Graphics Reference
In-Depth Information
Figure 19.1.
Linear one-dimensional elements and nodes.
the region of interest into a mesh of small subregions, but how these subregions are
used is quite different from difference methods. Furthermore, for the finite difference
method the mesh is defined by orthogonal rows and columns, whereas the finite
element method allows much more general meshes.
The mathematical foundations of the FEM actually date back to variational
methods introduced in the early 1900s. One can think of the FEM as modern
applications of the Ritz variational and the Galerkin weighted residual methods in
numerical analysis. The overall basic idea is that one subdivides problem domains
into small parts called
(finite) elements
with associated simple solutions. These ele-
ments are then assembled by means of interconnections at boundary points called
nodes
. The collection of elements and nodes is called a
finite element mesh
. The simple
solutions corresponding to the elements are functions of unknown values at the nodes.
Let us expand on this a bit.
Suppose that we have a one-dimensional problem defined by a differential equa-
tion whose solution is a function f(x) defined over some interval [a,b]. See Figure
19.1(a). With the FEM what we do is subdivide the interval into subintervals [x
i-1
,x
i
],
where a = x
0
< x
1
< ...< x
n-1
< x
n
= b. Over each interval we define an approximation
to the solution to our problem. The positions x
i
are the
nodes
and we assume that
the relevant information about f is known at those points. We are free to choose our
approximations to f(x) over each subinterval (the
elements
). Typically, we would use
some sort of polynomial approximation. For example, if we choose linear approxi-
mations, then all we need to know are the values f
i
= f(x
i
) to define the elements (Figure
19.1(b)). From those we can define the polygonal approximation to f(x) correspon-
ding to the elements shown in Figure 19.1(c). Assembling the local solutions gives the
global approximation shown in Figure 19.1(d). On the other hand, we might want to
get a smoother approximation and try to use higher-degree polynomials. Of course,
the higher the degree of the polynomial, the more computationally expensive the solu-
tion is. In higher-dimensional problems even cubic polynomial can already get very
