^{1}

^{2}

^{1}

The superconvergence in the finite element method is a phenomenon in which the fi-nite 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 L2- projections.

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 [^{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 establish a general superconvergence result for the nonconforming finite element approximations for second-order elliptic problems by L^{2}-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 [

Consider the second-order elliptic problem with the Dirichlet boundary condition which seeks

where

A variational formulation of (1) seeks

where

Let

The finite element space

The nonconforming finite element approximation problem (2) seeks

where

A well known error estimate for the finite element approximation solution

where C is a constant independent of the mesh size h.

To apply the superconvergence of finite element approximation, we assume that domain

where C is a constant independent of data f.

Let

Let ^{2}-projection from

The following lemma will provide an error estimate for

Lemma 1 Assume that the second-order elliptic problem (2) holds (5) with

where

Proof. Using the definition of

and

Then

Consider the following problem:

Multiplying the second-order elliptic Equation (1) by v and integrating it over

where n is the unit outward normal.

Subtract (3) from the above Equation (10) gives

Multiplying (9) by

The line integrals of the above equations are approximated in [

Using the Cauchy-Schwartz inequality, the approximation property (2), and line integral approximations (12) and (13) we have

Substituting

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

Proof. Since we assume the exact solution u is sufficiently smooth and by the de- finitions of

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

Using the triangle inequality and combining (17) and Lemma 1 we have

Hence the theorem has been proved.

The optimal

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

From the theoretical result (15) we have the following optimal error estimates:

and

From the results (19) and (20), theoretically, in L^{2} norm the L^{2}-projection to the existing numerical approximation does not improve the convergence rate but in ^{2}-projection to the existing numerical solution provides some superconver- gence.

The finite element partition

and

The numerical approximation is refined as

Using the

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

From ^{2}-projection to the existing numerical approxi- mation ^{2} norm and in ^{2} norm the convergence rate of

Example 2. Let the domain

From ^{2} fine triangle elements onto one coarse triangle element. Thus, as n increases, we can project ^{2}-projection to the existing numerical approximation ^{2} norm (see

iter | h | ||||
---|---|---|---|---|---|

1 | 2^{−3} | 0.1388e−1 | 0.3909e−3 | 0.8184e−2 | 0.3920e−3 |

2 | 3^{−3} | 0.4138e−2 | 0.3443e−4 | 0.1635e−2 | 0.3431e−4 |

3 | 4^{−3} | 0.1747e−2 | 0.6104e−5 | 0.5190e−3 | 0.6104e−5 |

4 | 5^{−3} | 0.8944e−3 | 0.1600e−5 | 0.2127e−3 | 0.1600e−5 |

5 | 6^{−3} | 0.5176e−3 | 0.5358e−6 | 0.1026e−3 | 0.5359e−6 |

0.9981 | 2.000 | 1.3287 | 2.0010 |

iter | h | ||||
---|---|---|---|---|---|

1 | 2^{−3} | 0.3933e−1 | 0.8429e−3 | 0.2214e−1 | 0.8453e−3 |

2 | 3^{−3} | 0.1189e−1 | 0.7404e−4 | 0.4387e−2 | 0.7408e−4 |

3 | 4^{−3} | 0.5019e−2 | 0.1317e−4 | 0.1392e−2 | 0.1318e−4 |

4 | 5^{−3} | 0.2570e−2 | 0.3454e−5 | 0.5708e−3 | 0.3455e−5 |

5 | 6^{−3} | 0.1487e−2 | 0.1156e−5 | 0.2754e−3 | 0.1157e−5 |

0.9983 | 1.9998 | 1.3311 | 2.0006 |

supports the theoretical result and confirms the superconvergence of NCFEM for the second-order elliptic problem.

The L^{2}-projection to the existing numerical approximation

rate in L^{2} norm. With the numerical experiments we can conclusively support the theoretical result and confirm the superconvergence of NCFEM for second-order elliptic problems by L^{2}-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.

Harris, A., Harris, S. and Rauls, D. (2016) Numerical Experiments Using MATLAB: Superconvergence of Nonconforming Finite Element Approximation for Second-Order Elliptic Problems. Applied Mathematics, 7, 2174-2182. http://dx.doi.org/10.4236/am.2016.717173