Graphics Reference
In-Depth Information
The Galerkin Method.
In this method we start with an approximation g(x) to a
solution f(x), where g(x) is a linear combination of functions g
i
(x) which are indexed
by the nodes:
()
=
()
+
()
++
()
gx
ag x
a g x
...
a g x
.
(19.3)
11
2 2
nn
The g
i
(x) are typically B-spline type functions that are piecewise polynomials and
vanish everywhere except on the elements adjacent to the ith node. They are called
the
global shape functions
. Let
(
)
=
()
-
()
Rxa
,
fx
gx
.
(19.4)
be the error function, also called the
residual
. It depends both on x and the unknowns
a
i
. The
method of weighted residuals
then tries to solve for the a
i
by solving
Ú
() (
)
=
wxRxa
,
0
.
(19.5)
D
where
D
is the domain of the problem and the w(x) are one or more suitable “weight-
ing” functions. If one applies the boundary conditions of the problem one gets a
system of linear equations in the a
i
that is then solved to get the solution (19.3). The
Galerkin method uses the n global shape functions as weighting functions, that is,
w
i
(x) = g
i
(x). Requiring (19.5) to hold for these functions then gives n equation in the
n unknown a
i
.
Many of the problems to which the FEM is applied are a case of finding approx-
imations to functions u that
(1) solve equations of the form
Au
= ,
where A is a linear differential operator satisfying boundary conditions, and
(2) minimize some linear functional of the form
()
=
()
<
F u
12
Au u
,
> - <
u f
,
>
,
where <,> is an inner product of two functions.
19.4
An Example
We shall work through a fairly standard example that is an application of the FEM
using the Galerkin method. Even though our one-dimensional example will be very
simple, it nevertheless shows how the FEM works and more complicated examples
do not involve anything different.
Our example is a one-dimensional heat conduction problem. See Figure 19.4.
What we have is a rod of constant cross section and length L. We assume a given heat