Numerical Experiments Using MATLAB: Superconvergence of Conforming Finite, Element Approximation for Second Order, Elliptic Problems ()
1. Introduction
Finite element method (FEM) is based on the premise that an approximation to any complex engineering problem can be reached by subdividing the problem into smaller and more manageable elements. Using FEMs partial differential equations that describe the behavior of structures can be reduced to a set of linear equations that can easily be solved using the standard techniques of matrix algebra. FEM is used in virtually every engineering discipline. The aerospace, automotive, biomedical, chemicals, electronics, energy, geotechnical, manufacturing, and plastics industries routinely apply finite element analysis. In addition, it is used not only for analyzing classical static structural problems, but also for such diverse areas as mass transport, heat transfer, dynamics, stability, and radiation problems.
The main objective of the superconvergence using various FEMs is to improve the accuracy of the existing approximation solution by applying certain post-processing techniques that are easy to implement. To obtain the superconvergence of FEMs, several methods have been proposed in the literature in the last thirty years. The method of local averaging has been a popular and useful technique in the study of superconvergence [1] - [9]. The underlying assumption of the existing superconvergence technique is that the finite element mesh has some special properties such as uniformity [7], local point-symmetry [8] [10], local translation-invariance [1] [8], or orthogonality [5] [11] [12] [13].
Zienkiewicz and Zhu [14] [15] introduced the patch recovery technique which provides some superconvergence for the gradient of the finite element solution by using a discrete least-squares fitting on a local patch with high order polynomials. The method of Zienkiewicz and Zhu has been computationally proved to be robust and efficient and to produce some superconvergence for the gradient of the finite element solution.
Wang proposed and analyzed superconvergence of the conforming finite element method (CFEM) by L2-projections. The main idea behind the L2-projections is to project the finite element solution to another finite element space with a coarse mesh and a higher order of polynomials.
The objective of this paper is to investigate the theoretical results in [16] for the conforming finite element approximations for second-order elliptic problems by L2-projection methods and to support the theoretical results with numerical experiments using MATLAB.
This paper is organized as follows. In Section 2, we present a review for the conforming finite element method for the second-order elliptic problem. In Section 3, we investigate the theoretical results in [16], the superconvergence of CFEM for the second-order elliptic problem by L2-projection methods. In section 4, we perform numerical experiments to support the theoretical results in [16]. Numerical experiments of superconvergence of CFEM are performed in MATLAB and its codes are posted at https://github.com/annaleeharris/Superconvergence-CFEM for anyone to use and to study.
2. CFEM for the Second-Order Elliptic Problem
Consider the second-order elliptic problem with the homogeneous Dirichlet boundary condition which seeks
satisfying
(1)
where
is the Laplacian operator,
is a bounded, connected, and open subset of R2,
is a Lipschitz continuous boundary, and a given function f is the external force.
A variational formulation of (1) seeks
such that
(2)
where
Let
be a quasi-uniform, i.e., it is regular and satisfies the inverse assumption [17], triangulation of
with
and let
be the space of polynomials of degree at most r with
on K. Assume that the polynomial space in the construction of
contains
. Define the finite element space
associated with
as
The finite element space
is assumed to satisfy the following approximation property for any
:
(3)
The finite element approximation problem (2) seeks
such that
(4)
where
A well known error estimate for the finite element approximation solution
is the following:
(5)
where C is a constant independent of the mesh size h.
Then from (3) and (5) we arrive at the following error estimate:
To apply the superconvergence of finite element approximation, we assume that domain
is so regular that it ensures a
, regularity for the solution of (2). In other words, for any
the problem (2) has a unique solution
satisfying the following a priori estimate:
(6)
where C is a constant independent of data g.
3. Superconvergence of CFEM
Let
be another finite element partition with coarse mesh size τ where
. Assume that τ and h have the following relation:
Let
be any finite element space consisting of piecewise polynomial of degree r associated with the partition
. Define
to be the L2-projection from
onto the finite element space
. The finite element space
is defined by
For the superconvergence of CFEM, the following theoretical results can be found in [16].
Lemma 1 Assume that the second-order elliptic problems (2) holds (6) with
and
. Then there exists a constant C independent of h and τ such that
(7)
where
and
.
Theorem 1 Assume that (6) holds true with
and
. If
is the finite element approximation of the exact solution
of (2), then there exists a constant C independent of h and τ such that
(8)
where
.
Theorem 2 Assume that (6) holds true with
and
. If
is the finite element approximation of the exact solution
of (2), then there exists a constant C independent of h and τ such that
(9)
where
.
From (8) and (9) α is selected to optimize the error estimates:
(10)
4. Numerical Experiments of Superconvergence of CFEM by L2-Projection Methods
In this section, we confirm the theoretical results in [16] with numerical experiments for second-order elliptic problems. Assume that the exact solution of the second-order elliptic problem has
regularity for some
and for simplicity, assume
,
, and
which gives
using the α Formula (10).
Then according to the theoretical results in [16], the best possible error estimates using the results (8) and (9) are given by
(11)
and
(12)
From the result (11), we do not see any superconvergence in L2 norm. However, from the result (12), we have some superconvergence for the gradient error estimate.
The finite element partition
is constructed by dividing the domain into an
rectangular mesh then dividing the rectangular mesh with the positive slope to form two triangles. The coarse finite element partition
is also constructed by dividing the domain into an
rectangular mesh then dividing the rectangular mesh with the positive slope to form two triangles. The finite element space
consists of the space of the linear polynomials
associated with the partition
and the dual finite element space
consists of the space of the quadratic polynomials
associated with the partition
. The finite element spaces
and
are defined by
and
The numerical approximation is refined as
, where
. Thus, the length of
and each τ element contains
elements. Using the difference in mesh size and a higher degree of polynomials we shall produce some superconvergence of CFEM for the second-order elliptic problems.
Example 1 Let the domain
and the exact solution is assumed to be
From Table 1, we observe that applying L2-projections to the existing numerical solution reduced the errors in L2 norm and in H1 norm. Surface plots of numerical solutions,
in fine meshes and
in coarse meshes, are shown in Figure 1. In L2 norm the error convergence rate of
and the error convergence rate of
are similar to the theoretical convergence rate, which is shown as
(see Figure 2). However, in H1 norm the error convergence rate of
is higher than the optimal error convergence rate of
and the error convergence rate of the numerical example,
,
Table 1. Numerical error approximation results using CFEM in Example 1,
.
(a) (b)
Figure 1. Surface plots of approximation solution using CFEM in Example 1,
. (L): Surface plot of
. (R): Surface of plot of
.
(a) (b)
Figure 2. Error convergence rates using CFEM in Example 1,
. (L):
norm error. (R):
norm error.
exceeds its theoretical error convergence rate, which is shown as
. As we expect from the theoretical results (11) and (12), the numerical example shows some superconvergence in H1 norm but not in L2 norm. The numerical Example 1 supports the theoretical results in [16] and confirms the superconvergence of CFEM for second-order elliptic problems.
Example 2 Let the domain
and the analytical solution to the problem is given as
From Table 2, we confirm that the numerical Example 2 supports the theoretical results in [16]. In L2 norm the error convergence rate of
is similar to the error convergence rate of
which is about the same as the theoretical result in (11), which is shown as
in Figure 3. The error convergence rate of
is about
and the error convergence rate of
is about
. In H1 norm the exact solution u clearly has some superconvergence. Figure 4 shows the surface plot of
in coarse meshes and the surface plot of
in fine meshes. The numerical Example 2
(a) (b)
Figure 3. Error convergence rates using CFEM in Example 2,
. (L):
norm error. (R):
norm error.
(a) (b)
Figure 4. Surface plots of approximation solution using CFEM in Example 2,
. (L): Surface plot of
. (R): Surface plot of
.
Table 2. Numerical error approximation results using CFEM in Example 2,
.
also supports the theoretical results in [16] and confirms the superconvergence of CFEM for second-order elliptic problems.
Example 3 Let the domain
and the analytical solution to the problem is given as
From Table 3, the numerical approximation results show that after the post-processing all the errors are reduced. The exact solution in L2 norm of
has the similar error convergence rate as
, which shown as
. In L2 norm, there is no improvement with the post-processing technique. See Figure 5, in H1 norm L2-projection method improved the convergence rate, which is shown as
for
. Figure 6 shows surface plots of
and
. The numerical Example 3 confirms the theoretical results in [16].
Example 4 Let the domain
and the exact solution is assumed to be
From Table 4, we confirm that the numerical Example 4 supports the theoretical results in [16]. In L2 norm the error convergence rate of
is similar to the error convergence rate of
which is about the same as the theoretical result,
. However, in H1 norm the exact solution u has some
(a) (b)
Figure 5. Error convergence rates using CFEM in Example 3,
. (L):
norm error. (R):
norm error.
(a) (b)
Figure 6. Surface plots of approximation solution using CFEM in Example 3,
. (L): Surface plot of
. (R): Surface plot of
.
superconvergence. The error convergence rate of
is about 34% faster than the error convergence rate of
and meets the theoretical minimum error convergence rate,
. See Figure 7, in L2 norm there is no difference in error convergence rates but in H1 norm applying L2-projection methods to the existing numerical approximations improved the errors and produced some superconvergence. Figure 8 shows surface plots of the numerical approximations of (2) before and after the post-processing.
Table 3. Numerical error approximation results using CFEM in Example 3,
.
Table 4. Numerical error approximation results using CFEM in Example 4,
.
(a) (b)
Figure 7. Error convergence rates using CFEM in Example 4,
. (L):
norm error. (R):
norm error.
(a) (b)
Figure 8. Surface plots of approximation using CFEM in Example 4,
. (L): Surface plot of
. (R): Surface plot of
.
With numerical experiments we support the theoretical results in [16] and confirm the superconvergence of CFEM for second-order elliptic problems.
5. Conclusion
The L2-projection to the existing numerical approximation
produced some superconvergence in H1 norm, convergence rate
, but did not affect the convergence rate in L2 norm. With the numerical experiments we can conclusively support the theoretical result and confirm the superconvergence of CFEM for second-order elliptic problems by L2-projection method.
Acknowledgements
We thank the Editor and the peer-reviewers for their comments. Research of Anna Harris is funded by the National Science Foundation Historical Black Colleges and Universities Undergraduate Program Research Initiative Award grant (#1505119). This support is greatly appreciated.