Conforming Finite Elements - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 60

§ 5. Some Standard Finite Elements

In practice, the spaces over which we solve the variational problems associated with
boundary-value problems are called finite element spaces . We partition the given
domain into (finitely many) subdomains, and consider functions which reduce to
a polynomial on each subdomain. The subdomains are called elements . For planar
problems, they can be triangles or quadrilaterals. For three-dimensional problems,
we can use tetrahedra, cubes, rectangular parallelepipeds, etc. For simplicity, we
restrict our discussion primarily to the two-dimensional case.
Here is a list of some of the important properties characterizing different finite
element spaces:
1. The kind of partition used on the domain: triangles or quadrilaterals. If all
elements are congruent, we say that the partition is regular .
2. In two variables, we refer to

c ik x i y k

P t : ={ u(x, y) = } ( 5 . 1 )

i + k ≤ t
i, k ≥ 0

as the set of polynomials of degree ≤ t. If all polynomials of degree ≤ t are

used, we call them finite elements with complete polynomials .
The restrictions of the polynomials to the edges of the triangles or quadrilat-
erals are polynomials in one variable. Sometimes we will require that their
degree be smaller than t (e.g., at most t − 1). Such a condition will be part

of the specification of the elements.
The admissible polynomial degrees in the elements or on their edges are a
local property.
3. Continuity and differentiability properties: A finite element is said to be a
C k element provided it is contained in C k () . 5 This property is of a global
character and is often concealed in interpolation conditions.
We remark that according to this scheme, the Courant triangles in Example 4.3
would be classified as linear triangular elements in C 0 () .
We use the terminology conforming finite element if the functions lie in the
Sobolev space in which the variational problem is posed. Nonconforming elements
will be studied in Chapter III.

5 The use of the terminology element may be somewhat confusing. We decompose the
domain into elements which are geometric objects, while the finite elements are actually
functions. However, we will deviate from this convention when discussing, e.g., C k elements
or linear elements, where the meaning is clear from the context.