¦ ¦
0
is called the Laplacean operator and appears in many models of physical and mechanical
phenomena. These models often lead to more general elliptic operators of the form
‚ ‚
—
¥ 6 6
0 0
¦ ¦ ¦ ¦
Y
t
k‚B ‚
¨ ¤ &¥
C
‚
where the scalar function depends on the coordinate and may represent some speci¬c
parameter of the medium, such as density, porosity, etc. At this point it may be useful to
recall some notation which is widely used in physics and mechanics. The operator can
§§ §§
be considered as a vector consisting of the components and . When applied to a
scalar function , this operator is nothing but the gradient operator, since it yields a vector
V 3
% %
§§ §§
with the components and as is shown in (2.4). The dot notation allows dot products
6
of vectors in to be de¬ned. These vectors can include partial differential operators. For
¤ % £ V V ‚ 3
¢
example, the dot product of with yields the scalar quantity,
%
0V
6V 3
6© — 0 ¦ ¦
¤ % ¢ ¤V
which is called the divergence of the vector function . Applying this divergence %
‚ ‚
operator to , where is a scalar function, yields the operator in (2.5). The ¥
3
V ¤ ¤ c ¤
divergence of the vector function is often denoted by div or . Thus,
6 — 00 ¤ ‚¢ ¤ 6
div ¦ ¦
”’©§ d¥ u„£
un u n
¢ ¢ ¡
£ § © ¥ ©¨¤© £ £
¡ ©¢¥ ¡
The closely related operator
‚ ‚
— 6
0
¥ 6 6
0 0
¦ ¦ ¦ ¦
y¤ ‚ B ‘t ¥
¨
&
C ¡
is a further generalization of the Laplacean operator in the case where the medium is
T‚ ‚
67 0 ¦
anisotropic and inhomogeneous. The coef¬cients depend on the space variable and
re¬‚ect the position as well as the directional dependence of the material properties, such as
porosity in the case of ¬‚uid ¬‚ow or dielectric constants in electrostatics. In fact, the above
B ‚ ¥
operator can be viewed as a particular case of , where is a matrix
C
which acts on the two components of .
¦ ¥
¨ ! 3 T$!¢ ¡! 4%
¨
B¢ ¥A
A¡ ¡ ¨
3©
!PG 'F
Many physical problems involve a combination of “diffusion” and “convection” phenom
ena. Such phenomena are modeled by the convectiondiffusion equation
w C k‚B ‚ 6 6 ™— 0
— ¡
V V V
—
0
¢ V
¦ ¦
or
¡ G ‚kB ‚ ¤
— C
V
—
¢ V V
the steadystate version of which can be written as
y¡ ¤ C w‚B c ‘t ¥
— ¨
3 (&
V V
Problems of this type are often used as model problems because they represent the simplest ¤
form of conservation of mass in ¬‚uid mechanics. Note that the vector is sometimes quite
large, which may cause some dif¬culties either to the discretization schemes or to the
iterative solution techniques.
! ©¤( &% ¨ ' ¢ ¥ ¥ £ ! ( £ ¥
£
§
) ©
c

The ¬nite difference method is based on local approximations of the partial derivatives in
a Partial Differential Equation, which are derived by low order Taylor series expansions.
The method is quite simple to de¬ne and rather easy to implement. Also, it is particularly
appealing for simple regions, such as rectangles, and when uniform meshes are used. The
matrices that result from these discretizations are often well structured, which means that
they typically consist of a few nonzero diagonals. Another advantage is that there are a
number of “fast solvers” for constant coef¬cient problems, which can deliver the solution