Chemistry Reference
In-Depth Information
where the vector
a
solves the minimization problem. We see that Eq. [22] is
very similar to the FD matrix Eq. [13], since the localized nature of the basis
leads to a highly banded matrix
K.
The last piece of the puzzle concerns how
to choose the polynomial basis functions
w
i
. Brenner and Scott
102
work
through a simple example using piecewise linear functions for the basis, and
for that basic case the analogy between the FD and FE formulations is appar-
ent. Higher order polynomial forms for the basis functions are described in the
review of Pask and Sterne.
62
Besides the basis set nature of the FE approach, the essential difference
between the FE and FD methods is manifested in Eq. [17] and the nature of the
boundary conditions. For the FE case,
the general boundary condition
f
ð
0
Þ¼
c
1
is required on one side of the domain, while a second boundary con-
f
0
ð
dition
c
2
is automatically implied by satisfaction of the variational
condition. (These two constants were assumed to be 0 for some of the discus-
sion above.) The first boundary condition is termed
essential
, while the second
is called
natural.
The FE method is called a
weak
formulation, in contrast to
the FD method, which is labeled a
strong
formulation (requiring both bound-
ary conditions from the start and twice differentiable functions). A clear state-
ment of these issues is given in the first chapter of Ref. 103, and the
equivalence of the strong and weak formulations is proven there. Most electro-
nic structure applications of FE methods have utilized zero or periodic bound-
ary conditions.
We can now see that a similar matrix representation is obtained for both
the FD and FE methods due to their near-local nature. If the grid is highly
structured and a low-order representation is assumed, the two representations
become virtually indistinguishable. The FE method has the advantages that it
is variational from the beginning, and it allows for greater flexibility in the
arrangement of the mesh. On the other hand, the FD method is nonvariational
in the sense of convergence from above, but the numerical representation is
highly structured and generally more banded (fewer terms to represent the
action of the Laplacian operator) than for the FE case. Thus, the preferred
choice between the FD and FE representations depends on the nature of the
problem, and the tastes of the practitioner.
1
Þ¼
Iterative Updates of the Functions, or Relaxation
To solve the Poisson equation iteratively, we make an initial guess for the
function values over the domain and then update Eq. [5] numerically. We first
write the spatial part of Eq. [5] in FD form:
qf
ð
x
i
Þ
qt
¼
1
h
2
½
f
ð
x
i
1
Þ
2
f
ð
x
i
Þþ
f
ð
x
i
þ1
Þþ
4
pr
ð
x
i
Þ
½
23
