galerkin approach utilizes basis functions, such as linear piecewise polynomials, to approximate the true solution. For example, the Galerkin approximation to the sample problem, (1), would require evaluating (19) for the specific grid formation and specific choice of basis function:
Difference quotients are then used to approximate the derivatives in (31). We note that if linear basis functions are utilized in (31), one obtains a formulation which corresponds exactly with the standard finite difference operator. Regardless of the difference scheme or order of basis function, the approximation results in a linear system of equations of the form , subject to the appropriate boundary conditions.
(See exercise 8.)