Comparison of finite difference method and finite element method

The performance of finite element (FEM) and finite difference equations (FDE) are compared. The smoothness of the approximate solution of FDE is determined as a power function of the order of the differential equation, while the FEM approximation must satisfy an equation in Sobolev space. Only modal values can be computed with FDE; spatial calculations are possible with FEM. FEM upwind versions are more suitable for convection-diffusion equations, and can derive the same discretized equations as FDE. An analysis of the truncation error of the two methods shows the accuracies cannot be compared except when an exact solution is available. The configuration of each method for irregular geometries is illustrated. It is noted that more algebraic manipulation is necessary to discretize the FEM equations than needed for the FDE. Finally, it is shown that a nonuniform grid will allow more accurate solutions than a uniform grid.