The conforming finite element method (CFEM) requires a strong continuity; hence it is
not easy to construct such finite elements for the complex partial differential equations. The nonconforming finite element method (NCFEM) is more appealing computationally due to better stability and flexibility properties compared to CFEM    . 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. 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 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  .
This paper is organized as follows. In Section 2, we present a review for the non- conforming finite element method for the second-order elliptic problem. In Section 3, we develop a general theory of superconvergence by following the idea presented in Wang  . In Section 4, we perform numerical experiments to support the theoretical results. Numerical experiements of superconvergence of NCFEM are performed in MATLAB and its codes are posted at https://github.com/annaleeharris/Superconvergence-NCFEM for anyone to use and to study.
2. NCFEM for the Second-Order Elliptic Problem
Consider the second-order elliptic problem with the Dirichlet boundary condition which seeks satisfying
where is the Laplacian operator, is a bounded, connected, and open subset of, is a Lipschitz continuous boundary, and a given function f is the external force.
A variational formulation of (1) seeks such that
Let be a quasi-uniform, i.e., it is regular and satisfies the inverse assumption  , triangulation of with. Let be the space of poly- nomials of degree at most k with on K. Let denote the union of the boun- daries of all elements and let be the collection of all interior edges. 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 pro- perty for any  :
The nonconforming finite element approximation problem (2) seeks such that
A well known error estimate for the finite element approximation solution is the following  :
where C is a constant independent of the mesh size h.
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
where C is a constant independent of data f.
3. Superconvergence of NCFEM
Let be another finite element partition with coarse mesh size where. Assume that 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 as follows:
The following lemma will provide an error estimate for.
Lemma 1 Assume that the second-order elliptic problem (2) holds (5) with and. Then there exists a constant C independent of h and such that
Proof. Using the definition of and, we have
Consider the following problem:
Multiplying the second-order elliptic Equation (1) by v and integrating it over give
where n is the unit outward normal.
Subtract (3) from the above Equation (10) gives
Multiplying (9) by, integrating it over, adding and subtracting, and using the result (11) we have
The line integrals of the above equations are approximated in  as follows:
Using the Cauchy-Schwartz inequality, the approximation property (2), and line integral approximations (12) and (13) we have
Substituting as by the regularity, applying the inverse in- equality to the term and using the definition of we have
Combining the above equation with the Equation (8) we have
which completes the proof of the lemma.
The following theorem provides an error estimate for.
Theorem 1 Assume that (5) 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 such that
Proof. Since we assume the exact solution u is sufficiently smooth and by the de- finitions of and, we have
Using the triangle inequality and combining (16) and Lemma 1 we obtain
which completes the error estimate of.
Similarly, we estimate.
Using the inverse inequality and the definitions of and we have
Using the triangle inequality and combining (17) and Lemma 1 we have
Hence the theorem has been proved.
The optimal is selected using Theorem 1 for the error estimates:
4. Numerical Experiments of Superconvergence of NCFEM by L2-Projection Methods
In this section, we present numerical experiments for second-order elliptic problems to support our theoretical results. Assume that the exact solution of the second-order elliptic problem has the regularity for some and for simplicity, assume
and which gives using the formula (18).
From the theoretical result (15) we have the following optimal error estimates:
From the results (19) and (20), theoretically, in L2 norm the L2-projection to the existing numerical approximation does not improve the convergence rate but in norm the L2-projection to the existing numerical solution provides some superconver- gence.
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 con- sists 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 as follows:
The numerical approximation is refined as where. The length of and each element contains elements.
Using the Equation (18) and our choice of and we have
Using the difference in mesh size and a higher degree of polynomials we shall produce some superconvergence of NCFEM for the second-order elliptic problems.
Example 1. Let the domain and the exact solution is assumed to be as follows:
From Table 1 we observe that the L2-projection to the existing numerical approxi- mation reduced the error estimates in L2 norm and in norm. In L2 norm the convergence rate of is similar to the convergence rate of which is the same as the theoretical result (19). The convergence rate of is about 33% faster than the convergence rate of in norm (see Figure 2). The surface plots of in coarse meshes and in fine meshes are shown in Figure 1. The numerical example 1 clearly supports the theoretical result and confirms the super- convergence of NCFEM for the second-order elliptic problem.
Example 2. Let the domain and let the analytical solution be given as
From Table 2, we can see that the numerical example 2 supports the theoretical result (15). See Figure 3, when and, we can project 32 fine triangle elements onto one coarse triangle element. Thus, as n increases, we can project more fine triangle elements to one coarse triangle element in which the process of refining elements produces better error estimates. The L2-projection to the existing numerical approximation produced some superconvergence in norm and did not affect the convergence rate in L2 norm (see Figure 4). The numerical example 2 also
Table 1. Numerical error approximation results using NCFEM in Example 1,.
Figure 1. Surface plots of approximation using NCFEM in Example 1,. (L): Surface plot of when. (M): Surface of plot of when. (R): Surface plot of when.
Figure 2. Error convergence rates using NCFEM in Example 1,. (L): L2 norm error; (R): norm error.
Table 2. Numerical error approximation results using NCFEM in Example 2,
Figure 3. Surface plots of approximation using NCFEM in Example 2, (L): Surface plot of when. (M): Surface of plot of when. (R): Surface plot of when.
supports the theoretical result and confirms the superconvergence of NCFEM for the second-order elliptic problem.
The L2-projection to the existing numerical approximation produced some super- convergence in norm, convergence rate, but did not affect the convergence
Figure 4. Error convergence rates using NCFEM in Example 2, (L): L2 norm error; (R): norm error.
rate in L2 norm. With the numerical experiments we can conclusively support the theoretical result and confirm the superconvergence of NCFEM for second-order elliptic problems by L2-projection method.
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.