Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems

Abstract

We study the strong superconvergence of a semidiscrete finite element scheme for linear parabolic problems on 𝑄=Ω×(0,𝑇], where Ω is a bounded domain in ℛ^𝑑(𝑑≤4) with piecewise smooth boundary. We establish the global two order superconvergence results for the error between the approximate solution and the Ritz projection of the exact solution of our model problem in 𝑊^1,𝑝(Ω) and 𝐿_𝑝(𝑄) with 2≤𝑝<∞ and the almost two order superconvergence in 𝑊^1,∞(Ω) and 𝐿_∞(𝑄). Results of the 𝑝=∞ case are also included in two space dimensions (𝑑=1 or 2). By applying the interpolated postprocessing technique, similar results are also obtained on the error between the interpolation of the approximate solution and the exact solution.

1. Introduction

Consider the followinginitial boundary value parabolic problem: 𝑢_𝑡+𝐴𝑢=𝑓,in𝑄=Ω×𝐽,𝐽=(0,𝑇],𝑢(𝑥,𝑡)=0,on𝜕Ω×𝐽,𝑢(𝑥,0)=𝑢₀,inΩ, (1.1) where Ω is a bounded domain in ℛ^𝑑(𝑑≤4) with piecewise smooth boundary 𝜕Ω, and 𝐴 is a second-order symmetric positive definite elliptic operator. Coefficients of 𝐴,𝑓(𝑥,𝑡) and 𝑢₀(𝑥) together with their derivatives up to certain order are bounded in order to guarantee our analysis. Note that our assumptions on 𝑢 do not have any restrictions, since it will be shown that approximate solutions considered below are uniformly close to the exact solution and thus only depend on the data of (1.1) in a neighborhood of 𝑢.

Superconvergence of finite element methods for parabolic problems has been studied in many works. For example, Thomée [1], Chen and Huang [2] studied superconvergence of the gradient in 𝐿₂ norm. In 1989, Thomée et al. [3] studied maximum-norm superconvergence of the gradient in piecewise linear finite element approximations of a parabolic problem. An analogous result was also obtained by Chen [4]. Moreover, Li and Wei [5] investigated global strong superconvergence of finite element schemes for a class of Sobolev equations in ℛ^𝑑(𝑑≥1), and two order superconvergence results are proved in 𝑊^1,𝑝(Ω) and 𝐿_𝑝(Ω) for 2≤𝑝<∞. In particular, Kwak et al. [6] studied superconvergence of a semi-discrete finite element scheme for parabolic problems in ℛ², in which superconvergence results in 𝑊^1,𝑝(Ω) and 𝐿_𝑝(Ω) are established for 2≤𝑝≤∞.

In this paper, we extend superconvergence results obtained in [6]. We derive the two order (2≤𝑝<∞) and the almost two order (𝑝=∞) global superconvergence estimates of 𝑈−𝑅_ℎ𝑢 in 𝑊^1,𝑝(Ω) and in 𝐿_𝑝(𝑄), where 𝑈 is the approximate solution, and 𝑅_ℎ𝑢 is the Ritz projection of the exact solution of (1.1). In addition to the results in [6], we establish two order superconvergence estimates in 𝐿_𝑝 norm for piecewise cubic or higher elements. Moreover, results of the 𝑝=∞ case are also included in two space dimensions (𝑑=1 or 2). As an application, by employing the interpolated finite element operators (cf. [7, 8]) to the approximate solution 𝑈 in the rectangular mesh, we obtain the two order global superconvergence of the error between 𝑢 and the interpolation of 𝑈. For a general domain Ω, we can also apply the optimal partition to most rectangular meshes to derive one and a half- order superconvergence.

The rest of this paper is organized as follows. Section 2 provides some preliminaries. Several useful lemmas are established in Section 3. In Sections 4 and 5, we derive the superconvergence in 𝑊^1,𝑝(Ω) and 𝐿_𝑝(𝑄) respectively. Finally, an application is presented in Section 6.

2. Preliminaries

We denote 𝑊^𝑚,𝑝(Ω) and 𝐻^𝑚(Ω), 𝑚≥0 and 1≤𝑝≤∞, the Sobolev spaces on Ω associated with the norms ‖⋅‖_𝑚,𝑝=‖⋅‖_𝑊^𝑚,𝑝_(Ω),‖⋅‖_𝑚=‖⋅‖_𝐻^𝑚_(Ω),‖⋅‖=‖⋅‖_𝐿2(Ω). (2.1)

If 𝑋 is a normed space with the norm ‖⋅‖_𝑋 and 𝜙∶𝐽→𝑋, then we define ‖𝜙‖^𝑝_{𝐿𝑝(0,𝑡;𝑋)}=^𝑡₀‖𝜙(𝜏)‖^𝑝_𝑋𝑑𝜏,‖𝜙‖^𝑝_{𝐿𝑝(𝐽;𝑋)}=^𝑇₀‖𝜙(𝜏)‖^𝑝_𝑋𝑑𝜏,‖𝜙‖_{𝐿∞(𝐽;𝑋)}=esssup_𝑡∈𝐽‖𝜙(𝑡)‖_𝑋. (2.2)

Moreover, we denote (𝜙,𝜓)=_Ω𝜙𝜓𝑑𝑥,(𝜙,𝜓)_𝑄=_𝑄𝜙𝜓𝑑𝑥𝑑𝑡=^𝑇₀(𝜙,𝜓)𝑑𝑡, (2.3) the inner product in 𝐿₂(Ω) (or 𝐿₂(Ω)²) and 𝐿₂(𝑄) (or 𝐿₂(𝑄)²), respectively. We also use Sobolev spaces 𝑊_𝑝^2𝑙,𝑙(𝑄) with norm ‖𝜙‖_𝑊𝑝^2𝑙,𝑙_(𝑄)=(_{𝑄|𝛼|+2𝑠≤2𝑙}|𝐷^𝛼_𝑥𝐷^𝑠_𝑡𝜙(𝑥,𝑡)|^𝑝𝑑𝑥𝑑𝑡)^1/𝑝. (2.4)

We use 𝐶 to denote a generic positive constant independent of ℎ that can take different values at different occurrences.

Let 𝒯 be a family of quasiuniform triangulation of Ω, and let 𝑆_ℎ⊂𝐻¹₀(Ω) be the 𝑘th (𝑘≥1) degree finite element space satisfying the following properties (cf. [9, 10]).

Lemma 2.1. For all 𝑘≥1,1≤𝑠≤𝑘+1 and 1≤𝑝≤∞, we have inf_{𝜒∈𝑆ℎ}{‖𝑣−𝜒‖_0,𝑝+ℎ‖𝑣−𝜒‖_1,𝑝}≤𝐶ℎ^𝑠‖𝑣‖_𝑠,𝑝,∀𝑣∈𝑊^𝑠,𝑝(Ω)∩𝐻¹₀(Ω). (2.5)

Lemma 2.2. For all 𝜒∈𝑆_ℎ, we have ‖𝜒‖_1,𝑝′≤𝐶ℎ^−𝑑/𝑝‖𝜒‖_1,1,2≤𝑝<∞,𝑝^′=𝑝,𝑝−1‖𝜒‖_0,∞≤𝐶ℎ^−𝑑/𝑝‖𝜒‖_0,𝑝,2≤𝑝<∞. (2.6)(2.7)

Given a function 𝑢∈𝑊^𝑘+1,𝑝(Ω)∩𝐻¹₀(Ω), we define its Ritz projection 𝑅_ℎ𝑢∈𝑆_ℎ that satisfies 𝐴(𝑅_ℎ𝑢−𝑢,𝜒)=0,∀𝜒∈𝑆_ℎ. (2.8) Then we get the following well-known estimate: ‖𝑅_ℎ𝑢−𝑢‖_0,𝑝+ℎ‖𝑅_ℎ𝑢−𝑢‖_1,𝑝≤𝐶ℎ^𝑘+1‖𝑢‖_𝑘+1,𝑝,1<𝑝<∞. (2.9)

Moreover, by using the duality argument and (2.9), it is true that for 𝑣∈𝑊^𝑙,𝑝′(Ω), |(𝑅_ℎ𝑢−𝑢,𝑣)|≤𝐶ℎ^𝑘+1+𝑙‖𝑢‖_𝑘+1,𝑝‖𝑣‖_𝑙,𝑝′,0≤𝑙≤𝑘−1, (2.10) where 1<𝑝<∞ and 1/𝑝+1/𝑝^′=1.

We now turn to the finite element scheme of (1.1).

Find a map 𝑈(𝑡)∶𝐽→𝑆_ℎ such that (𝑈_𝑡,𝜒)+𝐴(𝑈,𝜒)=(𝑓(𝑡),𝜒),𝜒∈𝑆_ℎ,𝑡∈𝐽,𝑈(0)=𝑈₀inΩ, (2.11) where 𝑈₀∈𝑆_ℎ is defined by 𝐴(𝑈₀,𝜒)=(𝑓(0),𝜒)−(𝑅_ℎ𝑢_𝑡(0),𝜒),𝜒∈𝑆_ℎ, (2.12) and 𝑢_𝑡(0)=𝑓(0)−𝐴𝑢₀ is given by (1.1).

3. Auxiliary Lemmas

To investigate the superconvergence of finite element approaches for parabolic problems, here and throughout the paper, we decompose the error as 𝑈−𝑢=(𝑈−𝑅_ℎ𝑢)+(𝑅_ℎ𝑢−𝑢)=𝜉+𝜂 and estimate 𝜉 in a superconvergent order.

We start with the superconvergence of initial value errors.

Lemma 3.1. Let 𝑢 and 𝑈 be solutions of (1.1) and (2.11), respectively. Then the following estimates are true: 𝑈_𝑡(0)=𝑅_ℎ𝑢_𝑡(0),‖𝜉(0)‖₁≤𝐶ℎ^𝑘+2‖𝑢_𝑡(0)‖_𝑘+1,𝑘>1,𝐶ℎ²‖𝑢_𝑡(0)‖₂,𝑘=1,‖𝜉(0)‖_0,𝑝≤𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝,2≤𝑝<∞,𝑘>2. (3.1)(3.2)(3.3)

Proof. From (2.12) and (2.11), we have for all 𝜒∈𝑆_ℎ(𝑅_ℎ𝑢_𝑡(0),𝜒)=(𝑓(0),𝜒)−𝐴(𝑈₀,𝜒)=(𝑈_𝑡(0),𝜒), (3.4) and thus (3.1) holds.
By the definition of 𝜉, (2.11), (2.8), (1.1), and the definition of 𝜂, we obtain that for all 𝜒∈𝑆_ℎ, (𝜉_𝑡,𝜒)+𝐴(𝜉,𝜒)=(𝑈_𝑡,𝜒)+𝐴(𝑈,𝜒)−(𝑅_ℎ𝑢_𝑡,𝜒)−𝐴(𝑅_ℎ𝑢,𝜒)=(𝑓,𝜒)−(𝑅_ℎ𝑢_𝑡,𝜒)−𝐴(𝑢,𝜒)=(𝑢_𝑡,𝜒)−(𝑅_ℎ𝑢_𝑡,𝜒)=−(𝜂_𝑡,𝜒). (3.5) Let 𝑡=0, and note that 𝜉_𝑡(0)=0, we obtain 𝐴(𝜉(0),𝜒)=−(𝜂_𝑡(0),𝜒),𝜒∈𝑆_ℎ. (3.6) Then by taking 𝜒=𝜉(0), it follows from (2.10) that ‖𝜉(0)‖²₁≤𝐶ℎ^𝑘+2‖𝑢_𝑡(0)‖_𝑘+1‖𝜉(0)‖₁,𝑘>1,𝐶ℎ²‖𝑢_𝑡(0)‖₂‖𝜉(0)‖,𝑘=1, (3.7) which implies (3.2).
Finally we turn to the proof of (3.3). To do so, we construct an auxiliary problem. Let Φ∈𝑊^1,𝑝′₀(Ω) satisfy 𝐴(𝑣,Φ)=(𝑣,𝜙),𝑣∈𝐻¹₀(Ω), (3.8) and hence by the regularity estimate, it holds that ‖Φ‖_2,𝑝′≤𝐶‖𝜙‖_0,𝑝′, (3.9) where 𝑝^′=𝑝/(𝑝−1).
Therefore, it follows from (3.8), (2.8), (3.6), (2.10), (2.9), and (3.9) that for 𝜙∈𝐿_𝑝′(Ω), (𝜉(0),𝜙)=𝐴(𝜉(0),Φ)=𝐴(𝜉(0),𝑅_ℎΦ)=(𝜂_𝑡(0),Φ−𝑅_ℎΦ)−(𝜂_𝑡(0),Φ)≤𝐶ℎ^𝑘+2‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖Φ−𝑅_ℎΦ‖_1,𝑝′+𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖Φ‖_2,𝑝′≤𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖Φ‖_2,𝑝′≤𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖𝜙‖_0,𝑝′, (3.10) which implies (3.3).

The following lemma gives superconvergence estimates for 𝜉_𝑡𝑡 and ∇𝜉.

Lemma 3.2. Let 𝑢 and 𝑈 be solutions of (1.1) and (2.11), respectively. Then for 𝑘>1, (^𝑡₀‖𝜉_𝑡𝑡‖²𝑑𝜏)^1/2+‖𝜉_𝑡‖₁≤𝐶ℎ^𝑘+2[‖𝑢_𝑡𝑡‖_𝑘+1+(^𝑡₀‖𝑢_𝑡𝑡𝑡‖²_𝑘+1𝑑𝜏)^1/2],𝑡∈𝐽. (3.11)

Proof. By differentiating (3.5) in time, we have (𝜉_𝑡𝑡,𝜒)+𝐴(𝜉_𝑡,𝜒)=−(𝜂_𝑡𝑡,𝜒),𝜒∈𝑆_ℎ. (3.12) Choosing 𝜒=𝜉_𝑡𝑡, (3.12) becomes ‖𝜉_𝑡𝑡‖²+12𝑑𝑑𝑡𝐴(𝜉_𝑡,𝜉_𝑡)=−(𝜂_𝑡𝑡,𝜉_𝑡𝑡). (3.13) Integrating both sides of (3.13) with respect to 𝑡 and applying the integration by parts argument, we obtain that from (3.1) and (2.10) ^𝑡₀‖𝜉_𝑡𝑡‖²1𝑑𝜏+2𝐴(𝜉_𝑡,𝜉_𝑡)=−^𝑡₀(𝜂_𝑡𝑡,𝜉_𝑡𝑡)𝑑𝜏=−(𝜂_𝑡𝑡,𝜉_𝑡)+^𝑡₀(𝜂_𝑡𝑡𝑡,𝜉_𝑡)𝑑𝜏≤𝐶ℎ^𝑘+2‖𝑢_𝑡𝑡‖_𝑘+1‖𝜉_𝑡‖₁+𝐶ℎ^𝑘+2𝑡₀‖𝑢_𝑡𝑡𝑡‖_𝑘+1‖𝜉_𝑡‖₁𝑑𝜏≤𝜀‖𝜉_𝑡‖²₁+𝐶[ℎ^2𝑘+4(‖𝑢_𝑡𝑡‖²_𝑘+1+^𝑡₀‖𝑢_𝑡𝑡𝑡‖²_𝑘+1𝑑𝜏)+^𝑡₀‖𝜉_𝑡‖²₁𝑑𝜏]. (3.14) Therefore, the proof is completed by eliminating 𝜀‖𝜉_𝑡‖²₁ and applying the Gronwall inequality.

Furthermore, the result below for 𝑘=1 can then be obtained by replacing (2.10) by (2.9) in the proof of Lemma 3.2.

Lemma 3.3. It holds that (^𝑡₀‖𝜉_𝑡𝑡‖²𝑑𝜏)^1/2+‖𝜉_𝑡‖₁≤𝐶ℎ²[‖𝑢_𝑡𝑡‖₂+(^𝑡₀‖𝑢_𝑡𝑡𝑡‖²₂𝑑𝜏)^1/2],𝑡∈𝐽. (3.15)

4. Superconvergence in 𝑊^1,𝑝(Ω)

In this Section, we derive the two order global superconvergence (2≤𝑝<∞) and the almost two order global superconvergence (𝑝=∞) estimates on 𝜉 in 𝑊^1,𝑝(Ω).

Theorem 4.1. Under the assumptions that 𝑢_𝑡∈𝑊^𝑘+1,𝑝(Ω),𝑢_𝑡𝑡∈𝐻^𝑘+1(Ω), and 𝑢_𝑡𝑡𝑡∈𝐿₂(0,𝑡;𝐻^𝑘+1(Ω)), we have for 𝑑≤4 and 2≤𝑝<∞, ‖𝜉‖_1,𝑝≤𝐶ℎ^𝑘+2,𝑘>1. (4.1)

Proof. We first introduce an auxiliary problem.
For 𝜓∈𝐿_𝑝′(Ω), let 𝜓_𝑥 be an arbitrary component of ∇𝜓, and let Ψ∈𝑊^1,𝑝′₀(Ω) be the solution of 𝐴(𝑣,Ψ)=−(𝑣,𝜓_𝑥),∀𝑣∈𝐻¹₀(Ω). (4.2) The following priori estimate holds: ‖Ψ‖_1,𝑝′≤𝐶‖𝜓‖_0,𝑝′. (4.3)
Let 𝑣=𝜉 in (4.2), it follows from the integration by parts argument, (2.8) and (3.5) that (𝜉_𝑥,𝜓)=𝐴(𝜉,Ψ)=𝐴(𝜉,𝑅_ℎΨ)=−(𝜂_𝑡+𝜉_𝑡,𝑅_ℎΨ). (4.4)
From (2.10), the stability of 𝑅_ℎ and (4.3), we obtain −(𝜂_𝑡,𝑅_ℎΨ)≤𝐶ℎ^𝑘+2‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑅_ℎΨ‖_1,𝑝′≤𝐶ℎ^𝑘+2‖𝑢_𝑡‖_𝑘+1,𝑝‖Ψ‖_1,𝑝′≤𝐶ℎ^𝑘+2‖𝑢_𝑡‖_𝑘+1,𝑝‖𝜓‖_0,𝑝′. (4.5)
On the other hand, for 𝑠=𝑠^′=2 and 𝑑=1 or 2, or 𝑠=2𝑑/(𝑑−2), 𝑠^′=𝑠/(𝑠−1), and 𝑑=3 or 4, Sobolev embedding inequalities (cf. [11]), Lemma 3.2, the stability of 𝑅_ℎ, and (4.3) imply that −(𝜉_𝑡,𝑅_ℎΨ)≤𝐶‖𝜉_𝑡‖_0,𝑠‖𝑅_ℎΨ‖_0,𝑠′≤𝐶‖𝜉_𝑡‖₁‖𝑅_ℎΨ‖_1,𝑝′≤𝐶ℎ^𝑘+2‖𝜓‖_0,𝑝′. (4.6)
Combining (4.4), (4.5), and (4.6), we have ‖𝜉_𝑥‖_0,𝑝=sup_{𝜓∈𝐿𝑝′(Ω)}(𝜉_𝑥,𝜓)‖𝜓‖_0,𝑝′≤𝐶ℎ^𝑘+2. (4.7)
Therefore, (4.1) follows from summing up all components 𝜉_𝑥 of ∇𝜉.

The following theorem can then be obtained immediately by using Lemma 3.2 and Theorem 4.1.

Theorem 4.2. Under the assumptions of Lemma 3.2 and Theorem 4.1, we have that for 𝑑≤4, ‖𝜉‖_1,𝑄≤𝐶ℎ^𝑘+2,𝑘>1. (4.8)

We now turn to the case of 𝑝=∞.

Theorem 4.3. Assume that 𝑢_𝑡∈𝐿_∞(𝐽;𝑊^𝑘+1,𝑝(Ω)),𝑢_𝑡𝑡∈𝐿_∞(𝐽;𝐻^𝑘+1(Ω)), and 𝑢_𝑡𝑡𝑡∈𝐿₂(𝐽;𝐻^𝑘+1(Ω)). Then for 𝑑=1 and 2, ‖𝜉‖_{𝐿∞(𝐽;𝑊}^1,∞_(Ω))≤𝐶ℎ^{𝑘+2−𝜀},𝑘>1, (4.9) where 𝑝 is large enough and 𝜀>𝑑/𝑝.

Proof. We first define the Green functions associated with the bilinear form 𝐴(⋅,⋅).
Let 𝐺^∗_𝑧∈𝐻¹₀(Ω) be the pre-Green function, and let 𝜕_𝑧𝐺^∗_𝑧 be the directional derivative of 𝐺^∗_𝑧 along some direction with respect to 𝑧. Let 𝐺^ℎ_𝑧,𝜕_𝑧𝐺^ℎ_𝑧∈𝑆_ℎ be the finite element approximations of 𝐺^∗_𝑧 and 𝜕_𝑧𝐺^∗_𝑧, respectively. Then we know that (cf. [1, 12]) ‖𝐺^ℎ_𝑧‖+‖𝐺^ℎ_𝑧‖_1,𝑞≤𝐶,𝑞<2,‖𝜕_𝑧𝐺^ℎ_𝑧‖²+‖𝜕_𝑧𝐺^ℎ_𝑧‖_1,11≤𝐶logℎ. (4.10)(4.11)
Now by definitions of Green functions, (3.5), Hölder's inequalities, (2.10), and (3.11), it is true that for all (𝑧,𝑡)∈𝑄, 𝜉(𝑧,𝑡)=𝐴(𝜉,𝐺^ℎ_𝑧)=−(𝜂_𝑡,𝐺^ℎ_𝑧)−(𝜉_𝑡,𝐺^ℎ_𝑧)≤𝐶ℎ^𝑘+2‖𝑢_𝑡‖_𝑘+1,𝑝‖𝐺^ℎ_𝑧‖_1,𝑝′+‖𝜉_𝑡‖‖𝐺^ℎ_𝑧‖≤𝐶ℎ^𝑘+2(‖𝐺^ℎ_𝑧‖_1,𝑝′+‖𝐺^ℎ_𝑧‖), (4.12) which together with (4.10) yields ‖𝜉‖_0,∞≤𝐶ℎ^𝑘+2. (4.13)
Similarly, by the inverse property (2.6) and (4.11), we have 𝜕_𝑧𝜉(𝑧,𝑡)=𝐴(𝜉,𝜕_𝑧𝐺^ℎ_𝑧)≤𝐶ℎ^𝑘+2(‖𝜕_𝑧𝐺^ℎ_𝑧‖_1,𝑝′+‖𝜕_𝑧𝐺^ℎ_𝑧‖)≤𝐶ℎ^𝑘+2(ℎ^−𝑑/𝑝‖𝜕_𝑧𝐺^ℎ_𝑧‖_1,1+‖𝜕_𝑧𝐺^ℎ_𝑧‖)≤𝐶ℎ^{𝑘+2−𝑑/𝑝}1logℎ, (4.14) which implies that, for 𝑝 large enough and ℎ sufficiently small, ‖∇𝜉‖_0,∞≤𝐶ℎ^{𝑘+2−𝜀}𝑑,𝜀>𝑝. (4.15)
Inequality (4.9) then follows from (4.13) and (4.15).

By the similar arguments used in the proof of Theorems 4.1–4.3 and Lemma 3.3, we obtain the following results.

Theorem 4.4. Under the assumptions of Theorems 4.1–4.3 with 𝑘=1, we have, for 𝑑≤4, ‖𝜉‖_1,𝑝≤𝐶ℎ²,‖𝜉‖_1,𝑄≤𝐶ℎ². (4.16)(4.17) Moreover, for 𝑑=1,2, ‖𝜉‖_{𝐿∞(𝐽;𝑊}^1,∞_(Ω))≤𝐶ℎ². (4.18)

5. Superconvergence in 𝐿_𝑝(𝑄)

In this section, we establish the strong superconvergence for 𝜉 in 𝐿_𝑝(𝑄) with 2≤𝑝≤∞.

We start with the following two order global superconvergence for 2≤𝑝<∞.

Theorem 5.1. Assume that 𝑢_𝑡(0)∈𝑊^𝑘+1,𝑝(Ω),𝑢_𝑡∈𝐿_𝑝(𝐽;𝑊^𝑘+1,𝑝(Ω)),𝑢_𝑡𝑡∈𝐿_𝑝(𝐽;𝐻^𝑘+1(Ω)), and 𝑢_𝑡𝑡𝑡∈𝐿_𝑝(𝐽;𝐿₂(0,𝑡;𝐻^𝑘+1(Ω))). Then, for 𝑑≤4 and 2≤𝑝<∞, it holds that ‖𝜉‖_0,𝑝,𝑄≤𝐶ℎ^𝑘+3,𝑘>2. (5.1)

Proof. First, we construct an adjoint problem of (1.1).
Let 𝑊∈𝐻¹₀(Ω) satisfy (𝑣,𝑊_𝑡)−𝐴(𝑣,𝑊)=(𝑣,𝑤),𝑣∈𝐻¹₀(Ω),𝑊(𝑇)=0inΩ. (5.2)(5.3)
By taking 𝑠=𝑇−𝑡, (5.2) and (5.3) can then be reduced to the weak form of (1.1) and thus we have the regularity estimate (cf. [2]) ‖𝑊‖_𝑊𝑝′^2,1_(𝑄)≤𝐶‖𝑤‖_{0,𝑝′,𝑄}. (5.4)
Let 𝑣=𝜉 in (5.2), it follows from (2.8) and (3.12) that (𝜉,𝑤)=(𝜉,𝑊_𝑡=𝑑)−𝐴(𝜉,𝑊)𝑑𝑡(𝜉,𝑊)−[(𝜉_𝑡=𝑑,𝑊)+𝐴(𝜉,𝑊)]𝑑𝑡(𝜉,𝑊)−[(𝜉_𝑡,𝑊)+𝐴(𝜉,𝑅_ℎ=𝑑𝑊)]𝑑𝑡(𝜉,𝑊)−[(𝜉_𝑡,𝑊)−(𝜂_𝑡,𝑅_ℎ𝑊)−(𝜉_𝑡,𝑅_ℎ=𝑑𝑊)]𝑑𝑡(𝜉,𝑊)+(𝜂_𝑡,𝑅_ℎ𝑊)+(𝜉_𝑡,𝑅_ℎ𝑊−𝑊). (5.5)
After integrating in 𝑡, we have (𝜉,𝑤)_𝑄=−(𝜉(0),𝑊(0))+^𝑇₀(𝜂_𝑡,𝑅_ℎ𝑊)𝑑𝑡+^𝑇₀(𝜉_𝑡,𝑅_ℎ𝑊−𝑊)𝑑𝑡. (5.6) Here the fact that 𝑊(𝑇)=0 was used.
Now we estimate the right-hand side of (5.6) term by term.
First of all, by Hölder's inequalities, (3.3), and the Sobolev embedding inequality, we obtain that −(𝜉(0),𝑊(0))≤‖𝜉(0)‖_0,𝑝‖𝑊(0)‖_0,𝑝′≤𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖𝑊‖_{𝐿∞(𝐽;𝐿𝑝′(Ω))}≤𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖𝑊‖_𝑊^1,1_{(𝐽;𝐿𝑝′(Ω))}≤𝐶ℎ^𝑘+3‖𝑢_𝑡(0)‖_𝑘+1,𝑝‖𝑊‖_𝑊𝑝′^2,1_(𝑄)≤𝐶ℎ^𝑘+3‖𝑊‖_𝑊𝑝′^2,1_(𝑄), (5.7) where ‖𝑊‖_𝑊^1,1_{(𝐽;𝐿𝑝′(Ω))}=^𝑇₀(‖𝑊‖_0,𝑝′+‖𝑊_𝑡‖_0,𝑝′)𝑑𝑡. (5.8)
Secondly, it follows from (2.9), (2.10) and Hölder's inequalities that ^𝑇₀(𝜂_𝑡,𝑅_ℎ𝑊)𝑑𝑡=^𝑇₀(𝜂_𝑡,𝑊)𝑑𝑡+^𝑇₀(𝜂_𝑡,𝑅_ℎ𝑊−𝑊)𝑑𝑡≤𝐶ℎ^𝑘+3𝑇₀‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑊‖_2,𝑝′𝑑𝑡+𝐶ℎ^𝑘+2𝑇₀‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑅_ℎ𝑊−𝑊‖_1,𝑝′𝑑𝑡≤𝐶ℎ^𝑘+3𝑇₀‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑊‖_2,𝑝′𝑑𝑡≤𝐶ℎ^𝑘+3(^𝑇₀‖𝑢_𝑡‖^𝑝_𝑘+1,𝑝𝑑𝑡)^1/𝑝‖𝑊‖_𝑊𝑝′^2,1_(𝑄)≤𝐶ℎ^𝑘+3‖𝑊‖_𝑊𝑝′^2,1_(𝑄). (5.9)
Finally, by Hölder's inequalities, Sobolev embedding inequalities and Lemma 3.2, we have ^𝑇₀(𝜉_𝑡,𝑅_ℎ𝑊−𝑊)𝑑𝑡≤^𝑇₀‖𝜉_𝑡‖_0,4‖𝑅_ℎ𝑊−𝑊‖_0,4/3𝑑𝑡≤𝐶ℎ^𝑇₀‖𝜉_𝑡‖₁‖𝑊‖_1,4/3𝑑𝑡≤𝐶ℎ^𝑇₀‖𝜉_𝑡‖₁‖𝑊‖_2,𝑝′𝑑𝑡≤𝐶ℎ(^𝑇₀‖𝜉_𝑡‖^𝑝₁𝑑𝑡)^1/𝑝‖𝑊‖_𝑊𝑝′^2,1_(𝑄)≤𝐶ℎ^𝑘+3‖𝑊‖_𝑊𝑝′^2,1_(𝑄). (5.10)
Therefore, (5.1) holds by combining all estimates together with (5.4) and (5.6).

We finally establish the almost two order global superconvergence in 𝐿_∞(𝑄). We define a function 𝑔(𝑡)∈𝐻¹₀(Ω), and its finite element approximation 𝑔_ℎ(𝑡)∈𝑆_ℎ satisfy that (𝑣,𝑔_𝑡)−𝐴(𝑣,𝑔)=0,𝑣∈𝐻¹₀(Ω),𝑔(𝑇)=𝛿_ℎinΩ,(𝜒,𝑔_ℎ𝑡)−𝐴(𝜒,𝑔_ℎ)=0,𝜒∈𝑆_ℎ𝑔_ℎ,(𝑇)=𝛿_ℎinΩ, (5.11)(5.12)(5.13)(5.14) where 𝛿_ℎ=𝛿^𝑧_ℎ(𝑥) is the discrete Delta function which satisfies (𝛿_ℎ,𝜒)=𝜒(𝑧),∀𝜒∈𝑆_ℎ. (5.15) Then the following estimate holds (cf. [2]): ‖𝑔_ℎ‖_0,1,𝑄+‖𝑔_ℎ𝑡‖_0,1,𝑄+‖𝜙²𝐷²_𝑥𝑔‖²_0,2,𝑄1≤𝐶logℎ, (5.16) where 𝜙 is the weight function defined by 𝜙(𝑡)=(|𝑥−𝑧|²+𝑡+𝛽²)^1/2,𝛽=𝛾ℎ,𝛾≫1. (5.17)

Furthermore, we have the following estimate.

Lemma 5.2. For 1<𝑞<2 and its conjugate index 𝑞^′, we have ‖𝑔‖_{𝐿𝑞(𝐽;𝑊2,𝑞(Ω))}≤𝐶ℎ^−4/𝑞′1(logℎ)^1/2. (5.18)

Proof. Using the Hölder inequality, it is easy to see that ‖𝐷²_𝑥𝑔‖_0,𝑞,𝑄≤(_𝑄𝜙^{−4𝑞/(2−𝑞)}𝑑𝑥𝑑𝑡)^{(2−𝑞)/2𝑞}‖𝜙²𝐷²_𝑥𝑔‖_0,2,𝑄. (5.19) Note that (cf. [2]) _𝑄𝜙^−𝜆𝛽𝑑𝑥𝑑𝑡≤𝐶^4−𝜆𝜆−4,𝜆>4, (5.20) the proof is then completed by the norm equivalence in 𝐻¹₀(Ω)∩𝑊^2,𝑞(Ω).

The following theorem gives the superconvergence of 𝜉 in 𝐿_∞(𝑄).

Theorem 5.3. Assume that 𝑢_𝑡(0)∈𝑊^𝑘+1,𝑝(Ω) and 𝑢_𝑡∈𝐿_𝑝(𝐽;𝑊^𝑘+1,𝑝(Ω)). Then, for 𝑑=1,2, ‖𝜉‖_0,∞,𝑄≤𝐶ℎ^{𝑘+3−𝜀},𝑘>2, (5.21) with 𝜀>4/𝑝 and 𝑝 large enough.

Proof. (5.13) and (3.5) yield that
𝐷_𝑡(𝜉,𝑔_ℎ)=(𝜉_𝑡,𝑔_ℎ)+(𝜉,𝑔_ℎ𝑡)=(𝜉_𝑡,𝑔_ℎ)+𝐴(𝜉,𝑔_ℎ)=−(𝜂_𝑡,𝑔_ℎ). (5.22) Then by integrating in 𝑡, it follows from (5.15) and (5.14) that 𝜉(𝑧,𝑇)=(𝜉(𝑇),𝛿_ℎ)=(𝜉(𝑇),𝑔_ℎ(𝑇))−(𝜉(0),𝑔_ℎ(0))+(𝜉(0),𝑔_ℎ(0))=−^𝑇₀(𝜂_𝑡,𝑔_ℎ)𝑑𝑡+(𝜉(0),𝑔_ℎ(0)). (5.23) On the one hand, (2.10) and Lemma 5.2 imply that, for 𝑝>2, −^𝑇₀(𝜂_𝑡,𝑔_ℎ)𝑑𝑡=^𝑇₀(𝜂_𝑡,𝑔−𝑔_ℎ)𝑑𝑡−^𝑇₀(𝜂_𝑡,𝑔)𝑑𝑡≤𝐶ℎ^𝑘+2𝑇₀‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑔−𝑔_ℎ‖_1,𝑝′𝑑𝑡+𝐶ℎ^𝑘+3𝑇₀‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑔‖_2,𝑝′𝑑𝑡≤𝐶ℎ^𝑘+3𝑇₀‖𝑢_𝑡‖_𝑘+1,𝑝‖𝑔‖_2,𝑝′𝑑𝑡≤𝐶ℎ^{𝑘+3−4/𝑝}(^𝑇₀‖𝑢_𝑡‖^𝑝_𝑘+1,𝑝𝑑𝑡)^1/𝑝‖𝑔‖_{𝐿𝑝′(𝐽;𝑊′2,𝑝(Ω))}≤𝐶ℎ^{𝑘+3−4/𝑝}1(logℎ)^1/2. (5.24) On the other hand, it follows from (2.7), the Sobolev embedding inequality, (3.3), and (5.16) that (𝜉(0),𝑔_ℎ(0))≤‖𝜉(0)‖_0,∞‖𝑔_ℎ(0)‖_0,1≤𝐶ℎ^−𝑑/𝑝‖𝜉(0)‖_0,𝑝‖𝑔_ℎ‖_{𝐿∞(𝐽;𝐿1(Ω))}≤𝐶ℎ^−𝑑/𝑝‖𝜉(0)‖_0,𝑝‖𝑔_ℎ‖_{𝑊1,1(𝐽;𝐿1(Ω))}≤𝐶ℎ^{𝑘+3−𝑑/𝑝}1logℎ. (5.25) Therefore, (5.21) follows from (5.23), (5.24), and (5.25).

6. An Application

In this section, we apply the interpolation postprocessing technique to improve the accuracy of the approximate solution 𝑈. Let 𝒯_ℎ be a quasi-uniform rectangular partition of Ω⊂ℛ², and let 𝑆_ℎ be the space of continuous piecewise polynomial: 𝑆_ℎ={𝑣∈𝐻¹₀(Ω)∶𝑣∈𝑄^𝑘(𝑇),𝑇∈𝒯_ℎ}, (6.1) where 𝑄^𝑘=span{𝑥^𝑖₁𝑥^𝑗₂∶0≤𝑖,𝑗≤𝑘},𝑘≥1. (6.2)

We introduce the higher interpolation operator 𝐼^𝑘+2_2ℎ, which satisfies the following properties (cf. [7, 8]), for 𝑘≥1,2≤𝑝≤∞,and𝑙=0,1, ‖𝑢−𝐼^𝑘+2_2ℎ𝑢‖_𝑙,𝑝≤𝐶ℎ^{𝑘+3−𝑙}‖𝑢‖_𝑘+3,𝑝,𝐼^𝑘+2_2ℎ𝑖^𝑘_ℎ=𝐼^𝑘+2_2ℎ,‖𝐼^𝑘+2_2ℎ𝜒‖_𝑙,𝑝≤𝐶‖𝜒‖_𝑙,𝑝,𝜒∈𝑆_ℎ, (6.3)(6.4)(6.5) where 𝑖^𝑘_ℎ is the finite element interpolation operator.

In addition, we assume that 𝐴𝑢=−Δ𝑢 in (1.1). By replacing the approximate solution 𝑈 by its interpolation 𝐼^𝑘+2_2ℎ𝑈, we can then establish the two order and the almost two order global superconvergence of 𝑢−𝐼^𝑘+2_2ℎ𝑈 in 𝑊^1,𝑝(Ω) and 𝐿_𝑝(𝑄) for 2≤𝑝≤∞.

Theorem 6.1. Under the assumptions of Theorems 4.1–4.4, 5.1 and 5.3, we have for 𝑢∈𝑊^𝑘+3,𝑝(Ω)∩𝐿_𝑝(𝐽;𝑊^𝑘+3,𝑝(Ω)) with 𝑘>1 and 𝑢∈𝑊^3,𝑝(Ω)∩𝐿_𝑝(𝐽;𝑊^3,𝑝(Ω)) with 𝑘=1, ‖𝑢−𝐼^𝑘+2_2ℎ𝑈‖_1,𝑝≤𝐶ℎ^𝑘+2,2≤𝑝<∞,𝑘>1,‖𝑢−𝐼^𝑘+2_2ℎ𝑈‖_1,𝑄≤𝐶ℎ^𝑘+2,𝑘>1,‖𝑢−𝐼^𝑘+2_2ℎ𝑈‖_{𝐿∞(𝐽;𝑊1,∞(Ω))}≤𝐶ℎ^{𝑘+2−𝜀},𝑘>1,‖𝑢−𝐼²_2ℎ𝑈‖_1,𝑝≤𝐶ℎ²,𝑘=1,‖𝑢−𝐼²_2ℎ𝑈‖_1,𝑄≤𝐶ℎ²,𝑘=1,‖𝑢−𝐼²_2ℎ𝑈‖_{𝐿∞(𝐽;𝑊1,∞(Ω))}≤𝐶ℎ²,𝑘=1,‖𝑢−𝐼^𝑘+2_2ℎ𝑈‖_0,𝑝,𝑄≤𝐶ℎ^𝑘+3,2≤𝑝<∞,𝑘>2,‖𝑢−𝐼^𝑘+2_2ℎ𝑈‖_0,∞,𝑄≤𝐶ℎ^{𝑘+3−𝜀},𝑘>2, (6.6) with 𝜀>2/𝑝 and 𝑝 large enough.

Proof. From (6.4), we have 𝑢−𝐼^𝑘+2_2ℎ𝑈=𝑢−𝐼^𝑘+2_2ℎ𝑢+𝐼^𝑘+2_2ℎ(𝑖^𝑘_ℎ𝑢−𝑅_ℎ𝑢)+𝐼^𝑘+2_2ℎ(𝑅_ℎ𝑢−𝑈), (6.7) which together with the triangular inequality and (6.5) yields that ‖𝑢−𝐼^𝑘+2_2ℎ𝑈‖_𝑙,𝑝≤‖𝑢−𝐼^𝑘+2_2ℎ𝑢‖_𝑙,𝑝+𝐶(‖𝑖^𝑘_ℎ𝑢−𝑅_ℎ𝑢‖_𝑙,𝑝+‖𝑅_ℎ𝑢−𝑈‖_𝑙,𝑝). (6.8)
Moreover, for 2≤𝑝≤∞, the following estimates hold (cf. [7, 8]): ‖𝑖^𝑘_ℎ𝑢−𝑅_ℎ𝑢‖_0,𝑝≤𝐶ℎ^𝑘+3‖u‖_𝑘+3,𝑝1(logℎ)^𝑝,𝑘>2,‖𝑖^𝑘_ℎ𝑢−𝑅_ℎ𝑢‖_1,𝑝≤𝐶ℎ^𝑘+2‖𝑢‖_𝑘+3,𝑝,𝑘>1,‖𝑖¹_ℎ𝑢−𝑅_ℎ𝑢‖_1,𝑝≤𝐶ℎ²‖𝑢‖_3,𝑝, (6.9) where 𝑝=0,when𝑝<∞,1,when𝑝=∞. (6.10)
Hence the proof is completed by (6.3) and the estimates for 𝜉 in Theorems 4.1–4.4, 5.1, and 5.3.