On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials.
We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.
Introduction
. In this paper, we consider the accuracy of finite volume element
methods for the following elliptic boundary value problem: Find
such that
where# is a bounded convex polygon in R 2 with boundary #
uniformly positive definite matrix in # and the source term
has enough regularity so that this boundary value problem has a unique
solution in a certain Sobolev space.
Finite volume (FV) methods have a long history as a class of important numerical
tools for solving di#erential equations. In the early literature [26, 27] they were
investigated as the so-called integral finite di#erence methods, and most of the results
were given in one-dimensional cases. FV methods also have been termed as box
schemes, generalized finite di#erence schemes, and integral-type schemes [20]. Generally
speaking, FV methods are numerical techniques that lie somewhere between finite
di#erence and finite element methods; they have a flexibility similar to that of finite
element methods for handling complicated solution domain geometries and boundary
conditions; and they have a simplicity for implementation comparable to finite di#er-
ence methods with triangulations of a simple structure. More important, numerical
solutions generated by FV methods usually have certain conservation features that
# Received by the editors March 10, 2000; accepted for publication (in revised form) May 25,
2001; published electronically January 30, 2002. This work was partially supported by NSF grant
DMS-9704621, U.S. Army grant ARO-39676-MA, and by the NSERC of Canada.
http://www.siam.org/journals/sinum/39-6/36887.html
Institute for Scientific Computation, Texas A&M University, College Station,
(ewing@isc.tamu.edu). This author was supported by NSF grants DMS-9626179, DMS-9706985,
DMS-9707930, NCR9710337, DMS-9972147, INT-9901498; EPA grant 825207; two generous awards
from Mobil Research and Development; and Texas Higher Education Coordinating Board Advanced
Research and Technology Program grants 010366-168 and 010366-0336.
# Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg,
VA 24061 (tlin@math.vt.edu).
- Department of Mathematics, University of Alberta, Edmonton, AL T6G 2G1 Canada
(ylin@hilbert.math.ualberta.ca).
are desirable in many applications. However, the analysis of FV methods lags far
behind that of finite element and finite di#erence methods. Readers are referred to
[3, 6, 17, 21, 22, 25] for some recent developments.
The FVE method considered in this paper is a variation of the FV method, which
also can be considered as a Petrov-Galerkin finite element method. Much has been
published on the accuracy of FVE methods using conforming linear finite elements.
Some early work published in the 1950s and 1960s can be found in [26, 27]. Later, the
authors of [20] and their colleagues obtained optimal-order H 1 error estimates and
superconvergence in a discrete H 1 norm. They also obtained L 2 error estimates of
the following form:
where u and u h comprise the solution of (1.1) and its FVE solution, respectively. Note
that the order in this estimate is optimal, but its regularity requirement on the exact
solution seems to be too high compared with that for finite element methods having
an optimal-order convergence rate when the exact solution is in W 2,p or H
Optimal-order H 1 estimates and superconvergence in a discrete H 1 norm also have
been given in [3, 17, 21, 22, 25] under various assumptions on the above form for
equations or triangulations.
More recently, the authors of [7, 8] presented a framework based on functional
analysis to analyze the FVE approximations. The authors in [11] obtained some new
error estimates by extending the techniques of [20]. The authors of [14, 15] considered
FVE approximations for parabolic integrodi#erential equations, covering the above
boundary value problems as a special case, in both one and two dimensions. All the
authors obtained optimal-order H 1 and W 1,# error estimates and superconvergence
in H 1 and W 1,# norms. In addition, they found an optimal-order L # error estimate
in the following form:
which is in fact an error estimate without any logarithmic factor. However, all the
estimates obtained by these authors require that the exact solution have H 3 regularity.
To the best of our knowledge, there have been no results indicating whether the
above W 3,p
regularity is necessary for the FVE solution with conforming linear
finite elements to have the optimal-order convergence rate. On the other hand, it
is well known that in many applications the exact solution of the boundary value
problem cannot have W 3,p or H 3 regularity. In fact, the regularities of the source
f , the coe#cient, and the solution domain all can abate the regularity of the
exact solution. A typical case is the regularity of the solution domain that may force
the exact solution not to be in W 3,p or H 3 even for the best possible coe#cient A
and source term f , such as constant functions.
It has been noticed that the regularity of the source term may a#ect the convergence
rate of an FVE solution. The counterexample in [18] showed that the FVE
solution with the conforming linear elements cannot have the optimal L 2 convergence
rate if the exact solution is in H 2 but the source term f is only in L 2 . On the other
hand, the author of [6] found an optimal error estimate for the FVE solution with the
nonconforming Crouzeix-Raviart linear element under the assumption that the exact
solution is in H 2 and the source term f is in H 1 , but did not state whether this H 1
regularity of f is necessary for the FVE method presented there.
The central aim of this paper is to show how, by both error estimates and coun-
terexamples, the regularity of the source term f can a#ect the convergence rate of the
FVE solution with conforming linear elements. The results indicate that, unlike the
finite element method, the H 2 regularity of the exact solution indeed cannot guarantee
the optimal convergence rate of the conforming linear FVE method if the source
term has a regularity worse than H 1 , assuming that the coe#cient is smooth enough.
Namely, we will present the following error estimate:
which leads to the optimal convergence rate of the FVE method only if f # H # with
# 1. Note first that, except for special cases such as when the dimension
of# is
one or the solution domain has a boundary smooth enough, the H 1 regularity of the
source term does not automatically imply the H 3 regularity of the exact solution. On
the other hand, the H 3 regularity of the exact solution will lead to the H 1 regularity
of the source term when the coe#cient is smooth enough, and this error estimate
reduces to one similar to estimates obtained in [11, 20]. Also, this error estimate is
optimal from the point of view of the best possible convergence rate and the regularity
of the exact solution. Moreover, counterexamples given in this paper indicate that
the regularity of the source term cannot be reduced. Hence, we believe this is a more
general error estimate than those in the literature.
In fact, the FVE method is a Petrov-Galerkin finite element method in which
the test functions are piecewise constant. As we will see later, the nonsmoothness in
the test function demands a stronger regularity of the source term than the Galerkin
finite element method. Also, our view of the FVE method as a Petrov-Galerkin finite
element method suggests that we treat the FVE method as a perturbation of the
Galerkin finite element method [6, 20] so that we can derive optimal-order L 2 , H 1 ,
and L # error estimates with a minimal regularity requirement just like finite element
methods except for the additional smoothness assumption on the source term f . This
error estimation framework also enables us to investigate superconvergence of the FVE
method in both H 1 and W 1,# norms using the regularized Green's functions [23, 29]
and to obtain the uniform convergence of the FVE method similar to that in [24]
for the finite element method. To summarize, we observe that the FVE method not
only preserves the local conservation of certain quantities of the solution (problem
but also has optimal-order convergence rates in all usual norms. The
additional smoothness requirement on the source term f is necessary due to the
formulation of the method.
The results of this paper can easily be extended to cover more complicated models.
For example, most of the results and analysis framework are still valid if the di#erential
equation contains a convection term # - (b u) (see [21] and [22]) and the symmetry
of the tensor coe#cient A(x) is not critical. Also, one may consider Neumann and
Robin boundary conditions on the whole or a part of the boundary #
In fact, the
FVE method was introduced in [2] as a consistent and systematic way to handle the
flux boundary conditions for finite di#erence methods. We also refer readers to [1, 19]
for FVE approximations of nonlinear problems, to [12] for an immersed FVE method
to treat boundary value problems with discontinuous coe#cients, and to [13] for the
mortar FVE methods with domain decomposition.
This paper is organized as follows. In section 2, we introduce some notation,
formulate our FVE approximations in piecewise linear finite element spaces defined
on a triangulation, and recall some basic estimates from the literature. All error
estimates are presented in the pertinent subsections of section 3. Section 4 is devoted
to counterexamples demonstrating that smoothness of the source term is necessary in
order for the FVE method to have the optimal-order convergence rate.
2. Preliminaries.
2.1. Basic notation. We will use the standard notation for Sobolev spaces
consisting of functions that have generalized derivatives of
order s in the spaces L
p(#7 The norm of W s,p is defined by
|#s
with the standard modification for In order to simplify the notation, we denote
s(# and skip the index
and# whenever possible; i.e., we will use
We denote by H 1
0(# the subspace of H
1(# of functions
vanishing on the boundary
# in the sense of traces. Finally, H
-1(# denotes the
space of all bounded linear functionals on H 1
0(#7 For a functional f # H
its
action on a function u
0(# is denoted by (f, u), which represents the duality
pairing between H
0(#1 To avoid confusion, we use (-) to denote both
the L
2(#89#401 product and the duality pairing between H
For the polygonal
domain# we now consider a quasi-uniform triangulation T h
consisting of closed triangle elements K such
K. We will use N h to
denote the set of all nodes or vertices of T h ,
is a vertex of element K # T h and p #
-#
and we let N 0
For a vertex x i # N h , we denote by #(i) the index set of
those vertices that, along with x i , are in some element of T h .
We then introduce a dual mesh T #
h based on T h ; the elements of T #
h are called
control volumes. There are various ways to introduce the dual mesh. Almost all
approaches can be described by the following general scheme: In each element K # T h
consisting of vertices x i , x j , and x k , select a point q in K, and select a point x ij on
each of the three edges x i x j of K. Then connect q to the points x ij by straight lines
# ij,K . Then for a vertex x i we let V i be the polygon whose edges are # ij,K in which x i
is a vertex of the element K. We call V i a control volume centered at x i . Obviously,
we have
and the dual mesh T #
h is then defined as the collection of these control volumes.

Figure

1 gives a sketch of a control volume centered at a vertex x i .
We call the control volume mesh T #
h regular or quasi-uniform if there exists a
positive constant C > 0 such that
here h is the maximum diameter of all elements
There are various ways to introduce a regular dual mesh T #
h depending on the
choices of the point q in an element K # T h and the points x ij on its edges. In
Fig. 1. Control volumes with barycenter as internal point and interface # ij of V i and V j .
this paper, we use a popular configuration in which q is chosen to be the barycenter
of an element K # T h , and the points x ij are chosen to be the midpoints of the
edges of K. This type of control volume can be introduced for any triangulation
T h and leads to relatively simple calculations for both two- and three-dimensional
problems. In addition, if T h is locally regular, i.e., there is a constant C such that
dual mesh
h is also locally regular. Other dual meshes also may be used. For example, the
analysis and results of this paper for all the error estimates in the H 1 norm are still
valid if the dual mesh is of the so-called Voronoi type [21].
2.2. The FVE method. We now let S h be the standard linear finite element
space defined on the triangulation T h ,
is linear for all K # T h and v|
and its dual volume element space S # h ,
h and v|
Obviously,
h } and S #
# i are the standard nodal basis functions associated with the node x i , and # i are the
characteristic functions of the volume V i . Let I h :
be the usual interpolation operators, i.e.,
I h
where
Then, the FVE approximation u h of (1.1) is defined as a solution to the following
problem: Find u h # S h such that
or
h .
Here the bilinear form a(u, v) is defined as follows:
where n is the outer-normal vector of the involved integration domain. Note that the
bilinear form a(u, v) has di#erent definition formulas according to the function spaces
involved. We hope that this will not lead to serious confusion but rather will simplify
tremendously the notation and the overall exposition of the material.
To describe features of the bilinear forms defined in (2.3), we first define some
discrete norms on S h and S # h ,
is the distance between x i and x j .
In the lemmas below, we assume that the lines of discontinuity (if any) of the
matrix A(x) are aligned with edges of the elements in the triangulation T h and that
the entries of the matrix A(x) are C 1 -functions over each element of T h .
Lemma 2.1 (see, e.g., [7, 21]). There exist two positive constants C 0 , C 1 > 0,
independent of h, such that
Lemma 2.2 (see, e.g., [7, 21]). There exist two positive constants C 0 , C 1 > 0,
independent of h and h 0 > 0, such that for all 0 < h # h 0 ,
3. Error estimates for the FVE method.
3.1. Optimal-order H 1 error estimates. We first consider the error of the
FVE solution u h in the H 1 norm. We start with the following two lemmas.
Lemma 3.1. For any u h , v h # S h , we have
with
j#Nh
and
Moreover, if A is in W
1,# , then there is a positive constant C > 0, independent
of h, such that
Proof. For the proof, see [12, 13].
Lemma 3.2. Assume that u h is the FVE solution defined by (2.1). Then we have
Proof. The proof follows directly from Lemma 3.1.
Theorem 3.3. Assume that u and u h are the solutions of (1.1) and (2.1), re-
spectively,
-1+# with 0 < # 1, and A # W
1,# .
Then we have
||f
||u|| 1+# .
Proof. By (3.1) and (1.1), we see that for #
Notice that from Lemma 2.2 and the approximation theory we have
the proof is then completed by combining these inequalities.
Remark. The main idea in the proof above is motivated by [6], which is somewhat
di#erent from those ideas in [3, 17, 20, 21, 25]. The approach is also more direct and
simpler because the key identity (3.2) allows us to employ the standard error estimation
procedures developed for finite element methods. In particular, the estimate for
||I h u - u h || is not needed in this proof. Moreover, the estimate here describes how
the regularities of the exact solution and the source term can independently a#ect the
accuracy of the FVE solution.
3.2. Optimal-order L 2 error estimates. In this section, we derive an optimal-
order L 2 error estimate for the FVE method with the minimal regularity assumption
for the exact solution u. This error estimate also will show how the error in the L 2
norm depends on the regularity of the source term.
The following lemma gives another key feature of the bilinear form in the FVE
method.
Lemma 3.4. Assume that u h , v h # S h . Then we have
Proof. It follows from Green's formula that
and
j#Nh
Then the proof is completed by taking the di#erence of these two identities.
Theorem 3.5. Assume that u and u h are the solutions of (1.1) and (2.1), respec-
tively, and u # H 2
2,# . Then there exists a
positive constant C > 0 such that
Proof. Let
0 be the solution of
# , and
# .
Then we have ||w|| 2 # ||u - u h || 0 . By Theorem 3.3 we have
Then by Lemma 3.4,
where the J i 's are defined for u h , w h # S h by
and the continuity of #u - n on each #K is used.
Since the dual mesh is formed by the barycenters, we have
(w h - I # h w h
so that
where f K is the average value of f on K. Similarly, using the fact that A # W 2,# ,
we have
For J 3 , according to the continuity of #u - n and the shape of the control volume,
we have
AK is a function designed in a piecewise manner such that for any edge E of a
triangle
and x c is the middle point of E. Since |A(x)-
AK | # h||A|| 1,# , we have from Theorem
3.3 that
Thus, it follows by taking w
therefore, we have
and the proof is completed.
Corollary 3.6. Assume that u # H
# with 0 < # 1, and
A # W
2,# . Then we have
Proof. Let f h be the L 2 projection of f into S h and consider S(u,
linear operator from H s
s > 0. For
any (u, f) # H s
-H -1+s , we let
-1+s .
Then, by Theorem 3.5, we have
Hence, according to the theory of interpolation spaces [4, 5], we have
which in fact is (3.7).
Remark. When the source term f is in H 1 , the order of convergence in Theorem
3.5 is optimal with respect to the approximation capability of finite element space.
Note that, in many applications, the H 1 regularity of f does not imply the W 3,# or
regularity of the exact solution required by the L 2 norm error estimates in the
literature. Moreover, counterexamples presented in the next subsection indicate that
the regularity assumption on f cannot be reduced. The result in Theorem 3.5 reveals
how the regularities of the exact solution and the source term can a#ect the error of
the FVE solution in the L 2 norm, and this is a more general result than those in the
literature.
3.3. Superconvergence in the H 1 norm. In a way similar to the finite element
solution with linear elements, we can show that the FVE solution has a certain
superconvergence in the H 1 norm when the exact solution has a stronger regularity
and the partition used has a better quality. Specifically, throughout this subsection
we assume that the involved partition for the FVE solution is uniform or piecewise
uniform without any interior meeting points. This requirement might be relaxed (see,
for example, [29]), but we would rather use this simpler assumption to present our
basic idea.
We first recall the following superconvergence estimates for the Lagrange interpolation
[9, 28, 29, 30] from finite element theory.
Lemma 3.7. Assume that u # W 3,p
0(# . We have
Theorem 3.8. Assume that f # H
2,# .
Then we have
Proof. It follows from Lemma 3.7 that
Following a similar argument used in the proof of Theorem 3.5, we see that
because I h u - u h is in S h . The result of this theorem follows by combining these two
inequalities.
We can use one of the applications of the above superconvergence property of the
FVE solution to obtain a maximum norm error estimate.
Corollary 3.9. Under the assumptions of Theorem 3.8 and u # W 2,#
3(# , we have
logh
Proof. The proof follows from Theorem 3.8 and from the approximation theory
stating that
logh
logh
We remark that this result is not optimal with respect to the regularity required
on the exact solution u. This excessive regularity can be removed according to the
result in the following subsection.
3.4. Error estimates in maximum norm. Now we turn to the L # norm and
W 1,# norm error estimates for the FVE solution. First, we recall from [10, 16, 23, 29]
the definition and estimates on the regularized Green's functions.
For a point
2(# to be the
solution of the equation
in# ,
is a smoothed #-function associated with the point z, which has the
following properties:
Let G z
h be the finite element approximation of the regularized Green's function,
i.e.,
a(G z
-G z
Following [29], for a given point z
# we define # z G z by
G z+#z
-G z
|#z|
for any fixed direction L in R 2 , where #z//L means that #z is parallel to L. Clearly,
# z G z satisfies
The finite element approximation # z G z
h of # z G z is then defined by
It is well known that the functions G z and # z G z have the following properties [29]:
For any w # H 1
where P h is an L 2 -projection operator on S h , i.e.,
Moreover, the following estimates have been established in the literature
[10, 16, 23, 29]:
#G z
-G z
# z G z
||G z
||# z G z
#u# 2,#
with constant C > 0 independent of h and z.
First, let us consider the W 1,# norm error estimate.
Theorem 3.10. Assume that u # W
# , and A # W
1,# .
Then there exist positive constants C > 0 and h 0 > 0 independent of u such that for
Proof. It follows from (3.8) that
For the second term on the right-hand side, we have
(f, # z G z
logh
For the third term, by the definition of E h given in Lemma 3.1 and the fact that # z G z
is a piecewise linear polynomial, we have
logh ||u|| 1,# .
Thus, we obtain
logh
so that we have for some h 0 > 0, such that 0 < h # h 0 ,
logh
Applying this inequality and (3.13) in
leads to the result of this theorem.
The following theorem gives a maximum norm error estimate for the FVE
solution.
Theorem 3.11. Assume that u # W
1,# , and A # W
2,# .
Then there exist constants C > 0 and h 0 > 0, independent of u, such that for all
Proof. We follow an idea similar to the proof of the previous theorem, but we
now use the regularized Green's function G z and its finite element approximation G z
as follows:
-G z
-G z
-G z
The functionals J 1 , J 2 , and J 3 above are defined in the same way as given in the proof
of Theorem 3.5. For J 1 (u h , G z
h ), from (3.6) we have
||G z
logh ||f || 1,# .
Similarly, we have
||G z
We know by Theorem 3.10 that
# Ch logh
Therefore, there exists a small h 0 > 0 such that for 0 < h # h 0 ,
logh
As for J 3 (u h , G z
), we note that G z
is a piecewise linear polynomial and
Thus, it is easy to see from Theorem 3.10 and (3.11) that
||G z
logh
Combining the estimates obtained above for the J i 's, we have
logh
This together with (3.13) completes the proof.
The following theorem gives a superconvergence property in the maximum norm
for the FVE solution.
Theorem 3.12. Under the same conditions as in Theorem 3.11, we have
Proof. It follows from the properties of # z G z
h and # z G z and from Lemma 3.7 that
||u|| 3,# z G z
We see from (3.6) and (3.11) that
logh ||f || 1,# .
When h > 0 is small, we also have
logh
For
h ), we have
|# z G z
because # z G z
h is piecewise linear in each element K # T h . Finally, the proof is
completed by combining the above estimates.
3.5. Uniform convergence for u in H 1
In many applications, the exact
solution u of (1.1) may be in the space H
1(#5 but not in H
1+# for any # > 0. In
this situation, the authors of [24] showed that for any # > 0, there exists h
such that for all 0 < h # h 0 , we have
for the Galerkin finite element solution u h # S h (or the Ritz projection of u into S h
of the exact solution of (1.1)). This implies that u h converges to u uniformly even
though there is no order of convergence for u h .
The following theorem shows that the FVE solution also has this uniform convergence
feature.
Theorem 3.13. Assume that A is uniformly continuous and f # L
. Let
0(# and u h # S h be the solutions of (1.1) and (2.1), respectively. Then for any
# > 0, there exists h such that for all 0 < h # h # , the following holds:
Proof. As in the proof in Theorem 3.3, we have
uniformly continuous in #
for any # 0 > 0, there exists h
. Thus, by Lemma 3.7 we can take h # (0, h 0 ) to obtain
is defined in Lemma 3.1. By Lemma 3.2, we have
Thus it follows from the triangle inequality that
Lemma 2 of [24] indicates that for any # 1 > 0, there exists h
Notice that the constant C > 0 above is independent of u, f , and A; therefore, the
theorem follows from the last two inequalities.
4. Counterexamples. In this section, we will present two examples to show
that, when the source term f(x, y) is only in L
2(#7 the FVE solution generally cannot
have the optimal second-order convergence rate even if the exact solution u(x, y) has
the usual H 2 regularity. The first example is based on theoretical error estimates,
while the second is presented through numerical computations. We also provide an
example to corroborate the optimal error estimate obtained in this paper under the
condition that the exact solution u is in H 2 and the source term f is in H 1 .
4.1. A one-dimensional example. First, we consider an example in one
dimension,
but is not in H 1 (0, 1) if 0 # < 1/2. Clearly this problem has an
exact solution,
x 2-#
which is in the space H 2 (0, 1).
Let T h be the uniform partition of the interval [0, 1] such that x
0, 1, . , N and x 1. Let S h be the piecewise
linear finite element space. Let u f # S h be the finite element solution of (4.1) defined
by
and let u h be the FVE solution. Then we have
with e
#h
Our main task is to show that there exists a constant C > 0 such that
This inequality and
together imply that the FVE solution cannot have the optimal L 2 norm convergence
rate for 0 < # < 1/2.
We start with the estimates of the error function e(x) at the nodes. Let G(x, y)
be the Green's function defined by
Then, we have
x k-1/2
xk
Now we will estimate the J l 's one by one under the assumptions that
# .
For J 1 and J 6 it easily follows from a simple calculation that
For J 5 , using the definition of x #
h and integration by parts, we have
x -# dx #
x 1-#
x k+1/2
x 1-# dx
# .
Note that
Thus there is a positive constant C 5 independent of h such that
because of the error estimate for the trapezoidal quadrature formula. Now consider
J 3 and J 4 . First rewrite J 3
x k-1/2
xk
x k-1/2
x k-1/2
xk
x k-1/2
x k-1/2
)xdx
Clearly, we have
and
Hence
For calculation similar to that for J 5 we have
x 1-#
x 1/2
x 1-# dx
# .
Letting applying the error formula for the trapezoidal quadrature
rule, we have
x 1/2
x 1/2
x
x 1/2
x 1/2
Hence
For J 7 we have
x k-1/2
xk
- 1. It is obvious that
Hence
Finally, it follows from the above estimates for the J i 's that there is a positive constant
independent of h, such that for all x k # [1/3, 2/3],
which in turn implies that
for all small h > 0 due to the equivalence of the discrete and continuous norms on
S h given in Lemma 2.1. This clearly indicates that the convergence rate of the FVE
solution for this example cannot be O(h 2
On the other hand, our discussion in subsection 3.2 shows that the FVE solution
can have the optimal convergence rate when the exact solution u is in H 2 and the
source term f is in H 1 . This is supported by the following example. We consider the
following boundary value problem:
where
a
a

Table
errors of the FVE solutions for various partition sizes h.
The boundary conditions are chosen so that
dt
is the exact solution to this boundary value problem. Note that u is piecewise smooth,
u # is continuous, but u # is discontinuous at Hence, in this example, the right-hand
side function f is H 1 (0, 1), but the exact solution to the boundary value problem
is only in H 2 (0, 1). The L 2 errors of the FVE solutions with linear finite elements
corresponding to various mesh sizes h are listed in Table 1. The involved calculations
were carried out such that is one of the mesh points in the partitions used.
Linear regression indicates that the data in this table satisfy
which suggests the optimal convergence rate, and the data are in agreement with the
error estimate obtain in subsection 3.2.
4.2. A two-dimensional example. We consider the following boundary value
problem:
where# is the unit square (0, 1) - (0, 1). It is easy to see that the exact solution to
this boundary value problem is
which is in H 2 but not in H
3(#2 On the other hand, the source term
5 is just in L
We have applied the FVE method (2.1) to generate the FVE solution u h (x, y)
to this boundary value problem by the usual uniform partition T h of the unit square
with the partition size h. Due to the lack of regularity in the source term, an exact

Table
Errors of the FVE solutions for various partition sizes h.
integration formula is used to carry out all the quadratures in (2.1) that involve the
source term f(x, y). In fact, we can show that for each triangle #A 1 A 2 A 2 with vertices
z 3
we have
f(x)dx
with
Note that this formula is valid only if the vertices of the triangle #A 1 A 2 A 2 have
distinct coordinate values. This is true when #A 1 A 2 A 2 is a triangle used in the
integration over a control volume.

Table

2 contains the errors of the FVE solutions for this boundary problem with
various typical partition sizes h. In this table,
is the usual L 2 error of an FVE solution u h (x, y). Obviously, the FVE solutions
in these computations do not seem to have the standard second-order convergence
because the error is not reduced by a factor of 4 when the partition size is reduced by
a factor of 2. Also see the counterexample in [18].
5. Conclusion. In this paper, we have considered the accuracy of FVE methods
for solving second-order elliptic boundary value problems. The approach presented
herein combines traditional finite element and finite di#erence methods as a variation
of the Galerkin finite element method, revealing regularities in the exact solution and
establishing that the source term can a#ect the accuracy of FVE methods. Optimal-
estimates and superconvergence also have been discussed. The
examples presented above show that the FVE method cannot have the standard O(h 2 )
convergence rate in the L 2 norm when the source term has the minimum regularity
in L 2 , even if the exact solution is in H 2 .



--R

A box scheme for coupled systems resulting from microsensor thermistor problems
A new finite-element formulation for convection-di#usion problems
Some error estimates for the box method
Notes Math.
The Analysis of
A finite Volume
On the finite Volume
On the accuracy of the finite Volume
Element analysis method and superconvergence
High Accuracy Theory of Finite Element Methods
estimates in L 2
The immersed finite Volume
The Mortar Finite Volume
Finite Volume
Finite Volume
Eine
On first and second order box schemes
On the finite Volume
Piecewise linear Petrov-Galerkin error estimates for the box method
Generalized Di
Finite Volume
Finite Volume
Some optimal error estimates for piecewise linear finite element approximations
Some new estimates for Ritz-Galerkin methods with minimal regularity assumptions

Homogeneous di
Homogeneous di
Superconvergence in Galerkin Finite Element Methods
Superconvergence Theory for Finite Element Methods
A survey of superconvergence techniques in finite element methods
--TR

--CTR
Zhoufeng Wang , Zhiyue Zhang, The characteristic finite volume element methods for the two-dimensional generalized nerve conduction equation, Neural, Parallel & Scientific Computations, v.15 n.1, p.27-44, March 2007
Rajen K. Sinha , Jrgen Geiser, Error estimates for finite volume element methods for convection-diffusion-reaction equations, Applied Numerical Mathematics, v.57 n.1, p.59-72, January 2007
Chunjia Bi , Hongxing Rui, Uniform convergence of finite volume element method with Crouzeix-Raviart element for non-self-adjoint and indefinite elliptic problems, Journal of Computational and Applied Mathematics, v.200 n.2, p.555-565, March, 2007
Guoliang He , Yinnian He, The finite volume method based on stabilized finite element for the stationary Navier-Stokes problem, Journal of Computational and Applied Mathematics, v.205 n.1, p.651-665, August, 2007
finite volume;error estimates;elliptic;counterexamples