CFD Online Discussion Forums - finite element vs. finite difference

(1). The finite-difference method uses Taylor's series expansion to approximate the derivatives terms in the governing equations. Results are then obtained by solving a set of resulting algebraic equations. (2). The finite-element method assumes that the solution within an element of volume will take certain functional form in the first place. It then uses minimization processes, variational principles, weighted residual methods to obtain the solutions in the assumed functional form. (3). The finite-difference method is consistent with the original definition of derivatives, and is widely used in all fields of physics. (4). The finite-element method has been very successful in the solid mechanics, structure analysis. The ability to handle complex structure and geometry is widely recognized. (5). The main difficulty with finite-element method has been the lack of sound physical foundation. The memory requirement for solving large matrix equations was also very high. (6). The finite-volume method tries to solve the integral form of the governing equation over a small finite volume. Approximations are applied to the integration in order to derive the algebraic equations. These are solved in the way similar to the finite-difference method. Using this approach and unstructured mesh similar to the finite-element method, problems with complex geometry can be handled easily. (7). Solutions to the highly non-linear, coupled, partial-differential equations are very hard to obtain. Therefore, any method is a good method, if good solution can be obtained. ( The war between the finite-difference and the finite-element methods was long over in 70's ). You are free to invent new methods to solve CFD problems.