Chapter 11
Numerical Methods: Finite Elements
In Chapter 5, we introduced the first, the oldest, and in many ways the simplest class
of numerical algorithms for approximating the solutions to partial differential equations:
finite differences. In the present chapter, we study the second of the two major numerical
paradigms: the finite element method. Finite elements are of more recent vintage, having
first appeared soon after the Second World War; historical details can be found in [
134
].
As a consequence of their flexibility and ability to readily handle complicated geometries,
finite elements have, in many situations, become the basic method of choice for solving
equilibrium boundary value problems governed by elliptic partial differential equations.
Extensions to dynamical problems are also extensively utilized, but these will not be treated
here due to lack of space.
Finite elements rely on a more sophisticated understanding of the partial differen
tial equation, in that, unlike finite differences, they are not obtained by simply replacing
derivatives by their numerical approximations. Rather, they can be based on an associated
minimization principle which, as we learned in Chapter 10, can be used to characterize the
unique solution to a positive definite boundary value problem. Restricting the minimizing
functional to an appropriately chosen finitedimensional subspace of functions reduces it
to a finitedimensional minimization problem, that can then be solved by numerical linear
algebra. When properly formulated, the resulting finitedimensional minimization problem
has a solution that well approximates the true minimizer.
After setting up the basic framework, we illustrate the basic constructions in the
context of boundary value problems for ordinary differential equations.
The following
section then extends finite element analysis to boundary value problems associated with
the twodimensional Laplace and Poisson equations. A rigorous justification and proof of
convergence of the finite element approximations requires further analysis, and we refer
the interested reader to [
134
,
151
]. In this chapter, we shall concentrate on understanding
how to formulate and implement the finite element method in practical situations.
11.1.
Minimization and Finite Elements.
To explain the key ideas behind the finite element method, we return to the abstract
framework that was developed in Chapter 10. Recall that we can characterize the solution
to a positive definite boundary value problem as the function
u
⋆
∈
U
, belonging to a cer
tain infinitedimensional function space, that minimizes an associated quadratic functional
Q
:
U
→
R
.
This sets the stage for the first key idea of the finite element method.
Instead of
trying to minimize the functional
Q
over the entire infinitedimensional function space,
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2012
Peter J. Olver
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we will seek to minimize it over a
finitedimensional subspace
W
⊂
U
.
The effect is to
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 Fall '10
 Olver
 Differential Equations, Numerical Analysis, Equations, Partial Differential Equations, Finite Element Method, Partial differential equation, Peter J. Olver

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