Finite Element Methods - Discretization - General Form of The Finite Element Method

General Form of The Finite Element Method

In general, the finite element method is characterized by the following process.

• One chooses a grid for . In the preceding treatment, the grid consisted of triangles, but one can also use squares or curvilinear polygons.
• Then, one chooses basis functions. In our discussion, we used piecewise linear basis functions, but it is also common to use piecewise polynomial basis functions.

A separate consideration is the smoothness of the basis functions. For second order elliptic boundary value problems, piecewise polynomial basis function that are merely continuous suffice (i.e., the derivatives are discontinuous.) For higher order partial differential equations, one must use smoother basis functions. For instance, for a fourth order problem such as, one may use piecewise quadratic basis functions that are .

Another consideration is the relation of the finite dimensional space to its infinite dimensional counterpart, in the examples above . A conforming element method is one in which the space is a subspace of the element space for the continuous problem. The example above is such a method. If this condition is not satisfied, we obtain a nonconforming element method, an example of which is the space of piecewise linear functions over the mesh which are continuous at each edge midpoint. Since these functions are in general discontinuous along the edges, this finite dimensional space is not a subspace of the original .

Typically, one has an algorithm for taking a given mesh and subdividing it. If the main method for increasing precision is to subdivide the mesh, one has an h-method (h is customarily the diameter of the largest element in the mesh.) In this manner, if one shows that the error with a grid is bounded above by, for some and, then one has an order p method. Under certain hypotheses (for instance, if the domain is convex), a piecewise polynomial of order method will have an error of order .

If instead of making h smaller, one increases the degree of the polynomials used in the basis function, one has a p-method. If one combines these two refinement types, one obtains an hp-method (hp-FEM). In the hp-FEM, the polynomial degrees can vary from element to element. High order methods with large uniform p are called spectral finite element methods (SFEM). These are not to be confused with spectral methods.

For vector partial differential equations, the basis functions may take values in .