Superconvergence of nonconforming finite element approximation for the second order elliptic problem

In this paper a general superconvergence of nonconforming finite element method for the second order elliptic problem is derived. In order to verify and support the theoretical results numerical examples are given.


Introduction
"The superconvergence of finite element (FE) solutions is an interesting and useful phenomenon in the scientific computing of real world problems and has become an area of active research in recent years" [4]. Wang [9] proposed and analyzed the L 2 -projection method for the least-squares conforming finite element method on the second order elliptic problem.The goal of this article is to derive a general superconvergence of nonconforming FE with its application for elliptic problem by applying certain postprocessing.

‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬
The paper is organized as follows. In section two, we give a preliminaries for the nonconforming finite element. In section three, we derived some results to improve the existing accuracy with framework for the procedure that we given in section two. Finally, in section four, several numerical examples are tested to support and confirm the theoretical results derived in section three.

Preliminaries
We shall consider the following elliptic problem with Dirichlet boundary condition. This model problem seeks an unknown function u such that . ∥ s,Ω , s ≥ 0. Let L 2 (Ω) be a coincides space of square integrable functions, [4].
where ‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬ Let h be a partition of the domain Ω into Ω = ⋃ K∈ h . Suppose that h is quasiuniform, that is it the inverse assumption is satisfy and regular [8], Let ℰ h denote the union of the all boundaries of all elements K of h and The nonconforming FE space associated with the mesh h is defined as

Superconvergence by -projection
The L 2 -Projection is a postprocessing technique defined by Wang [9] for Galerkin methods. The idea is to project the FE solution to another finite element space with a different mesh size. Let τ be a coarse mesh where τ ≫ h. Suppose that and h satisfying: with α ∈ (0,1).
We will be seen that α plays an important role in the postprocessing. For now, let V τ ⊂ H s−2 (Ω), for the exact solution u. Define Q τ to be the L 2projectors from L 2 (Ω) onto the FE space V τ .
We shall give the structure of the coarse mesh by L 2 -projection method.
In order to compute the Q τ u h , we first note that can be written as the linear combination where c j , is N unknown coefficients, and substituting 3.3 into 3.2, we finally get the follwing system which given by: the linear system of 3.4 can be written as

‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬
where ψ i is the local basis function of element K The element matrix of force vector (the entries of the 6 × 1) is given by: Also, the matrix of element K (the entries of the 6 × 6) given by: , and the force vector where M −1 is the inverse of M.
We will prove the following lemma that we need it later.
Where is independent of ℎ.
The following theorem can be found in [11]. respectively. Then, there is a constant C, such that where is an independent of h.
The superconvergence analysis requires certain regularity for the second order elliptic problem. To this end, suppose that the domain is so regular, that is H s , s ≥ 1 regularity for solution , that is ∥ ∥ ≤∥ Define the finite element space V τ ⊂ H s−2 (Ω), for .
Next, we will prove the following lemma. Lemma 3.3: Suppose that (3.6) hold with 1 ≤ s ≤ k + 1 and V τ ⊂ H s−2 (Ω). Then, there is a constant, C such that Where independent of h and τ.

‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬
Proof: From the definition of ‖. ‖ and Q τ , we have Using the definition of the L 2 -projection for Q τ , we obtain
Next, we estimate ‖∇ τ (u − Q τ u h )‖ by following the similar idea in [4], then

This completes the proof of the theorem
The above error estimate is optimized if α is selected as 3.20 Solving α from above yields 3.21

Numerical Examples
We present several numerical experiments to conform Theorem 3.4.
The triangulation of h is describe by: (1) the domain is divided into an n 3 × n 3 rectangular partition and (2) the diagonal line is connecting with the positive slope. Let h = 1 n 3 as the mesh size. In the implementation, the quantity Q τ u h is L 2 -projection of u h to V τ associate with τ . We define V τ as follows: V τ = {v ∈ L 2 (Ω): v|K ∈ P 2 (K), ∀K ∈ τ }. By using Theorem 3.4, we get the order O�h  Table 4.1 shows that after the use of the postprocessing, the errors are reduced. The error in the H 1 -norm has a higher order, which is shown as (ℎ ), = 0.9950 1.3624     From the results shown in Table 4.2, it is easy to note that the solution u in the H 1 -norm has the faster convergence, (see Figure 4.3). and (b), shows that the approximate solutions u h and Q τ u h . This is in agreement with the previously stated theory.

‫اﻟﻌﺎﺷر‬ ‫اﻟدوﻟﻲ‬ ‫اﻟﻌﻠﻣﻲ‬ ‫اﻟﻣؤﺗﻣر‬ ‫واﺳط‬ ‫ﺟﺎﻣﻌﺔ‬ / ‫اﻟﺗرﺑﯾﺔ‬ ‫ﻛﻠﯾﺔ‬
(ℎ ), = 0.9944 1.3460     From the data shown in Table 4.4, it is clear that the exact solution u in the H 1 -norm has the superconvergence, (see Figure 4.7). and (b), shows that the approximate solutions u h and Q τ u h . This is in agreement with the previously stated theory.