Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients

Computers and Mathematics with Applications xxx (xxxx) xxx

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Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients ∗

Wenming He , Xiong Liu, Jin Xiao Department of Mathematics, Lingnan Normal University, Zhanjiang, Guangdong, 524048, China

article

info

a b s t r a c t Assume that u(x) satisfies the problem Lu(x) ≡ − ∂∂x (aij ∂∂xu ) = f (x), ∀x ∈ Ω , u(x) = i j 0, ∀x ∈ ∂ Ω . In this article, using interpolation postprocessing technique, we will investigate the local ultraconvergence of the primal variable and the derivative of finite element approximation of u(x) using piecewise polynomials of degrees bi-k (k ≥ 3) over a rectangular partition. Assume that k ≥ 3 is odd and x0 is an interior vertex satisfying ρ (x0 , ∂ Ω ) ≥ c. Using the new interpolation postprocessing formula presented in this study, we show that the primal variable and the derivative of the post-processed finite element solution using piecewise of degrees bi-k (k ≥ 3) at x0 converge to the primal variable and the derivative of the exact solution with order O(hk+3 |ln h|) under suitable regularity and mesh conditions, respectively. Finally, we use numerical experiments to illustrate our theoretical findings. © 2019 Elsevier Ltd. All rights reserved.

Article history: Received 28 February 2019 Received in revised form 9 September 2019 Accepted 21 November 2019 Available online xxxx Keywords: The local symmetric theory Primal variable Derivative Constant coefficients Interpolation operator

1. Introduction Consider the following model problem

{

Lu(x) ≡ −

∑2

∂

i,j=1 ∂ xi

(

aij

∂ u(x) ∂ xj

)

= f (x), in Ω ,

u(x) = 0,

on ∂ Ω .

(1.1)

Here Ω ⊂ ℜ2 is a bounded rectangle, and A = (aij ) is a uniformly positive constant matrix in the sense that there exists δ > 0 such that aij ξi ξj ≥ δξi ξi ,

∀ξ ∈ ℜ2 .

Note that throughout the paper, the Einstein convention is used: the summation will be taken over all repeated indices. Since 1970s, the investigation of superconvergence/ultraconvergence properties of finite element method for the problem (1.1) has been a research hot spot in finite element field (see [1–14]). In this article, we will investigate the local ultraconvergence of finite element method over a rectangular partition. First let us introduce some related works. Assume that Rh u is the finite element approximation of u(x) using piecewise ∗ Corresponding author. E-mail addresses: [email protected] (W. He), [email protected] (X. Liu), [email protected] (J. Xiao). https://doi.org/10.1016/j.camwa.2019.11.016 0898-1221/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

2

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

polynomials of degree at most k in each variable over a rectangular partition Th and x0 is a vertex of Th . Lin et al. (see [15]) observed that, if u ∈ W k+2,∞ (Ω ), then

|(u − Rh u)(x0 )| ≤ chk+2 |ln h|∥u∥W 2k,∞ (Ω ) . Furthermore, Chen (see [16,17]) observed that, if (1.1) is the Poisson equation, then there holds

|(u − Rh u)(x0 )| ≤ ch2k |ln h|∥u∥W 2k,∞ (Ω ) . It was observed that, if the underlying partition has some local symmetry property (see [18–20]), then the corresponding finite element solution holds certain natural superconvergence properties. Let x0 be a local symmetric vertex. That is to say there exists some radius d > 0 satisfying the underlying mesh is symmetric in the neighborhood B(x0 , d) = {x : |x0 − x| ≤ d}. They obtained that, if k ≥ 3 is odd, then the discrete gradient of the finite element solution using piecewise polynomials of degrees k (k ≥ 1) converges to the gradient of the exact solution with degree O(hk+1−ε ) at x0 , and if k is even, then the finite element solution converges to the exact solution with order O(hk+2−ε ) at x0 , where ε > 0 can be arbitrary small. Postprocessing the finite element solution on a certain local symmetric rectangular partition is a good idea to get higher local superconvergence/ultraconvergence result. See e.g., [5,21] and finite element references therein. For example, Lin (see [21]) introduced a special interpolation postprocessing operator and got that, if k is odd and x0 is an interior vertex, then the gradient of the postprocessing finite element solution on a uniform rectangular mesh converges with order O(hk+2 |ln h|) at x0 . Zhang et al. (see [8]) observed that, if k is even, then the recovered gradient (see e.g. SPR by [22,23] and PPR by [8]) of finite element solution on some uniform rectangular partitions converges with order O(hk+2 |ln h|) as well. M. Asadzadeh et al. (see [1]) showed that the discrete gradient of some extrapolated finite element solution superconverges with order O(hk+1 |ln h|) under suitable local symmetric mesh. W. He et al. (see [24]) presented a special 2k interpolation postprocessing operator Π2kh and showed that, if k is even, then the derivative of post-processing finite element solution using piecewise polynomials of degrees k (k ≥ 2) for second-degree elliptic problem with variable coefficients superconverges with order O(hk+2 |ln h|). We also observe that there are many works investigating the superconvergence of finite element method for more complex equations (see [25–27] et al.). In this article, the interpolation postprocessing technique is used to investigate the local ultraconvergence of the primal variable and the derivative for finite element method using piecewise polynomials of degrees bi-k over a rectangular partition for the problem (1.1) where k ≥ 3 is odd. The rest of the article is organized as follows. In Section 2, we will introduce some notations and definitions. In Section 3, we will investigate the local ultraconvergence properties of the finite element method using piecewise polynomials of degrees k for the problem (1.1). Numerical experiments are presented in Section 4. 2. Preliminary In this article, some standard notations for Sobolev spaces and their norms are introduced. ρ (x, ∂ Ω ) means the distance between the points x and ∂ Ω . B(x, r) denote the open ball centered at point x with radius r. Assume that the bilinear form aK (., .) and the linear functional FK (.) are defined by aK (ψ, φ ) =

∫ A∇ψ.∇φ dx K

and F K (φ ) =

∫

f φ dx, K

respectively, where domain K ⊂ ℜ2 is a bounded domain. Let H01 (K ) = {v ∈ H 1 (K ) : v = 0 on the boundary of K }, aΩ (., .) and FΩ (.) be denoted by a(., .) and F (.) for simplicity. The weak form of the problem (1.1) is to find u ∈ H01 (Ω ) satisfying the problem a(u, φ ) = F (φ ),

∀φ ∈ H01 (Ω ).

Assume that z ∈ K . We define the Green’s function GKz ∈ W 1,p (K ), 1 ≤ p < 2 by aK (GKz , ψ ) = ψ (z),

1,p1

∀ψ ∈ W0

where p1 = +∞ if p = 1, and p1 =

(K ).

p p−1

(2.2)

if 1 < p < 2.

K In particular, we denote GΩ z by Gz . Assume that Th is a conforming quasi-uniform partition of K with grid size h and NhK denotes the set of all vertices of K . We define finite element spaces

Sh (K ) = {v ∈ C (K ) : v|e ∈ Qk

∀e ∈ ThK }

for tensor product elements., where Qk is the space of polynomials of degree up to k in every variable. Let Sh0 (K ) = Sh (K ) ∩ H01 (K ) and the finite element projectors RKh : H01 (K ) → Sh0 (K ) be defined by aK (ψ − RKh ψ, φ ) = 0,

∀φ ∈ Sh0 (K )

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

3

In particular, introduce the discrete Green’s function RKh GKz (x) by

∀ψ ∈ Sh0 (K ).

aK (RKh GKz , ψ ) = ψ (z), Ω

Ω

Ω

Ω

(2.3)

Ω

In particular, let Th , Nh , Rh and Rh Gz be denoted by Th , Nh , Rh and Rh Gz , respectively. We assume that every interior mesh vertex x0 in Th is a local center of symmetry of the mesh (see, e.g., [20]); i.e. for a sufficiently large r, if φ ∈ Sh (B(x0 , r)), then φ (x0 −(x−x0 )) ∈ Sh (B(x0 , r)). In this paper, Ih denotes the Lagrange interpolation operator using piecewise polynomials of degrees bi-k over Th . The rest of this study is organized as follows. The interpolation postprocessing technique is used to investigate the local ultraconvergence of the derivative of finite element method using piecewise polynomials of degrees bi-k over a rectangular partition for the problem (1.1) in Section 2. In Section 3, we will use numerical examples to illustrate our main result. 3. Main results Assume that m ≥ 3 is odd and Th is a uniform rectangular partition over Ω . Let τ0 be a rectangle constituting of the elements in the Th such that each edge of τ0 only contains k + m + 1 vertices of Th . We denote by x0 the center of τ0 . Note that x0 ∈ Nh where Nh denotes the set of all vertices of Th . We introduce a bi-(k + m) degree interpolation operator Π(kk++mm)h over τ0 by letting Π(kk++mm)h v ∈ Qk+m satisfying

Π(kk++mm)h v (x) = v (x),

∀ x ∈ τ0 ∩ Nh .

(3.4)

One observes that, if v (x) is a bi-(k + m) degree polynomial function defined in τ0 , then

∂Π(kk++mm)h v (x) ∂ xi

=

∂v (x) , ∂ xi

i = 1, 2.

Next we explain how to compute m = 3, then

(3.5)

k+m ∂Π(k +m)h Rh u(x0 )

∂ xi

for the case k = 3. Assume that β1 = (1, 0), β2 = (0, 1). We have, if

∂Π(kk++mm)h Rh u(x0 ) =

∂ xi 15[Rh u(x0 + hβi ) − Rh u(x0 − hβi )] − 3[Rh u(x0 + 2hβi ) − Rh u(x0 − 2hβi )]

+

Rh u(x0 + 3hβi ) − Rh u(x0 − 3hβi ) 60h

20h

,

and if m = 5, then

∂Π(kk++mm)h Rh u(x0 ) =

∂ xi 0.8[Rh u(x0 + hβi ) − Rh u(x0 − hβi )] − 0.2[Rh u(x0 + 2hβi ) − Rh u(x0 − 2hβi )]

+

4[Rh u(x0 + 3hβi ) − Rh u(x0 − 3hβi )] 105h

h

−

[Rh u(x0 + 4hβi ) − Rh u(x0 − 4hβi )] 280h

.

In this section, our main result is as follows. Theorem 3.1. Assume that k ≥ 3 is odd and u(x) is the solution for the problem (1.1). Assume also that τ ⊂ Ω is a rectangle with center x0 satisfying ρ (x0 , ∂τ ) ≥ c. Under the assumption that u(x) ∈ H k+1 (Ω ) ∩ W k+4,∞ (τ ), there holds

|(u − Rh u)(x0 )| ≤ chk+3 |ln h|(∥u∥W k+3,∞ (τ ) + ∥u∥H k+1 (Ω ) ),

(3.6)

|∇ (u − Π(kk++mm)h Rh u)(x0 )| ≤ chk+3 |ln h|(∥u∥W k+4,∞ (τ ) + ∥u∥H k+1 (Ω ) ).

(3.7)

and

To prove Theorem 3.1, we need to introduce some lemmas. Lemma 3.2. Let k1 =

√∑ e∈Thτ

Ω

Ω

k+m . 2

Assume that Ωl = {x + hlβi | x ∈ Ω } (−k1 ≤ l ≤ k1 ) and Φl = Ωl ∩ Ω . Then there holds

∥Rh l Gx0l − Rh Gx0 ∥2H 3 (e) ≤ ch.

(3.8)

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

4

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx Ω

Ω

Ω

Proof. Assume that Exl 0 (x) = (Gx0l − Gx0 )(x). We split (Rh l Gx0l − Rh Gx0 )(x) into Ω

Ω

(Rh l Gx0l − Rh Gx0 )(x) Ω

Φ

Ω

Φ

Ω

Φ

Φ

= (Rh l − Rh l )Gx0l (x) + (Rh l Gx0l − Rh l Gx0 )(x) + (Rh l − Rh )Gx0 (x) Ω

Φ

Φ

Ω

Φ

= Rh l Exl 0 (x) + (Rh l − Rh l )Gx0l (x) + (Rh l − Rh )Gx0 (x).

(3.9)

We need to estimate the three items of the right-hand side. Let d be a positive constant such that B(x0 , 2d) ⊂ Ω , we first estimate ∥Exl 0 ∥W 4,∞ (B(x0 ,d)) . He et al. (see [28]) showed that, for any r ≥ c, there holds

∥Gx0 ∥W 4,∞ (Ω ∖B(x0 ,r)) ≤ c ,

(3.10)

and Ω

∥Gx0l ∥W 4,∞ (Ω ∖B(x0 ,r)) ≤ c .

(3.11)

Note that ρ (∂ Φl , ∂ Ω ) = h|l| and ρ (∂ Φl , ∂ Ωl ) = h|l|. From (3.10) and (3.11) it follows that Ω

∥Exl 0 ∥L∞ (∂ Φl ) ≤ ∥Gx0l ∥L∞ (∂ Φl ) + ∥Gx0 ∥L∞ (∂ Φl ) ≤ ch.

(3.12)

Noticing the fact that B(x0 , 2d) ⊂ Ω , and there holds LExl 0 (x) = 0,

∀x ∈ Φl .

(3.13)

By (3.12) and (3.13), we obtain

∥Exl 0 ∥W 4,∞ (B(x0 ,d)) ≤ ∥Exl 0 ∥L∞ (∂ Φl ) ≤ ch. Next, we proceed to estimate ∥

(3.14)

Exl 0 W 4,∞ (Φl \B(x0 ,d)) .

∥

Assume that x ∈ Ω \ B(x0 , d). We have

Ω

Exl 0 (x) = Gx0 (x) − Gx0l (x) Ω

Ω

= Gx0l+hlβi (x + hlβi ) − Gx0l (x) Ω

Ω

Ω

Ω

= [Gx0l+hlβi (x + hlβi ) − Gx0l+hlβi (x)] + [Gx0l+hlβi (x) − Gx0l (x)] = Exl 0 ,1 (x) + Exl 0 ,2 (x).

(3.15) Ω

We need to estimate Exl ,1 (x) and Exl ,2 (x), respectively. Let G(y, x) = Gy l (x). From [29] it follows that, if ρ (y, ∂ Ω ) ≥ c, then 0 0 there holds

⏐ α ⏐ ⏐ ∂ G(y, x) ⏐ −|α| ⏐ ⏐ ⏐ ∂ xα1 ∂ yα2 ⏐ ≤ c |x − y| ,

(3.16)

where α = (α1 , α2 ). By (3.16), we obtain Ω

∥Exl 0 ,1 ∥W 4,∞ (Φl \B(x0 ,d)) ≤ ch∥Gx0l ∥W 5,∞ (Φl \B(x0 ,d)) ≤ ch.

(3.17)

Similarly,

∥Exl 0 ,2 ∥W 4,∞ (Φl \B(x0 ,d)) ≤ ch.

(3.18)

Inserting the estimates (3.17) and (3.18) into (3.15), we have

∥Exl 0 ∥W 4,∞ (Φl \B(x0 ,d)) ≤ ch.

(3.19)

Combining (3.14) and (3.19) gives

∥Exl 0 ∥H 4 (Φl ) ≤ ch.

(3.20)

This implies

√∑ e∈Thτ

≤

Φ

∥Rh l Exl 0 ∥2H 3 (e)

√∑ e∈Thτ

Φ

∥(Rh l − Ih )Exl 0 ∥2H 3 (e) +

√∑ e∈Thτ

∥Ih Exl 0 ∥2H 3 (e)

Φ

≤ ch−2 ∥(Rh l − Ih )Exl 0 ∥H 1 (τ ) + ch ≤ ch−2 h3 ∥Exl 0 ∥H 4 (Φl ) + ch ≤ ch2 + ch ≤ ch.

(3.21)

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx Ω

Φ

5

Ω

Next we estimate (Rh l − Rh l )Gx0l (x). One observes that, for all x ∈ Φl ∖ B(x0 , d), there holds Ω

Φ

Ω

|(Rh l − Rh l )Gx0l (x)| Ωl

Ω

Φ

Ω

≤ |(Rh − Ih )Gx0l (x)| + |(Rh l − Ih )Gx0l (x)| ≤ ch4 + ch4 ≤ ch4 .

(3.22)

Note that, for any v ∈ Sh0 (Φl ), there holds Ω

Φ

Ω

aΦl ((Rh l − Rh l )Gx0l , v ) Ω

Ω

Φ

Ω

= aΦl (Rh l Gx0l , v ) − aΦl (Rh l Gx0l , v ) = v (x0 ) − v (x0 ) = 0.

(3.23)

By the arguments of Schatz et al. in [18,19] and (3.23), we have Ω

Φ

Ω

∥(Rh l − Rh l )Gx0l ∥H 1 (B(x0 ,d)) Ω

Φ

Ω

≤ c ∥(Rh l − Rh l )Gx0l ∥L2 (Φl ∖B(x0 , 3d )) ≤ ch4 .

(3.24)

2

Similarly to (3.22), we have Ω

Φ

Ω

∥(Rh l − Rh l )Gx0l ∥H 1 (Φl ∖B(x0 ,d)) ≤ ch3 .

(3.25)

From (3.24) and (3.25) it follows that Ω

Φ

Ω

∥(Rh l − Rh l )Gx0l ∥H 1 (Φl ) ≤ ch3 . This implies

√∑ e∈Thτ

Ω

Φ

Ω

∥(Rh l − Rh l )Gx0l ∥2H 3 (e) ≤ ch.

(3.26)

Similarly, we get

√∑ e∈Thτ

Φ

∥(Rh l − Rh )Gx0 ∥2H 3 (e) ≤ ch.

(3.27)

Plugging (3.21), (3.26) and (3.27) into (3.9), we get the desired result (3.8). Lemma 3.3. Assume that ρ (xˆ , x0 ) ≤ ch. Then there holds

∥(Gx0 − Rh Gx0 ) − (Gxˆ − Rh Gxˆ )∥H 1 (Ω ∖B(x0 ,d)) ≤ ch4 .

(3.28)

Proof. Set exˆ (x) = (Gx0 − Gxˆ )(x).

(3.29)

Let Ω ∖ B(x0 , ⊂ Ω1 ⊂ Ω ∖ B(x0 , be decomposed into d ) 2

d ) 4

satisfy Ω1 =

⋃

e∈Th ,e⊂Ω1

e. One observes that (Gx0 − Rh Gx0 )(x) − (Gxˆ − Rh Gxˆ )(x) can

(Gx0 − Rh Gx0 )(x) − (Gxˆ − Rh Gxˆ )(x) = (exˆ − Rh exˆ )(x) Ω

Ω

= (exˆ − Rh 1 exˆ )(x) + (Rh 1 exˆ − Rh exˆ )(x).

(3.30)

We need to estimate the two items of the right-hand side. Similarly to (3.17), we have

∥exˆ ∥H 4 (Ω1 ) ≤ ch. Consequently, we have Ω

∥exˆ − Rh 1 exˆ ∥H 1 (Ω1 ) ≤ ch3 ∥exˆ ∥H 4 (Ω1 ) ≤ ch4 .

(3.31)

Ω1

Next we estimate (Rh exˆ − Rh exˆ )(x). One observes that Ω

Ω

∥Rh 1 exˆ − Rh exˆ ∥L2 (Ω1 ) ≤ ∥Ih exˆ − Rh exˆ ∥L2 (Ω1 ) + ∥Rh 1 exˆ − Ih exˆ ∥L2 (Ω1 ) ≤ ch4 ∥exˆ ∥H 4 (Ω1 ) + ch4 ∥exˆ ∥H 4 (Ω1 ) ≤ ch4 . Note that, for any v ∈ Ω1

Sh0 (Ω1 ),

(3.32)

there holds

aΩ1 (Rh exˆ − Rh exˆ , v ) = 0.

(3.33)

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

6

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

By the arguments of Schatz et al. in [18,19] and (3.33), we have Ω

Ω

∥Rh 1 exˆ − Rh exˆ ∥H 1 (Ω ∖B(x0 ,d)) ≤ c ∥Rh 1 exˆ − Rh exˆ ∥L2 (Ω1 ) Ω

≤ c ∥Rh 1 exˆ − exˆ ∥L2 (Ω1 ) + ∥exˆ − Rh exˆ ∥L2 (Ω1 ) ≤ ch4 + ch4 ≤ ch4 .

(3.34)

Plugging (3.31) and (3.34) into (3.30), we get the desired result (3.28). Let d be a positive constant such that B(x0 , 2d) ⊂ τ . Assume that θ (x) ∈ C ∞ (Ω ) is a cutoff function satisfying θ (x) = 1 if y ∈ B(x0 , 2d ), and θ (x) = 0 if y ∈ Ω ∖ B(x0 , d), and ∥θ∥W J ,∞ (Ω ) ≤ c for any positive integer J. We split u(x) into u(x) = u1 (x) + u2 (x),

(3.35) τ

where u1 (x) = θ (x)u(x). Let k1 = k+2m , β1 = (1, 0), β2 = (0, 1). Assume that xˆ l = x0 + hlβi ∈ Nh 0 (−k1 ≤ l ≤ k1 ). To prove Theorem 3.1, we need to estimate [u1 (x0 ) − Rh u1 (x0 )] − [u1 (xˆ l ) − Rh u1 (xˆ l )] and [u2 (x0 ) − Rh u2 (x0 )] − [u2 (xˆ l ) − Rh u2 (xˆ l )], respectively. Lemma 3.4. Under the same assumptions of Theorem 3.1, there holds

|[u1 (x0 ) − Rh u1 (x0 )] − [u1 (xˆ l ) − Rh u1 (xˆ l )]| ≤ chk+4 |ln h|∥u∥W k+4,∞ (τ ) .

(3.36)

Proof. Introduce Ωl by

Ωl = {x − hlβi | x ∈ Ω }.

(3.37)

Set u1 (x) = u1 (x + hlβi ),

u˜ 1 (x) = u1 (x) − u1 (x).

(3.38)

Notice the fact that u1 (x0 ) = u1 (xˆ ), and Ω

Rh l u1 (x0 ) = Rh u1 (xˆ ). By the above two equalities, we split [u1 (x0 ) − Rh u1 (x0 )] − [u1 (xˆ l ) − Rh u1 (xˆ l )] into

[u1 (x0 ) − Rh u1 (x0 )] − [u1 (xˆ l ) − Rh u1 (xˆ l )] Ω

= [u1 (x0 ) − Rh u1 (x0 )] − [u1 (x0 ) − Rh l u1 (x0 )] Ω

= [u1 (x0 ) − Rh u1 (x0 )] − [u1 (x0 ) − Rh u1 (x0 )] − [Rh u1 (x0 ) − Rh l u1 (x0 )] Ω

= [˜u1 (x0 ) − Rh u˜ 1 (x0 )] − [Rh u1 (x0 ) − Rh l u1 (x0 )].

(3.39)

We need to estimate the two items of the right-hand side. Without loss of generality, we assume that τ is a rectangle with center x0 . To estimate u˜ 1 (x0 ) − Rh u˜ 1 (x0 ), we split u˜ 1 (x0 ) − Rh u˜ 1 (x0 ) into u˜ 1 (x0 ) − Rh u˜ 1 (x0 ) = [˜u1 (x0 ) − Rτh u˜ 1 (x0 )] + [Rτh u˜ 1 (x0 ) − Rh u˜ 1 (x0 )].

(3.40)

Assume that w1 (x) satisfies w1 (x) = u˜ 1 (2x0 − x) if x ∈ B(x0 , d), and w1 (x) = 0 if x ∈ τ ∖ B(x0 , d). Set

w(x) =

1 2

[˜u1 (x) + w1 (x)].

(3.41)

It is easy to see that u˜ 1 (x0 ) = w (x0 ).

(3.42)

Next we prove Rτh w1 (x0 ) = Rτh u˜ 1 (x0 ). Note that

∂ Rτh Gx0 (2x0 − x) ∂ Rτ Gx (x) =− h 0 . ∂ xi ∂ xi

(3.43)

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

7

Let y = 2x0 − x. Using (3.43), we get τ

Rh w1 (x0 ) =

∫ aij

τ

∂ Rτh w1 (x) ∂ Rτh Gx0 (x) dx ∂ xi ∂ xj

∂w1 (x) ∂ Rτh Gx0 (x) = aij dx ∂ xi ∂ xj ] [ ] ∫τ [ ∂ Rτ Gx (2x0 − x) ∂ u˜ 1 (2x0 − x) = aij − × − h 0 dx ∂ xi ∂ xj τ ∫ ∂ u˜ 1 (2x0 − x) ∂ Rτh Gx0 (2x0 − x) = aij dx ∂ xi ∂ xj ∫τ ∂ u˜ 1 (y) ∂ Rτh Gx0 (y) dy = Rτh u˜ 1 (x0 ), = aij ∂ yi ∂ yj τ

∫

which implies Rτh w (x0 ) = Rτh u˜ 1 (x0 ).

(3.44)

By (3.42) and (3.44), we have u˜ 1 (x0 ) − Rτh u˜ 1 (x0 ) = w (x0 ) − Rτh w (x0 ).

(3.45) τ

In the following, we will estimate w (x0 ) − Rh w (x0 ). Lin et al. (see [21]) showed

|a(w − Ih w, Rτh Gτx0 )| ≤ chk+3

∑∫ e

e⊂τ

|Dk+2 w(x)D3 Rτh Gτx0 (x)|dx.

(3.46)

Note that k is odd. From (3.41) it follows that Dk+2 w (x0 ) = 0.

(3.47)

By (3.46) and (3.47), we have

w(x0 ) − Rτh w(x0 ) = aτ (w − Rτh w, Rτh Gτx0 ) = a(w − Ih w, Rτh Gτx0 ) ∑∫ ≤ chk+3 |Dk+2 w(x)D3 Rτh Gτx0 (x)|dx e

e⊂τ

≤ chk+3

∑∫ e

e⊂τ

≤ ch

k+3

∑∫ e⊂τ

e

In the following, we will estimate

∑

∑∫ e

e⊂τ

∫ e

e⊂B(x0 ,2h)

e

e⊂τ ∖B(x0 ,h)

∫

∑

+

e

e⊂τ ∖B(x0 ,h)

∑

≤

e⊂τ

∫ e

(3.48)

|x − x0 ||D3 Rτh Gτx0 (x)|dx. One observes that

|x − x0 ||D3 Rτh Gτx0 (x)|dx

∫

∑

+

|Dk+3 w(x)||x − x0 ||D3 Rτh Gτx0 (x)|dx.

|x − x0 ||D3 Rτh Gτx0 (x)|dx

∑

≤

|[Dk+2 w(x) − Dk+2 w(x0 )]D3 Rτh Gτx0 (x)|dx

e⊂B(x0 ,2h)

+ ch−2

∫ e

|x − x0 ||D3 (Rτh Gτx0 − Ih Gτx0 )(x)|dx

|x − x0 ||D3 Ih Gτx0 (x)|dx

chh−1 |D2 Rτh Gτx0 (x)|dx

∑

|x − x0 ||D(Rτh Gτx0 − Ih Gτx0 )(x)|dx

e⊂τ ∖B(x0 ,h)

∫ +c τ ∖B(x0 ,h)

|x − x0 ||x − x0 |−3 dx

≤ c |ln h| + ch−2 h2 |ln h| + c |ln h| ≤ c |ln h|,

(3.49)

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

8

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

where we have used the estimates (see [30])

∑

∥D2 Gτx0 ∥L1 (e) ≤ c |ln h|,

(3.50)

e⊂τ

and (see [28])

|∇ (Rτh Gτx0 − Ih Gτx0 )(x)| ≤ ch2 |x − x0 |−3 .

(3.51)

By (3.48) and (3.49), we have

|w(x0 ) − Rτh w(x0 )| ∑∫ ≤ chk+3 |Dk+3 w(x)||x − x0 ||D3 Rτh Gτx0 (x)|dx e⊂τ

≤ ch

k+3

e

∥˜u1 ∥W k+3,∞ (τ ) |ln h| ≤ chk+4 ∥u1 ∥W k+4,∞ (τ ) |ln h|

≤ chk+4 |ln h|∥u∥W k+4,∞ (τ ) .

(3.52)

This, together with (3.45), gives

|˜u1 (x0 ) − Rτh u˜ 1 (x0 )| ≤ chk+4 |ln h|∥u∥W k+4,∞ (τ ) .

(3.53)

τ

Next we estimate Rh u˜ 1 (x0 ) − Rh u˜ 1 (x0 ). Note that Rτh u˜ 1 (x0 ) − Ih u˜ 1 (x0 ) = aτ (u˜ 1 − Ih u˜ 1 , Rτh Gτx0 ),

(3.54)

Rh u˜ 1 (x0 ) − Ih u˜ 1 (x0 ) = a(u˜ 1 − Ih u˜ 1 , Rh Gx0 ) = aτ (u˜ 1 − Ih u˜ 1 , Rh Gx0 ).

(3.55)

and

By (3.8), (3.54) and (3.55), we have

|Rτh u˜ 1 (x0 ) − Rh u˜ 1 (x0 )| = |aτ (u˜ 1 − Ih u˜ 1 , Rτh Gτx0 − Rh Gx0 )| ∑∫ |Dk+2 u˜ 1 (x)D3 (Rτh Gτx0 − Rh Gx0 )(x)|dx ≤ chk+3 e⊂τ

e

≤ chk+4 ∥u∥W k+3,∞ (τ )

∑∫

Ω

Ω

|D3 (Rh Gx0 − Rh l Gx0l )(x)|dx e

e⊂τ

≤ chk+4 ∥u∥W k+3,∞ (τ ) .

(3.56)

Inserting (3.53) and (3.56) into (3.40), we obtain

|˜u1 (x0 ) − Rh u˜ 1 (x0 )| ≤ chk+4 |ln h|∥u∥W k+4,∞ (τ ) .

(3.57)

Ω

Next we estimate Rh u1 (x0 ) − Rh l u1 (x0 ). Note that Rh u1 (x0 ) − Ih u1 (x0 ) = a(u1 − Ih u1 , Rh Gx0 ) = aτ (u1 − Ih u1 , Rh Gx0 )

(3.58)

and Ω

Ω

Ω

Ω

Ω

Rh l u1 (x0 ) − Ih u1 (x0 ) = aΩl (u1 − Ih u1 , Rh l Gx0l ) = aτ (u1 − Ih u1 , Rh l Gx0l ).

(3.59)

From the above two equalities and (3.8) it follows that Ω

|(Rh u1 (x0 ) − Rh l u1 (x0 ))| Ω

Ω

= |aτ (u1 − Ih u1 , Rh Gx0 − Rh l Gx0l )| ∑∫ Ω Ω k+3 ≤ ch |Dk+2 u1 (x)D3 (Rh Gx0 − Rh l Gx0l )(x)|dx e⊂τ

≤ ch

k+3

e

∥u1 ∥W k+2,∞ (τ )

∑∫ e⊂τ

k+4

≤ ch

Ω

Ω

|D3 (Rh Gx0 − Rh l Gx0l )(x)|dx e

∥u∥W k+2,∞ (τ ) .

(3.60)

Combining (3.39), (3.57) and (3.60), we have

|[u1 (x0 ) − Rh u1 (x0 )] − [u1 (xˆ l ) − Rh u1 (xˆ l )]| ≤ chk+4 |ln h|∥u∥W k+4,∞ (τ ) .

(3.61)

This completes the proof. Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

9

Lemma 3.5. Under the same assumptions of Theorem 3.1, there holds

|[u2 (x0 ) − Rh u2 (x0 )] − [u2 (xˆ l ) − Rh u2 (xˆ l )]| ≤ chk+4 ∥u∥H k+1 (Ω ) .

(3.62)

Proof. One observes that (u2 − Rh u2 )(x0 ) = a(u2 − Rh u2 , Gx0 − Rh Gx0 ) = a(u2 − Ih u2 , Gx0 − Rh Gx0 )

= aΩ ∖B(x0 , d ) (u2 − Ih u2 , Gx0 − Rh Gx0 ),

(3.63)

2

and (u2 − Rh u2 )(xˆ ) = aΩ ∖B(x

d 0, 2 )

(u2 − Ih u2 , Gxˆ − Rh Gxˆ ).

(3.64)

By (3.28), (3.63) and (3.64), we have

|(u2 − Rh u2 )(x0 ) − (u2 − Rh u2 )(xˆ )| = |aΩ ∖B(x0 , d ) (u2 − Ih u2 , Gx0 − Rh Gx0 − Gxˆ + Rh Gxˆ )| 2

≤ c ∥u2 − Ih u2 ∥H 1 (Ω ∖B(x0 , d )) ∥Gx0 − Rh Gx0 − Gxˆ + Rh Gxˆ ∥H 1 (Ω ∖B(x0 , d )) 2

2

≤ chk ∥u2 ∥H k+1 (Ω ) ch4 ≤ chk+4 ∥u∥H k+1 (Ω ) . The proof is complete. Based on the above lemmas, we are now in a position to give a proof of Theorem 3.1. We first prove (3.6). Let u1 (x) and u2 (x) be defined as above. We split (u − Rh u)(x0 ) into (u − Rh u)(x0 ) = (u1 − Rh u1 )(x0 ) + (u2 − Rh u2 )(x0 ).

(3.65)

To estimate (u1 − Rh u1 )(x0 ), we decompose it into (u1 − Rh u1 )(x0 ) = (u1 − Rτh u1 )(x0 ) + (Rτh u1 − Rh u1 )(x0 ).

(3.66)

Similarly to (3.53) and (3.56), we have

|(u1 − Rτh u1 )(x0 )| ≤ chk+3 |ln h|∥u∥W k+3,∞ (τ ) , and

|(Rτh u1 − Rh u1 )(x0 )| ≤ chk+3 |ln h|∥u∥W k+3,∞ (τ ) . Substituting the above two estimates into (3.66) gives

|(u1 − Rh u1 )(x0 )| ≤ chk+3 |ln h|∥u∥W k+3,∞ (τ ) .

(3.67)

Next we estimate (u2 − Rh u2 )(x0 ). Note that d ≥ c. By (3.63), we have

|(u2 − Rh u2 )(x0 )| ≤ chk ∥u∥H k+1 (Ω ) ∥Gx0 − Rh Gx0 ∥H 1 (Ω ∖B(x0 , d )) 2

≤ chk ∥u∥H k+1 (Ω ) ch3 ≤ chk+3 ∥u∥H k+1 (Ω ) ,

(3.68)

where we have used the estimate (see [28])

∥Gx0 − Rh Gx0 ∥W 1,∞ (Ω ∖B(x0 ,r)) ≤ chk r −k−2 . Combining (3.67) and (3.68) gives the desired result (3.6). Next we prove (3.7). Assume that, for all v ∈ C 1 ,

∂ (Π(kk++mm)h v )(x0 ) ∂ xi We decompose

∂ u(x0 ) ∂ xi

=

−

k1 1 ∑

h

k+m ∂ (Π(k +m)h v )(x0 )

∂ xi

can be described as

αl v (xˆ l ).

(3.69)

l=−k1

k+m ∂ (Π(k +m)h Rh u)(x0 )

∂ xi

into

∂ u(x0 ) ∂ (Π − ∂ xi ∂ xi [ ] [ ] k+m ∂ ( Π ∂ (Π(kk++mm)h u)(x0 ) ∂ (Π(kk++mm)h Rh u)(x0 ) ∂ u(x0 ) (k+m)h u)(x0 ) = − + − ∂ xi ∂ xi ∂ xi ∂ xi ⎡ ⎤ ⎡ ⎤ k1 k1 k1 ∑ ∂ u(x0 ) 1 ∑ 1 ∑ − αl u(xˆ l )⎦ + ⎣ αl u(xˆ l ) − αl Rh u(xˆ l )⎦ . =⎣ ∂ xi h h k+m (k+m)h Rh u)(x0 )

l=−k1

l=−k1

(3.70)

l=−k1

Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

10

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx Table 1 k = 3. h = 1/16

h = 1/20

h = 1/24

h = 1/28

h = 1/32

∥u − Rh u∥∞,h

0.21e−008

0.55e−009

0.18e−009

0.73e−010

0.33e−010

∥u − Rh u∥∞,h

0.35e−001

0.35e−001

0.35e−001

0.35e−001

0.35e−001

1.3e−006

3.4e−007

1.1e−007

4.48e−008

2.01e−008

2.2e+001

2.2e+001

2.2e+001

2.2e+001

2.2e+001

1.8e−008

0.37e−008

0.11e−008

0.39e−009

0.16e−009

0.30

0.24

0.21

0.19

0.176

h6

∥∇ (u − Π(kk++33)h Rh u)∥∞,h ∥∇ (u − Π(kk++33)h Rh u)∥∞,h h6

∥∇ (u − Π

k+5 (k+5)h Rh u) ∞,h

∥

∥∇ (u − Π

k+5 (k+5)h Rh u) ∞,h

∥

h6

We need to estimate the two items of the right-hand side. By (3.69), we have

⏐ ⏐ ⏐ ⏐ k1 ⏐ ⏐ ∂ u(x0 ) 1 ∑ 2k1 ⏐ ⏐ ˆ ∥u∥W 2k1 +1,∞ (τ ) ≤ chk+3 ∥u∥W k+4,∞ (τ ) . α u( x ) − l l ⏐ ≤ ch ⏐ ∂ xi h ⏐ ⏐ l=−k1

(3.71)

Next we estimate [u(x0 ) − Rh u(x0 )] − [u(xˆ l ) − Rh u(xˆ l )]. By (3.36) and (3.62), we have

|[u(x0 ) − Rh u(x0 )] − [u(xˆ l ) − Rh u(xˆ l )]| ≤ |[u1 (x0 ) − Rh u1 (x0 )] − [u1 (xˆ l ) − Rh u1 (xˆ l )]| + |[u2 (x0 ) − Rh u2 (x0 )] − [u2 (xˆ l ) − Rh u2 (xˆ l )]| ≤ chk+4 |ln h|∥u∥W k+4,∞ (τ ) + chk+4 ∥u∥H k+1 (Ω ) . Inserting the above two estimates into (3.70), we have

⏐ ⏐ ⏐ ∂ u(x ) ∂ (Π k+m Rh u)(x0 ) ⏐ ⏐ ⏐ 0 (k+m)h − ⏐ ⏐ ⏐ ∂ xi ⏐ ∂ xi ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ ⏐ k1 ⏐ k1 k1 ∑ ⏐ ∂ u(x0 ) 1 ∑ ⏐ ⏐ 1⏐∑ ≤ ⏐⏐ αl u(xˆ l ) − αl Rh u(xˆ l )⏐⏐ − αl u(xˆ l )⏐⏐ + ⏐⏐ h ⏐ ∂ xi ⏐ ⏐ h ⏐l=−k1 l=−k1 l=−k1 ≤ chk+3 ∥u∥W k+4,∞ (τ ) + chk+3 |ln h|∥u∥W k+4,∞ (τ ) + chk+3 ∥u∥H k+1 (Ω ) ≤ chk+3 |ln h|∥u∥W k+4,∞ (τ ) + chk+3 ∥u∥H k+1 (Ω ) .

(3.72)

We complete the proof. 4. Numerical examples Consider (1.1) with Ω = [0, 1]2 and the coefficients a11 = 1,

a22 = 1,

a12 = a21 = 0.75.

The problem admits the exact solution u(x) = sin(π x1 ) sin(π x2 ). In this section, we will investigate (3.7) by numerical experiments. For simplicity, the underlying mesh is chosen as a uniform one which consists of equal-sized isosceles right-angled triangles. Note that the estimate (3.7) is only valid for an interior vertex x0 . We test our results only in the following vertices. We first consider the case that the distance between x0 and ∂ Ω is greater than 41 . Let 1 3 1 3 Nh = Nh ∩ [ , ] × [ , ]. 4 4 4 4 Correspondingly, we introduce the discrete norm ∥.∥∞,h by ∥v∥∞,h = maxxj ∈Nh | v (xj )| We will investigate our estimates for the case k = 3. Notice the fact that once k and h are given, the corresponding finite element solution Rh u can be obtained using the standard finite element method. Depicted in Table 1 are our numerical ultraconvergence results corresponding to the finite element method using piecewise polynomials of bicubic for the case that x0 ∈ Nh0 . Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.

W. He, X. Liu and J. Xiao / Computers and Mathematics with Applications xxx (xxxx) xxx

11

From Table 1 it follows that the primal variable of the finite element and the derivative of the post-processed finite element solutions approximate the primal variable and the derivative of u with order O(h6 ) at an interior vertex, which shows that Theorem 3.1 is valid. One also observes that

has much higher accuracy than

∂ xi 4[u(x0 +3hβi )−u(x0 −3hβi )] 0.8[u(x0 +hβi )−u(x0 −hβi )]−0.2[u(x0 +2hβi )−u(x0 −2hβi )] h 105h 15[u(x0 +hβi )−u(x0 −hβi )]−3[u(x0 +2hβi )−u(x0 −2hβi )] u(x0 +3hβi )−u(x0 −3hβi ) . 20h 60h

important reason is that ∂ u(x ) more approximates ∂ x 0 than i

k+5 ∂Π(k +5)h Rh u(x0 )

+

−

k+3 ∂Π(k +3)h Rh u(x0 )

∂ xi [u(x0 +4hβi )−u(x0 −4hβi )] 280h

. An

much

+

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Please cite this article as: W. He, X. Liu and J. Xiao, Local ultraconvergence of high order finite element method by interpolation postprocessing technique for elliptic problems with constant coefficients, Computers and Mathematics with Applications (2019), https://doi.org/10.1016/j.camwa.2019.11.016.