Page 386 - Process Modelling and Simulation With Finite Element Methods

A MATLAB/FEMLAB Primer for Vector Calculus 3 73

V2$ = 0 (A20)

is an example where the Laplacian is known (zero) but the function @ is to be
found. FEMLAB routinely computes the first derivatives of a dependent
variable, but not necessarily the second, directly. But as we already know how
to compute both div and grad separately, computing div(grad) is a matter of
using auxiliary dependent variables vl,v2,v3 that are assigned values in the last
example of
Fl=vl-ux; F2zv2-u~; F3zv3-u~
so that

v2u = vlx + v2y + v3z (A21)

Scalar and Vector Potentials

Quite a lot of space in vector calculus books is devoted to the topics of scalar
and vector potentials.
A scalar potential @ for a vector field F is a scalar function for which V@=F.
The textbooks show that this is only possible if, and only if, curl F=O.
Similarly, a vector potential A for a vector field F is a vector function for which
F=curl A. Again, the textbooks show that this is only possible, if and only if,
div F=O.
Scalar and vector potentials are useful for simplifying pde systems that are
either irrotational or divergence free (solenoidal). In the case of fluid flow,
either inviscid or completely viscous flow are simplified dramatically by such
potentials. One might ask, can FEMLAB help in the task of identifying these
potentials? In the case of 2-D flows, we already saw that the streamfunction acts
like a vector potential (3.3), so the answer is a qualified yes. For many years in
both electrodynamics and hydrodynamics, the hunt for vector potentials or scalar
potentials to simplify calculations was paramount - many analyses end is
solving, even approximately, for such a potential. Yet whether sufficient
symmetries exist in a given modeling situation to use scalar and vector potentials
to simplify the calculations is now almost a moot point. General pde engines
like FEMLAB can compute numerical approximations to the primitive variables
in the most general cases, limited only by their CPU requirements. So the virtue
of finding such simplifications is a reduction of CPU usage, for which we must
still pay the price of numerical differentiation to arrive at the primitive variables
(using our grad and curl recipes) if detailed solutions are required.
It is perhaps a sobering note to end our Appendix on that general purpose
numerical solvers like FEMLAB limit the need for many of the complexities of