Fundamentals of Finite Element Method (FEM)

The main idea behind FDM is the approximation of the derivative operations ∂u/∂x and ∂2u/∂x2 by the difference quotients Δu(x)/Δx and Δ2u(x)/Δx2, which reduces the partial differential equation to a set of algebraic equations. The application of FDM has two serious limitations. First, the regular steps of h _x , h _y , h _z which construct an array of grid nodes in the x, y, z directions are not suitable for a field with a rapidly changing gradient or for problems having a curved boundary. Second, different formulae must be derived for specific interfaces between different media and for the various shapes of boundaries.