Numerical Experiments Using MATLAB : Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems

The superconvergence in the finite element method is a phenomenon in which the finite element approximation converges to the exact solution at a rate higher than the optimal order error estimate. Wang proposed and analyzed superconvergence of the conforming finite element method by L2-projections. However, since the conforming finite element method (CFEM) requires a strong continuity, it is not easy to construct such finite elements for the complex partial differential equations. Thus, the nonconforming finite element method (NCFEM) is more appealing computationally due to better stability and flexibility properties compared to CFEM. The objective of this paper is to establish a general superconvergence result for the nonconforming finite element approximations for second-order elliptic problems by L2-projection methods by applying the idea presented in Wang. MATLAB codes are published at https://github.com/annaleeharris/Superconvergence-NCFEM for anyone to use and to study. The results of numerical experiments show great promise for the robustness, reliability, flexibility and accuracy of superconvergence in NCFEM by L2projections.


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, manufac- 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 L 2 -projections.The main idea behind the L 2 -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 L 2 -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 L 2 -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-CFEMfor anyone to use and to study.

CFEM for the Second-Order Elliptic Problem
Consider the second-order elliptic problem with the homogeneous Dirichlet boundary condition which seeks ( ) where A variational formulation of (1) seeks ( ) where  be a quasi-uniform, i.e., it is regular and satisfies the inverse assumption [17], triangulation of Ω with ( ) , h diam K h K ≤ ∈ and let ( ) r P K be the space of polynomials of degree at most r with 0 r ≥ on K. Assume that the polynomial space in the construction of h V contains ( ), The finite element space h V is assumed to satisfy the following approxima- tion property for any ( ) ( ) The finite element approximation problem (2) A well known error estimate for the finite element approximation solution h u is the following: 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 Ω is so regular that it ensures a , 1 s H s ≥ , regularity for the solu- tion of (2).In other words, for any ( ) where C is a constant independent of data g.

Superconvergence of CFEM
Let τ  be another finite element partition with coarse mesh size τ where h τ  .Assume that τ and h have the following relation: Let V τ be any finite element space consisting of piecewise polynomial of de- gree r associated with the partition τ  .Define Q τ to be the L 2 -projection from ( )

2
L Ω onto the finite element space V τ .The finite element space V τ is de- fined 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 Then there exists a constant C independent of h and τ such that where ( ) ( ) is the finite element approximation of the exact solution ( ) , then there exists a constant C independent of h and τ such that ( ) where ( ) Theorem 2 Assume that (6) holds true with 1 , then there exists a constant C independent of h and τ such that ( ) where ( ) From ( 8) and (9) α is selected to optimize the error estimates: ( )

Numerical Experiments of Superconvergence of CFEM by L 2 -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 s H regularity for some 1 2 s ≤ ≤ and for simplicity, assume 2 s = , and 2 r = which gives 2 3 α = 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 ( ) ( ) From the result (11), we do not see any superconvergence in L 2 norm.However, from the result (12), we have some superconvergence for the gradient error estimate.
The finite element partition h  is constructed by dividing the domain into an 3 3  n n × rectangular mesh then dividing the rectangular mesh with the positive slope to form two triangles.The coarse finite element partition τ  is also constructed by dividing the domain into an 2 2  n n × rectangular mesh then dividing the rectangular mesh with the positive slope to form two triangles.The finite element space h V consists of the space of the linear polynomials ( ) associated with the partition h  and the dual finite element space V τ consists of the space of the quadratic polynomials ( ) P K associated with the partition τ  .The finite element spaces h V and V τ are defined by The numerical approximation is refined as

=
and each τ element contains 2 n h ele- ments.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 ( ) ( ) cos 0.5π sin π .u y y x = From Table 1, we observe that applying L 2 -projections to the existing numerical solution reduced the errors in L 2 norm and in H 1 norm.Surface plots of numerical solutions, h u in fine meshes and O h (see Figure 2).However, in H 1 norm the error con- vergence rate of is higher than the optimal error convergence rate of iter h ( ) 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 H 1 norm but not in L 2 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

. u x x y y = − −
From Table 2, we confirm that the numerical Example 2 supports the theoretical results in [16].In L 2 norm the error convergence rate of which is about the same as the theoretical result in (11), which is shown as O h in Figure 3.The error convergence rate of

O h
and the error convergence rate of In H 1 norm the exact solution u clearly has some superconvergence.Figure 4 shows the surface plot of  Table 2. Numerical error approximation results using CFEM in Example 2, ( ) ( )  O h .In L 2 norm, there is no improvement with the post-processing tech- nique.See Figure 5, in H 1 norm L 2 -projection method improved the convergence rate, which is shown as ( ) . Figure 6 shows surface plots of From Table 4, we confirm that the numerical Example 4 supports the theoretical results in [16].In L 2 norm the error convergence rate of

O h
. See Figure 7, in L 2 norm there is no difference in convergence rates but in H 1 norm applying L 2 -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.

(
)( ) ( )   With numerical experiments we support the theoretical results in [16] and confirm the superconvergence of CFEM for second-order elliptic problems.

Conclusion
The L 2 -projection to the existing numerical approximation h u produced some superconvergence in H 1 norm, convergence rate 1.3 ≥ , but did not affect the convergence rate in L 2 norm.With the numerical experiments we can conclusively support the theoretical result and confirm the superconvergence of CFEM for second-order elliptic problems by L 2 -projection method.

Figure 1 .
Figure 1.In L 2 norm the error convergence rate of

Figure 1 .Figure 2 . 2 L
Figure 1.Surface plots of approximation solution using CFEM in Example 1, ( ) cos 0.5π sin(π ) u y y x = .(L): Surface plot of h u .(R): Surface of plot of and the surface plot of h u in fine meshes.The numerical Example 2 Applied Mathematics

Figure 8 .
Figure 8. Surface plots of approximation using CFEM in Example 4, ( ) ( ) sin 2π cos 1.5π u x x y y = .(L): Surface plot of h u .(R): Surface plot of is the Laplacian operator, Ω is a bounded, connected, and open A. Harris et al.

Table 1 .
Numerical error approximation results using CFEM in Example 1, From Table3, the numerical approximation results show that after the post-processing all the errors are reduced.The exact solution in L 2 norm of DOI: 10.4236/am.2018.96047698 Applied Mathematics