1 Introduction

Consider the following parabolic integro-differential equation^[1]:

where Ω∈R² is a bounded convex polygonal domain with Lipschitz continuous boundary, X = (x,y),T ∈ (0,+∞) is a set value,and u(X) is a given smooth function. The problem (1) is derived from many practical problems in physics and engineering,such as heat conduction in material with memory,the viscoelastic fluid model,and the heat transfer problem in nuclear reactor. A lot of numerical simulation methods have been put forward for (1),such as the finite difference method^[2],the finite element method (FEM)^{[3, 4, 5, 6, 7, 8, 9, 10]},the finite volume method^[11],and the spline collocation method^[12].

As we know,the H¹-Galerkin mixed finite element method (MFEM) was firstly proposed in ^[13]. Compared with the standard MFEM,this method has a big advantage. The approximation spaces can be chosen freely without the restriction of the Ladyzhenskaya-Babuska-Brezzi (LBB) condition,and the quasi-uniformity on the meshes is not required. Subsequently,this method was further applied to regularized long-wave equations^[14],second order hyperbolic differential equations^[15],pseudo-hyperbolic equations^[16],and hyperbolic type integro-differential equations^[17]. As for problem (1),Pani and Fairweather^[18] discussed the error estimates by means of Ritz projection for semi-discrete and fully discrete schemes. Chen et al.^[19] researched the nonlinear integro-differential equations and obtained the optimal error estimates. Shi and Wang^[20] analyzed the approximations of nonconforming EQ₁^{rot [21]} and Crouzeix-Raviart type finite elements^[22] for anisotropic meshes.

The main purpose of this article is to propose a highly efficient H¹-Galerkin MFEM for problem (1) with the simplest linear triangular element and to investigate the properties of superconvergence and extrapolation. Firstly,some important results about the integral estimation (see Lemma 2.1 below) and asymptotic expansions (see Lemmas 2.2-2.4 below) are proved. Secondly, the superclose and superconvergence of order O(h²) for both the original variable u and flux p are deduced through the interpolation post processing technique. At the same time,by virtue of the asymptotic expansions and construct a suitable auxiliary problem,the extrapolation solutions with order O(h3) are obtained for the above two variables. Finally,we give a numerical example to verify the correctness of the theoretical analysis and the effectiveness of the proposed method.

2 H¹-Galerkin MFEM and some lemmas

Assume that the domain is a rectangle. Th = {e} is an isosceles right triangular mesh of Ω. h is the length of the edge parallelling to the x-axis or the y-axis (see Fig. 1).

The spaces used in this paper are defined as

where ∇· denotes the divergence of functions. Hm( ) is the Sobolev space with norm . is divergence space and equipped with norm: .

Let p = (p1,p2) = ∇u. The H¹-GalerkinMFEMfor the problem (1) is to seek such that

where uvdxdy.

Then,the semi-discrete H¹-Galerkin MFEM for (2) is to find a pair (uh,ph) ∈ M^h × V^h satisfying

where Ih is the associated interpolation operator over M^h. The existence and uniqueness of the solution to problem (3) can be found in Ref. ^[20].

Now,we start to prove the following lemmas w^hich play an important role in the superconvergence and extrapolations.

Lemma 2.1 Assume that p ∈ (H4( ))2. For all w^h ∈ V^h,there holds

Proof It has been proved in Ref. ^[23] that

Now,we introduce another elements’ combination different from Ref. ^[23]. Let e1 and e2 be adjacent elements,E = e1 ∪ e2 (see Fig. 2), and be2 are the reference elements of e1 and e2,respectively,and b E = ∪ be2 (see Fig. 3). Then,there exists an affine mapping F: b E → E

Then,with the similar manner as Ref. ^[23],we can prove that

From (5)-(9),Green’s formula,and the inverse inequality,the proof is completed.

Lemma 2.2 If ,we have

Proof Suppose that ei (i = 1,2,3,4) are adjacent elements as shown in Fig. 4. is the reference element of e1. F is defined in (7).

We define the linear interpolation operator bI on : (bai) = (bai),where bai (i = 1,2,3) are three vertices of ,and the functional is

where Pi() denotes the space of polynomial on with degree less than or equal to i.

It is easy to check that for all ∈ H³(),there holds

.

Obviously,when .

If is a polynomial of degree 2 on ,the corresponding interpolations are shown in Table 1.

By direct computing,for all ∈ P2(),b w^h 1 ∈ P1(),we derive

and the Bramble-Hilbert lemma gives

.

Then,via the scaling technique,we have

where.

Similarly,there holds

Note that w^h 1x is continuous on the common edge of e1 and e2,w^h 1y is continuous on the common edge of e1 and e3,and w^hen Th is an isosceles right triangular mesh,is continuous on the common edge of e1 and e4. From the boundary condition ,(11), and (12),we have

Next,we prove the following three formulas:

Consider the functional on b E = ∪ be2,

.

As b w^h 1bx is a constant and continuous on the common edge of and be2,a direct computing shows that

.

From the Bramble-Hilbert lemma and the scaling technique,(14) is obtained,and (15) and (16) can be derived in the same way.

A combination of (13)-(16) gives

The proof is completed.

Similar to the proof of Lemma 2.1,we can get Lemmas 2.3 and 2.4.

Lemma 2.3 If p ∈ (H3( ))2 and v^h ∈ M^h,we have

Lemma 2.4 Assume that u ∈ H4( ),p ∈ (H5( ))2. For all v^h ∈ M^h,w^h ∈ V^h,there hold

)

3 Superclose and superconvergence analysis

Theorem 3.1 Suppose that (u,p) and (uh,ph) are solutions to (2) and (3),respectively,

where

Proof Let . We have the following error equations from (2) and (3):

Let v^h = ρ in (22). Then,by Ref. ^[24],we have ,w^hich together with the interpolation theory and Young’s inequality leads to

Adding (η,w^h) to both sides of the second formula in (22) and letting w^h=η_t,by derivative transfer techniques,there holds

First of all,via the interpolation theory and Young’s inequality,

Then,Lemma 2.1 yields

Substituting the above estimations into (24),we have

Integrating from 0 to t and noticing that η(X,0) = 0,we derive Similarly,

Thus,

Applying Gronwall’s lemma, there holds w^hich is the desired result of (21),and (20) comes from the combination of (23) and (26). The proof is completed.

In order to derive the superconvergence,we construct the post-processing interpolation operator I2h as in Ref. ^[23],such that

Theorem 3.2 Under the assumptions of Theorem 3.1,we have k u − I2huh k16 Ch2

Proof By the triangle inequality,Theorem 3.1,and (27),we have

Similarly,we can derive (29),w^hich ends the proof.

4 Extrapolations

Theorem 4.1 Suppose that (u,p) and (uh,ph) are solutions to (2) and (3),respectively, where ∈ M^h × V ^h such that

Proof Let be the exact solution and the corresponding finite element solution of the following auxiliary problem:

where

Applying the Cauchy-Schwarz inequality,there hold

Thus,Gh(v) and Lh(w) are the bounded linear functionals on H¹ 0 ( ) and H(div, ),respectively.

From Theorem 3.1,we know that ,satisfying

By Lemmas 2.2-2.4,the right hand sides of (22) can be written as Combining (22),(32),(35),and (36),we have

Similar to Theorem 3.1,we can get The proof is completed.

In order to obtain the global extrapolation,combine the adjacent nine elements into one large elemente (see Fig. 5),and construct a post-processing interpolation operator Π_3h³ on satisfying

Theorem 4.2 Under the assumptions of Theorem 4.1,let and . Then,there hold

Proof From Theorem 4.1,(33),(34),and (38),we have

Similarly,we can get (40). The proof is completed.

5 Numerical results

We consider problem (1) with

It can be verified that the exact solution to (1) is

. Then, we have

We divide the domain into m × m equal rectangles,and then divide each rectangle into two triangles (see Fig. 1).

For convenience,we just plot the exact solutions u,p,and the finite element solutions uh, ph at t = 0.1 (see Fig. 6,Fig. 7,Fig. 8).

The convergence,supercloseness,superconvergence,and extrapolation results are listed in Table 2,Table 3,Table 4,Table 5,Table 6,Table 7,Table 8,Table 9.

At the same time,we describe the error reduction results in Fig. 9,Fig. 10,Fig. 11,Fig. 12,where u1,u2,and u3 denote ,and ,respectively.

It can be seen from Tables 2-9 that are convergent at rate of O(h), are convergent at rate of O(h²),and are convergent at rate of O(h3),w^hich coincide with our theoretical analysis.