Page 78 - Process Modelling and Simulation With Finite Element Methods

Partial Differential Equations and the Finite Element Method 65

associated law of physics - from the wavefunction in Schrodinger’s equation to
classical electrodynamics. Thus, classification and solution of second order
spatial temporal systems in 2-D and 3-D are of wide applicability and
importance in the sciences and engineering. For this reason, and that finite
element methods (FEM) are intrinsically well-suited to treating second order
systems, FEM are techniques with wide applicability.
In this chapter, we focus on second order systems in 2-D and 3-D. There
are three canonical exemplar systems that are nearly uniformly treated in the
standard textbooks. We shall not disappoint. They are:
a2u
+
Laplace’s equation (elliptic): - 7
0
=
ax2 ay
au a2u
Diffusion equation (parabolic): - -
=
at ax2
a2u aZu
Wave equation (hyperbolic): - - (2.3)
=
at2 ax2
The terms elliptic, parabolic and hyperbolic are traditional guides to the features
of a PDE system from characterization of the linear terms by reference to the
general linear, second-order partial differential equations in one dependent and
two independent variables:
a2u a2u aZu au au
~-+2b- +c?+d-+e-+fu+g =O (2.4)
ax2 axay ay ax ay

where the coefficients are functions of the independent variables x and y only, or
constant. The three canonical forms are determined by the following criterion:

elliptic: b2-ac<0 (2.54
parabolic: b2 -ac = 0 (2.5b)

hyperbolic: b2 - uc > 0 (2.5c)
These classifications serve as a rough guide to the information flow in the
domain. For instance, in elliptic equations, information from the boundaries is
propagated instantaneously to all interior points. Thus, elliptic equations are
termed “non-local”, meaning that information from far away influences the given
position, versus “local”, where only information from nearby influences the field
variable. In parabolic systems, information “diffuses”, i.e. it spreads out in all
directions. In hyperbolic systems, information “propagates”, i.e. there is a