Comparison To The Finite Difference Method
The finite difference method (FDM) is an alternative way of approximating solutions of PDEs. The differences between FEM and FDM are:
- The most attractive feature of the FEM is its ability to handle complicated geometries (and boundaries) with relative ease. While FDM in its basic form is restricted to handle rectangular shapes and simple alterations thereof, the handling of geometries in FEM is theoretically straightforward.
- The most attractive feature of finite differences is that it can be very easy to implement.
- There are several ways one could consider the FDM a special case of the FEM approach. E.g., first order FEM is identical to FDM for Poisson's equation, if the problem is discretized by a regular rectangular mesh with each rectangle divided into two triangles.
- There are reasons to consider the mathematical foundation of the finite element approximation more sound, for instance, because the quality of the approximation between grid points is poor in FDM.
- The quality of a FEM approximation is often higher than in the corresponding FDM approach, but this is extremely problem-dependent and several examples to the contrary can be provided.
Generally, FEM is the method of choice in all types of analysis in structural mechanics (i.e. solving for deformation and stresses in solid bodies or dynamics of structures) while computational fluid dynamics (CFD) tends to use FDM or other methods like finite volume method (FVM). CFD problems usually require discretization of the problem into a large number of cells/gridpoints (millions and more), therefore cost of the solution favors simpler, lower order approximation within each cell. This is especially true for 'external flow' problems, like air flow around the car or airplane, or weather simulation.
Read more about this topic: Finite Element Method
Famous quotes containing the words comparison to the, comparison, finite, difference and/or method:
“In comparison to the French Revolution, the American Revolution has come to seem a parochial and rather dull event. This, despite the fact that the American Revolution was successful—realizing the purposes of the revolutionaries and establishing a durable political regime—while the French Revolution was a resounding failure, devouring its own children and leading to an imperial despotism, followed by an eventual restoration of the monarchy.”
—Irving Kristol (b. 1920)
“It is comparison than makes people miserable.”
—Chinese proverb.
“For it is only the finite that has wrought and suffered; the infinite lies stretched in smiling repose.”
—Ralph Waldo Emerson (1803–1882)
“...there is a difference between being convinced and being stubborn. I’m not certain what the difference is, but I do know that if you butt your head against a stone wall long enough, at some point you realize the wall is stone and that your head is flesh and blood.”
—Maya Angelou (b. 1928)
““I have usually found that there was method in his madness.”
“Some folk might say there was madness in his method.””
—Sir Arthur Conan Doyle (1859–1930)