The Finite Element Method - Computer Graphics and Geometric Modeling - page 743

Figure 19.3. Two-dimensional elements.

The Variational Method. Here one tries to get a solution to the differential equa-
tion by translating it into a minimization problem for an energy function F that is a
linear functional defined on a function space W. One has a problem of the form

() ()

Find a function u Œ W so that F u £ Fv for all functions v Œ W . (19.1)

In general, the functions v correspond to continuously changing quantities such as
displacements of elastic bodies, temperature, etc., and F is an energy function asso-
ciated to the problem. The space W is usually infinite dimensional and so one replaces
it with a finite dimensional approximation W c generated by some “simple” functions.
The original problem (19.1) then becomes

() ()

Find a function u c Œ W so that F u c £ Fv for all functions v Œ W

c .

(19.2)

c

The choice of space W c is influenced by such factors as the particular formulation of
the variational problem, the desired accuracy of the solution, the regularity of the
exact solution, etc. It often consists of piecewise polynomial functions. The problem
in (19.2) then basically becomes one of solving a large system of linear or nonlinear
equations.
One issue here is whether the new solution u c in (19.2) is an adequate approxi-
mation to the actual solution u in (19.1). (Actually, there is also the mathematical
problem as to whether the variational problem (19.1) in fact has a solution in W,
because not all do since the space may not be closed. In our case, we are only inter-
ested in approximations and can choose a closed set W c .)
The basic steps in the FEM using the variational approach are:

(1) Translate the problem involving the differential equation into a variational one.
(2) Discretize using the FEM. This amount to specifying the space W c .
(3) Solve the discrete problem.

Doing (1) may involve defining an artificial functional for the problem. Solving the
variational problem may require less continuity than that of the actual solution. See
[MitW78].