Finite Element and Finite Difference Methods 19.3 The Equivalence of FEM and FDM Methods 1041
3. The good news is that both FDM and FEM must be
converged to more accurate solutions if the grid size
or mesh size h is close to zero. To utilize this na-
ture, the convergence test to evaluate the accuracy
is possible in discretized methods, although the ma-
trix size becomes bigger with a tiny grid or mesh
size. It should be noted that the grid (mesh) size
doesn’t have to be uniform. If the accuracy is to
be improved locally, for example, around a singu-
lar point, it is a good idea to use a smaller mesh size
around the targeted local domain. However, a rapid
change of mesh size distribution usually leads to bad
results. An adaptive mesh in FEM and a boundary-
fitted grid in FDM provide some solution to it. This
important matter will be discussed in Sect. 19.9.
4. The obtained finite element function u
h
(x) [for ex-
ample, in (19.35)] are continuous even between
elements, however, the finite element derivative
function ∂u
h
(x)/∂x is not continuous, which means
the derivative value at a node with two elements (in
the 1-D case) is not unique. To obtain the derivative
value, such as heat flux in heat conduction problems
and stress in solid mechanics (see Sect. 19.4), post-
processing is inevitable and is treated in Sect. 19.9.
5. Since a function is not defined within a grid in the fi-
nite difference method, there are no problems in the
derivations of the derivative value at a node. How-
ever you may have another problem regarding how to
select a proper definition among (19.16–19.18). Usu-
ally, the central difference scheme has a good rep-
utation due to second-order accuracy, but you need
some tricks in the multidimensional case [19.1].
6. Mathematically, FEM is governed by function anal-
ysis [19.3] since the concept of norm can be intro-
duced due to the shape of weak form. Using function
analysis, a new established finite element formula
can be evaluated in advance of real calculations in
a normal sense in functional analysis.
7. As was discussed, a finite element domain is packed
with element-wise finite element functions. Since the
field is represented by the summation of FE func-
tions, FEM is very strong at local resolution, for
example, in stress distributions in solid mechanics,
if the mesh is fine enough for the analysis. On the
contrary, if the mesh is poor (too coarse for the
problem), it is possible to obtain unexpected oscil-
lations, for example, in fluid dynamics. The stabi-
lized FEM, such as the streamline-upwind Petrov–
Galerkin method (SUPG) and the Galerkin/least
squares scheme (GLS), is one solution for it. A finite
difference domain is not filled for any functions that
are strong for unexpected oscillations, but not suit-
able for local resolution in solid mechanics.
8. As a general summary, FEM is good for arbitrary-
shaped domains, treatment of second boundary con-
ditions, solid mechanics, and parallel computations,
while FDM is only valid for regular shaped domains,
but suitable for fluid mechanics. The rough summary
table, considering other related numerical schemes,
is given in Table 19.1 in Sect. 19.9.
19.3 The Equivalence of FEM and FDM Methods
A reader might be wondering whether the treatment of
FDM and FEM are equivalent to each other. In this sec-
tion, we’ll consider the equivalence of these two forms
by explaining the variational method.
According to linear algebra [19.4], the following
three forms are equivalent, where K, f and u are a posi-
tive, definite symmetric matrix, a given force vector and
a unknown vector, respectively
(Form 1)
Ku− f =0 ,
(19.37)
(Form 2) w ·(Ku − f) =0 , ∀w ∈ R
n
,
(19.38)
(Form 3) F(v) −F(u) ≥0 , ∀v ∈R
n
,
(19.39)
where K =K
t
,
F(v) =
1
2
v ·Kv −v · f .
We now extend the discussion to the boundary-value
problem below [19.5]. Again consider the same prob-
lem as treated in Sect. 19.2 by (19.20)and(19.21),
which is Form 1 shown below. Form 2 is the weak
form already explained in (19.27). Form 3 is called the
variational form, which means the solution u is the min-
imizer of the function F(v).
(Form 1) k
∂
2
u
∂x
2
=−f in (0, 1) , u(1) =a ,
∂u
∂x
(0) =b ; (19.40)
(Form 2)
1
0
k
∂u
∂x
∂w
∂x
− fw
dz −kw(0)b =0 ,
∀w with w(1) =0 ; (19.41)
Part E 19.3