Abstract
In this paper, we investigate local ultraconvergence properties of the high-order finite element method (FEM) for second order elliptic problems with variable coefficients. Under suitable regularity and mesh conditions, we show that at an interior vertex, which is away from the boundary with a fixed distance, the gradient of the post-precessed kth \((k\ge 2)\) order finite element solution converges to the gradient of the exact solution with order \(\mathcal{O}(h^{k+2} (\mathrm{ln} h)^3)\). The proof of this ultraconvergence property depends on a new interpolating operator, some new estimates for the discrete Green’s function, a symmetry theory derived in [26], and the Richardson extrapolation technique in [20]. Numerical experiments are performed to demonstrate our theoretical findings.
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1 Introduction
In this paper, we consider the ultraconvergence of FE approximation for the following elliptic problem
where \(\Omega \subset \mathfrak {R}^{n}~(n=2,3)\) is a bounded polygon \((n=2)\) or polyhedron \((n=3)\), and \(A=(a_{ij})\in (C^\infty (\Omega ))^{n^2}\) is a uniformly positive definite matrix in the sense that there exists \(\delta >0\) satisfying
Note that throughout the paper, the Einstein convention is used: the summation will be taken over all repeated indices.
The study of optimal convergence and superconverence/ultraconvergence properties has been an area of active research, see [1, 2, 4–9, 13–15] and [17–37] for an uncompleted list of references). For instances, Bank and Xu (see [5]) proved that, for linear finite element, the recovered gradient \(Q_{h}\nabla u^{h}\) converges with order \(O(h^{1+\min (1,\sigma )}|\log h|^{\frac{1}{2}})\), where \(Q_{h}\) is a global \(L^{2}\) projection, the underlying mesh \(\mathcal {T}_{h}\) is quasi-uniform and satisfies the so-called \((\alpha ,\sigma )\)- parallelogram property. Huang and Xu (see [15]) obtained that, for second-degree finite element, the recovered gradient \(Q_{h}\nabla u^{h}\) converges with order \(O(h^{2+\min (1,\sigma )}|\log h|^{\frac{1}{2}})\).
It is known that when the underlying mesh has a certain local symmetry property, the corresponding finite element solution has some natrual superconvergence properties. For instances, Schatz, Sloan and Wahlbin discovered in [22, 23, 27] that at a local symmetric vertex \(x_{0}\), when k is even, the finite element solution converges to the exact solution with order \(\mathcal{O}(h^{k+2-\varepsilon })\), where \(\varepsilon >0\) can be arbitrary small; when k is odd, the discrete gradient of the finite element solution converges to the gradient of the exact solution with order \(\mathcal{O}(h^{k+1-\varepsilon })\), where a vertex \(x_0\) is called local symmetric if there exists some radius \(d>0\) such that the underlying mesh is symmetric in the neighborhood \(B(x_0,d)=\{y:|x_0-y|\le d\}\). The natural superconvergence or even ultraconvergence of the finite element solution has been also investigated by other techniques such as the so-called weak estimate, see [6, 14, 18, 19, 36] for an uncompleted list of references.
To obtain better superconvergence/ultraconvergence result, one natural idea is to post-process the finite element solution on some local symmetric mesh, see e.g., [6, 7, 13, 18–20] and references therein. Along this direction, Lin found in [19] that the gradient of a special interpolation of some odd order finite element solution on a uniform rectangular mesh converges with order \(\mathcal{O}(h^{k+2}|\ln h|\)) at an interior vertex. Zhang et al. discovered in [30, 33] that the recovered gradient (see e.g. SPR by [34, 37] and PPR by [30]) of some even order finite element solution on some uniform rectangular meshes converges with order \(\mathcal{O}(h^{k+2}|\ln h|) \) as well. M. Asadzadeh, A. Schatz, and W. Wendland showed that the discrete gradient of some extrapolated finite element solution superconverges with order \(\mathcal{O}(h^{k+1}|\ln h|)\) under suitable local symmetric mesh.
All aforementioned ultraconvergence results are valid only for elliptic equations with constant coefficients. Recall that the classical superconvergence analysis for variable-coefficient problems is done by estimating the difference between the variable coefficient bilinear form and its corresponding (piecewise) constant coefficient bilinear form. Since this difference is only a one-order-higher term, we can not obtain ultraconvergence results of the corresponding finite element solution for variable-coefficient problems. In other words, we can not use this approach to prove the ultraconvergence of the post-processed FE solution for variable-coefficient problems.
In this paper, we propose a novel local interpolation operator to post-process finite element solutions for variable-coefficient problems. Unlike the classical interpolation operator defined in [19, 20], our local operator interpolates the value of the original finite element solution at all vertices of the underlying mesh in a patch instead of interpolating all nodes in a relatively smaller-sized patch.
To prove the ultraconvergence property of our post-processed FE solution, we first investigate properties of the so-called discrete Green’s function in the whole domain \(\mathfrak {R}^n\). We found the difference between two errors, one is the FE projection error of the Green’s function with variable coefficients, another is the FE projection error of the Green’s function with constant coefficients by fixing one point value of the corresponding variable coefficients, is of order almost \(O(h^{2})\). Combining with some further nice properties for the FE projection error of the Green’s function (see Theorem 2.1 for the details), we show that for even k and translation invariant mesh, between two interior vertices \(y_{1}\) and \(y_{2}\) satisfying \(|y_{1}-y_{2}|\lesssim h\), there holds
where u and \(u^h\) are the exact and finite element solutions, respectively, of the following problem
The inequality (1.2) plays a critical role in the proof of our main result
where k is even and \(\varPi _{2kh}^{2k}\) is our local operator which interpolates a continuous function to a polynomial of order 2k in a mesh-patch of size 2kh.
Note that the above estimate is only valid for the case that the degree k is even. To obtain the same ultra-convergence property for the case that k is odd, we need to do some special treatment. Towards this end, we first extrapolate the finite element solution to obtain
where P is the extrapolation operator defined in Section 4. Subsequently, we obtain
Based on (1.3) and (1.5) for \(\mathfrak {R}^{n}\), we establish similar estimates on the bounded region \(\Omega \) for the problem (1.1) with help of interior analysis ( see [26] et al.) and negative norm estimate (see [21]).
Other than variable coefficients, we also would like to emphasize that our results are valid for any locally symmetric mesh, particularly for simplical meshes. Comparing with the best known gradient superconvergence result for variable coefficients by Schatz-Sloan-Wahlbin [23, 26]
and by Asadzadeh et al. [2]
where \(0<\delta ^{'}<1\), our results raise the superocnvergence order to \(h^{k+2}|\ln h|^{3}\), an ultraconvergence result.
The rest of this paper is outlined as follows. In Sect. 2, we discuss the discrete Green’s functions in the whole domain \(\mathfrak {R}^n\) over a uniform conforming partitions. In Sects. 3 and 4, we investigate the ultraconvergence of the finite element solution for the problems in the whole domain \(\mathfrak {R}^n\) over a uniform conforming partitions, where Sect. 3 is for the even order and Sect. 4 is for the odd order finite element solution. In Sect. 5, we apply our theory to problem (1.1). The numerical experiments supporting our theory are presented in Sect. 6.
2 Discrete Green’s functions in \(\mathfrak {R}^n\)
This section is dedicated to a discussion of the finite element approximation properties of the Green’s function. For any positive definite coefficient matrix \(B=(b_{ij})_{n\times n}\), we define the associated bilinear form
where \(H_0^1(\mathfrak {R}^n)=\{v\in H^1(\mathfrak {R}^n)| v \text {\ has\ a compact \ support}\}\). In particular, we denote \(a(\cdot ,\cdot )=a_A(\cdot ,\cdot )\) for simplicity. Let the Green function \(G^B_{z}\) be defined by
For a given point \(z\in \mathfrak {R}^n\), we define the shift of A by \(A_z=(a_{ij}(y+z))\) and the constant matrix \(\overline{A}_{z}=(a_{ij}(z))\). For simplicity, we denote
Let \(\mathcal T_h\) be a uniform conforming partitions of \(\mathfrak {R}^n\) and \(\mathcal N_h\) be the set of all vertices of \(\mathcal T_h\). We assume that \(\mathcal T_h\) is symmetric in the sense that each vertex \(y\in \mathcal N_h\) is a symmetric center of the mesh \(\mathcal T_h\). That is to say, for all \(z\in \mathcal N_h\), \(2z-y\in \mathcal N_h\). Let
be the associated finite element space of degree k and let \(S_h^0=S_h\cap H_0^1(\mathfrak {R}^n)\), we introduce the finite element projector \(R_{h}^{B}:H_{0}^{1}(\mathfrak {R}^n)\rightarrow S_h^0\) for all \(\psi \in S_{h}^{0}(\mathfrak {R}^n)\) by
In particular, we denote \(R_h=R_h^{A},\quad \tilde{R}_{h}^{z}=R_{h}^{A_{z}},\quad \overline{R}_{h}^{z}=R_h^{\overline{A}_z}\).
The following approximation property
is shown in [24]. Usually, the estimate (2.3) can be used to derive the vertex-wise convergence or superconvergence of \(|(u-R_h u)(x_{0})|, x_0\in \mathcal N_h\), where \(u\in H_0^1(\mathfrak {R}^n)\) is the solution of
and \(R_hu=u^h\in S_h^0\) is the finite element solution satisfying
To derive the ultraconvergence of \(R_h u\), we need to discuss further approximation properties of the discrete Green’s functions \(R_hG_z\). Precisely, in this section, we shall estimate the following three quantities defined for all \(y,z\in \mathfrak {R}^n\) by
which depend on the smoothness of variable coefficient \(A=(a_{ij})\).
We have the following Theorem.
Theorem 2.1
Let \(a_{ij}\in C^{\infty }(\mathfrak {R}^n), 1\le i,j\le n\). Then
Moreover if \(|z|\lesssim h\), then we have
and
Before proceeding the proof of Theorem 2.1, we first introduce some lemmas.
Lemma 2.2
Let \(a_{ij}\in C^{\infty }(\mathfrak {R}^n), 1\le i,j\le n\) and \(E(y)=G_{z}(y)-\overline{G}_{z}(y), y\in \mathfrak {R}^n\). Then
Consequently,
Proof
We first show (2.13) for the case \(l=1\). We denote \(\chi _{t}(s)=\frac{\partial E(s)}{\partial s_{t}}, t=1,\ldots ,n\). One observes that
The fact that \(w(z)=a(G_z,w)=a_{\overline{A}_{z}}(\overline{G}_z,w), \forall w\in W^{1,q}(\mathfrak {R}^n)\cap H_{0}^{1}(\mathfrak {R}^n), \forall q>n\) yields that
and thus
Combining (2.15) and (2.16), we have
Let \(y\in \mathfrak {R}^{n}{\setminus } B(z,d)\). Then
We first estimate \(J_{1}(y)\). For all \(d>0\), let \(d_{0}=\max \{2d, 1\}\) and \(\Omega _{0}=B(z,d_{0})\cup B(y,d_{0})\). Then
and the fact that \(a_{ij}\in C^\infty \) yields that
When \(y\in \mathfrak {R}^n\setminus B(z,d), s\in B(z,\frac{d}{2})\), we have \(|y-s|\ge \frac{d}{2}\) and thus
Similarly,
A straightforward calculation yields that
and
Then by (2.19), we have
Similarly,
Then, by (2.18), we get the desired result (2.13) for the case \(l=1\). We turn now to the proof of (2.13) for \(l\ge 2\). From (2.17) it follows that, for any positive integer m,
where we have used the estimate (see [11] and [12])
where \(m^{'}\) is a positive integer. Assume that the weighted Sobolev space \(\kappa _{a}^{T}(B(s,R))\) is a normed linear spaces if equipped with the norms \(\Vert \mu \Vert _{\kappa _{a}^{T}(B(s,R))}=\sum \nolimits _{|\alpha |\le T}\left( \int _{B(s,R)}\left( |D^{\alpha }\mu (x)|\rho (x,M)^{|\alpha |-a}\right) ^{2}dx\right) ^{\frac{1}{2}}\). By Theorem 6.5 (see [3]), (2.21) and (2.13) for the case \(l=1\), we obtain, for all positive integer \(m\ge 2\),
This implies
By (2.22), we have
Consequently,
This gives the desired result (2.13).
Next we show (2.14). For any \(y\in \mathfrak {R}^n\),
where \(I_h^k\) is the standard interpolating operator from \(H^1_0(\mathfrak {R}^n)\) to \(S^0_h\). Letting
we have
We next estimate \(B_1\) and \(B_2\) for \(y\in \mathfrak {R}^n\setminus B(z,2d)\) for some \(d\ge c_1h>h\). Since \(y\in \mathfrak {R}^n\setminus B(z,2d)\), we have \(s\in \mathfrak {R}^n\smallsetminus B(y,d)\) if \(s\in B(z,d)\). Note that
Therefore, by (2.3), we have
where we have used (2.3), (2.13) for the case \(l=k+1\) and the fact that \(d> h\) in the last inequality.
Similarly,
Substituting (2.25) and (2.26) into (2.24) and noticing \(h<d\), we obtain
Thus
This implies
We turn now to the estimation of \(\Vert E-R_hE\Vert _{W^{1,1}(B(z,2c_{1}h))}\). One observes that, for all \(y\in B(z,2c_1h)\),
Using the inverse estimate, we have, if \(n=2\),
and if \(n=3\),
Combining the above three estimates, we have
Furthermore, by (2.29) and the inverse estimate, we have, for \(n=2,3\),
This implies
Consequently,
The estimate (2.14) is a direct result of (2.28) and (2.31). \(\square \)
Lemma 2.3
If all \(a_{ij}\in C^{\infty }(\mathfrak {R}^n), 1\le i,j\le n\), then
Proof
By the definition of \(R_h\) and \(\overline{R}^{z}_{h}\), we have that for any \(v\in S_{0}^{h}(\mathfrak {R}^n)\),
Then,
Consequently,
where for \(1\le l\le n\), \(R_{h}g_{l}\) is the discrete derivative of Green’s function defined by
There exists the following estimates (see [35, 36])
where \(d>c_{0}h\). In the following, we use (2.3), (2.34), (2.35) to prove (2.32). Let \(d_{0}=c_{0}h\). Note that
Combining this estimate and (2.35), we have
We next turn to an improved estimate of \(\Vert (R_h-\overline{R}_h^{z})\overline{G}_{z}\Vert _{W^{1,\infty }(\mathfrak {R}^n\setminus B(z,d))}\) for all \(d\ge d_0=c_0h\). By (2.34), we have that for any \(y\in \mathfrak {R}^n\setminus B(z,d)\),
with
On the other hand, by the fact of \(a_{ij}\in C^\infty \) and (2.3),
This estimate, together with (2.35), gives
Inserting (2.39) and (2.40) into (2.38),
Consequently,
As an immediate consequence of (2.37) and (2.41), we obtain
from which the proof is completed. \(\square \)
Based on Lemmas 2.2 and 2.3, we are now ready to prove Theorem 2.1.
Proof of Theorem 2.1
First, noticing the definition of \(\alpha \), the estimate (2.9) is a direct consequence of the Lemmas 2.3 and 2.4.
Next we show (2.10). Let \( E_{0}(y)=G_{z}(y+z), E_{1}(y)=G_{0}(y)-E_{0}(y), y\in \mathfrak {R}^n\). Noticing the fact that \(\tilde{R}_h^{z}E_{0}(y)=(R_{h}G_{z})(y+z)\), one observes that
By the same arguments in the proof of (2.13), we have that, for all \(|z|\lesssim h\) and all \(d>0\),
Then by the same reasoning to show (2.14), we can prove
Moreover, similarly to the proof of (2.32), we can show that
Plugging the estimates (2.43) and (2.44) into the equality (2.42), we get the desired result (2.10).
By the same reasoning, we obtain the estimate (2.11) .
Next we show (2.12). Note that
and
Then by letting
and noticing the fact that \(\tilde{R}_{h}^{z}\beta _{z}(y)=R_{h}(G_{z}-\overline{G}_{z})(y+z)\), we have
We next estimate the three terms of the right-hand of the above equality. First, note that
we have
Note that
Similarly to (2.13), it follows from (2.47) that, for all \(d>0\),
and
Furthermore, by the same arguments in the proof of (2.14), we have from the above estimate that
We now estimate the second term of the right-hand side of (2.45). Note that, for any \(v\in S_{h}^{0}(\mathfrak {R}^n)\),
and
The above two equalities yields
Similarly to (2.32), by (2.3), (2.10), (2.47) and (2.49), we have
Similarly, we have
Then (2.12) follows by combining (2.45), (2.48), (2.50) and (2.51). \(\square \)
3 Ultraconvergence in the case k is even
In this section, we discuss the ultraconvergence of the finite element solution of (1.1). It is known that the weak solution \(u\in H_{0}^{1}(\mathfrak {R}^n)\) of (1.1) satisfies the following variational form
and the finite element solution \(u^h\in S_h^0\) satisfies
Note that by the definition (2.2), we actually have \(u^h=R_h u\).
Let \(\tau _0\) be a parallelogram or parallelepiped constituting of the elements in the \({\mathcal {T}_h}\) such that each edge of \(\tau _0\) contains \(2k+1\) vertices of \(T_h\). We denote by \(y_0\) the center of \(\tau _0\). Note that the fact \({\mathcal {T}_h}\) yields \(y_0\in \mathcal N_h\). We introduce a \(2k-\)degree interpolation operator \(\varPi _{2kh}^{2k}\) over \(\tau _{0}\) by letting \(\varPi _{2kh}^{2k}v\in (P_{2k})^n\) satisfying
Next we present the main result of this section.
Theorem 3.1
Let \(k\ge 2\) be even. If \(a_{ij}\in C^{\infty }(\mathfrak {R}^n), 1\le i,j\le n\) and \(u\in W^{k+3,\infty }(\mathfrak {R}^n)\) with a compact support, then
The rest of this section is dedicated to the proof of Theorem 3.1. Without loss of generality, we assume \(y_0=0\). Assume that \(\Delta y\in \tau _{0}\cap \mathcal N_{h}\). In the process of proving (3.4), we mainly apply the following estimate
Let \(d>0\) be a constant such that \(\mathrm{Supp} u\subset B(0,2d)\). Let \(\mu \in C^\infty _0(\mathfrak {R}^n)\) satisfy
Moreover, let \(u^J\in \mathbb P_{k+1}\) satisfy
We split u(y) as
with \(u_1=\mu u^J\). It is easy to check that both \(u_1, u_2\) have a compact support and \({\nabla }^{l}u_{1}(0)=0, \forall y\in B(0,d),|l|\ge k+2\) and \(u_{1}(y)=0\) and that \({\nabla }^{k+1}u_{2}(0)=0\). Next, we estimate \(|\nabla (u_1-\varPi _{2kh}^{2k}(R_hu_1))(0)|\) and \(|\nabla (u_2-\varPi _{2kh}^{2k} (R_hu_2))(0)|\) separately.
In the following, we estimate \(|(u_1-R_{h}u_1)(0)-(u_1-R_{h}u_1)(\Delta y )|\). For all \(\Delta y\in \tau _0\cap \mathcal N_h\),
The following Lemmas 3.2 and 3.3 estimate the two terms of the right-hand side of the above equality, respectively.
Lemma 3.2
Under the assumptions of Theorem 3.1, we have that for all \(\Delta y\in \tau _0\cap \mathcal N_h\),
Proof
We first present \((u_1-\overline{R}_{h}^{0}u_1)(0)\) in its integral form. Let the linear operators \(\zeta \) and \(\zeta _{1}\) be defined for all \(v\in C^{0}(\mathfrak {R}^n)\) by
Apparently,
Moreover, the fact \(u_1\) is a polynomial of order \(k+1\) in B(0, d) and that k is even yield
we conclude that \(\zeta (u_1)\) is a polynomial of order k in B(0, d). That is
This equality, together with (3.8), implies
Next we present \((u_1-\overline{R}_{h}^{\Delta y }u_1)({\Delta y})\). We also have
Since \(|\Delta y |\lesssim h\), we have that \(y+\Delta y\in B(0,d)\) for all \(y\in B(0,d/2)\). That is, for all given \(\Delta y\in \tau _0\cap N_h\), \(\zeta _1(u)\) is also a polynomial of degree \(k+1\). On the other hand, the fact that k is even yields
Then we have
This equality, together with (3.11), shows
Set \(\varsigma (y)=\frac{1}{2}(u_1(y)-u_1(y+2\Delta y ))\). One observes that
Using (3.10), (3.13) and (3.14), one observes that \((u_1-\overline{R}_{h}^{0}u_1)(0)-(u_1-\overline{R}_{h}^{{\Delta y} }u_1)(\Delta y )\) can be split into
where \(\alpha _{2}(y,z)\) is defined as (2.8). We next estimate \(W_i, i=1,2,3\) separately. To estimate \(W_{1}\), using (2.3) and the following estimate
we have
We turn now to the estimation of \(W_{2}\). Note that
Inserting (2.3), (3.14) and (3.18) into (3.15), we obtain
Similarly, by (2.11), we obtain
Summing up (3.15), (3.17), (3.19) and (3.20), we have the desired result (3.6). \(\square \)
Lemma 3.3
Assume that \(u\in W^{k+3,\infty }(\mathfrak {R}^n)\). Then
Proof
Note that
and
We observe that \((\overline{R}_{h}^{0}u_1-R_{h}u_1)(0)\) can be split into
where \(\alpha (y,z)\) is defined as (2.6). Moreover, since
and
We obtain
We first estimate \(I_{1}-J_{1}\). Let \(\alpha _{1}(y,z)\) be defined by (2.7). We have that
We first estimate \(Z_{1}^{1}\). Note that
Combining (2.3) and (3.26) gives
We turn now to the estimation of \(Z_{1}^{2}\). Set \(\varsigma _{1}(y)=u_1(y)-u_1(y+\Delta y )\). One observes that
By (2.3), (3.26) and (3.28), we arrive at
Note that the combination of (2.10) and (3.26) implies
Inserting the above two estimates and (3.27) into (3.25) yields
Next we estimate \(I_{2}-J_{2}\). One observes that \(I_{2}-J_{2}\) can be decomposed into
We need estimate the three items of the right-hand side. Summing up (2.9), (2.12), (3.26) and (3.28), we also get, for \(l=1,2,3\),
Substituting the above estimate into (3.30), we obtain
The desired result (3.21) follows from (3.24), (3.29) and (3.31). \(\square \)
We are now in a perfect position to give an estimate for \(\nabla (u_1-\varPi _{2kh}^{2k}(R_{h}u_1))(0)\).
Theorem 3.4
Under the assumptions of Theorem 3.1, we have
Proof
The estimate of \(|\nabla (u_1-\varPi _{2kh}^{2k}(R_{h}u_1))(0)|\) can be reduced to the boundedness of \(|(u_1-R_{h}u_1)(0)-(u_1-R_{h}u_1)(\Delta y )|\) for all \(\Delta y\in \tau _0\cap \mathcal N_h\). In fact,
where \(I_hu_1\in S_h\) is the interpolation of \(u_1\). We first estimate \(\nabla (u_1-\varPi _{2kh}^{2k}(I_{h}u_1))(0)\). One has
Next we estimate \(\nabla \varPi _{2kh}^{2k}(R_{h}u_1-I_{h}u_1)(0)\). Combining (3.6) and (3.21) gives
Set \(\psi (y)=(R_{h}u_1-I_{h}u_1)(y)-(R_{h}u_1-I_{h}u_1)(0)\) for all \(y\in \tau _{0}\). Then, by the inverse estimate,
Then (3.32) can be obtained by combining the estimates (3.33) and (3.34). \(\square \)
Next, we turn to the estimation of \(|\nabla (u_2-\varPi _{2kh}^{2k}(R_{h}u_2))(0)|\).
Theorem 3.5
Under the same assumptions of Theorem 3.1,
Proof
By the same arguments in the proof of Theorem 3.4, the estimate of \(|\nabla (u_2-\varPi _{2kh}^{2k}(R_{h}u_2))(0)|\) can be reduced to the boundedness of \(|(u_2-R_{h}u_2)(0)-(u_2-R_{h}u_2)(\Delta y )|\) for all \(\Delta y\in \tau _0\cap \mathcal N_h\). We decompose
and we will estimate the above two terms separately. We first present \((u_2-\overline{R}_{h}^{0}u_2)(0)-(u_2-\overline{R}_{h}^{\Delta y}u_2)(\Delta y)\) in its integral form. Similarly to Theorem 3.1, we have
and
Then
Noticing the facts that \(\nabla ^{k+1}u_2(0)=0, \mathrm{Supp} u_2\subset B(0,2d)\), (2.3) and (3.26), we have
By \(\nabla ^{k+1}u_2(0)=0\) and (2.10), we have
We next estimate \(K_{2}\). For any \(y\in \mathfrak {R}^{n}\),
This equality implies
Then, we have
Let \(\mu (y)=(u_{2}(y)-u_{2}(y+\Delta y))-(u_{2}(-y)-u_{2}(-y+\Delta y))\). Combining (3.37) and (3.40) gives
Again by \(\nabla ^{k+1}u_{2}(0)=0\), we have
Plugging (2.3) and (3.42) into (3.41) and noticing that \(\mathrm{Supp} u_2\subset B(0,2d)\), we have
Combining (3.38), (3.39) and (3.43), we have that
Similarly to (3.21), we have
Therefore,
By the same arguments in the proof of (3.32), using (3.44), we get the desired result (3.35).
Finally, we are ready to prove Theorem 3.1.
Proof of Theorem 3.1
By the definition of \(u_1\) and \(u_2\), we have that
Then (3.4) is an immediate consequence of Theorems 3.4 and 3.5.
4 Ultraconvergence in the case k is odd
When k is odd, the equality (3.9) is not necessary valid. Therefore, the reasoning in the previous section can not be generalized to an arbitrary integer k. In other words, the inequality (3.4) is not necessary valid. To obtain the similar result for the case k is odd, we first need to extrapolate the finite element solution. Precisely, let \(\mathcal T_{h/2}\) be obtained by decomposing each element of \(\mathcal T_h\) into \(2^{n}\) equal-sized elements. We assume that \(\mathcal {T}_{h/2}\) is symmetric in the sense that each vertex \(y\in \mathcal N_{h/2}\) is a symmetric center of the mesh \(\mathcal T_{h/2}\). We denote by \(u_{h/2}\) the finite element solution corresponding to the mesh \(\mathcal T_{h/2}\). We define the Richardson extrapolating function \(Pu^h\in S_{h}\) by letting
Theorem 4.1
Assume that \(k\ge 3\) is odd. Let \(a_{ij}\in C^{\infty }(\mathfrak {R}^n), 1\le i,j\le n\) and \(u\in W^{k+3,\infty }(\mathfrak {R}^{n})\). Then
As in the previous section, we also decompose \(u=u_1+u_2\) and estimate the errors for \(u_1\) and \(u_2\) separately.
Lemma 4.2
Under the assumptions of Theorem 4.1,
Proof
We first present \((u_{1}-P(\overline{R}_{h}^{0}u_{1}))(0)\) in its integral form. One observes that
Set
Similarly to (4.4), by (4.5), we have
Since \(u_1\) is a polynomial of order \(k+1\) in B(0, d). One observes that \(\chi _{1}\) is a polynomial of order k in B(0, d). By (4.4) and (4.6), we have
Similarly, we have
where \(\chi _{2}(y)=2^{k+1} u_{1}(\frac{y+\Delta y }{2}), \chi _{3}(y)=u_{1}(y)-\chi _{2}(y)\). Combining the above two estimates, we obtain
where \(\chi _{4}(y)=\chi _{1}(y)-\chi _{3}(y+\Delta y)\) and \(\alpha _{2}(y,z)\) is defined as (2.8).
Next we estimate \(S_i, i=1,2,3.\) First, noticing (2.3) and (3.16), we obtain
Secondly, the estimate (2.3) and the fact that
yield
Finally, by (2.11),
Then (4.3) follows by substituting (4.7), (4.8), (4.9) into (4.10). \(\square \)
Based on Lemmas 3.3, 4.2 and Theorem 3.4, we are now in a position to show Theorem 4.1.
Proof of Theorem 4.1
A straightforward calculation yields that
with
First by (4.3),
Secondly, by Lemma 3.3,
Finally, it follows from (3.35) that
Inserting the above three estimates into the equality (4.11), we have
Combining this estimate and the inverse inequality, we get the desired result (4.2) by the same arguments in the proof of (3.32). \(\square \)
5 Ultraconvergence in a bounded domain
In this section, we apply the previous theory to the problem (1.1). Let \(\mathcal T_{h}^{\Omega }\) be a quasi-uniform conforming partition of \(\Omega \) satisfying the following property: there exists a parallelogram or parallelepipe \( \tau \subset \Omega \) such that : 1) \(\tau \) is the union of some elements in \(\mathcal T_h^\Omega ,\) 2) \(\tau \) has a size \(h_\tau \simeq 1,\) 3) \(\mathcal T_{h}^\Omega \cap \tau =\mathcal T_{h}\cap \tau \). Here \(\mathcal T_{h}\) is a mesh in the whole domain \(\mathfrak {R}^n\) defined in Sect. 2, \(\mathcal T_h^\Omega \cap \tau =\{\tau '\in \mathcal T_h^\Omega :\tau '\subset \tau \}\) and \(\mathcal T_h\cap \tau =\{\tau '\in \mathcal T_h:\tau '\subset \tau \}\). Let
be the associated standard finite element space of degree k and let \(S_h^0(\Omega )=S_h(\Omega )\cap H_0^1(\Omega )\), we introduce the finite element projector \(R_{h}^{\Omega }:H_{0}^{1}(\Omega )\rightarrow S_h^0(\Omega )\) for all \(\psi \in S_{h}^{0}(\Omega )\) by
where the associated bilinear form is defined by
Based on Theorem 3.1, we have the following result.
Theorem 5.1
Let \(a_{ij}\in C^{\infty }(\Omega ), 1\le i,j\le n\) and \(\varPi _{2kh}^{2k}\) be defined as in Section 3, and \(\mathcal N_h^{\tau }\) be the set of all vertices of \(\overline{\mathcal T}_{h}^\Omega \cap \tau \). Let \(k\ge 2\) be an even. Assume that \(y_{0}\in \mathcal N_{h}^{\Omega }\) is away from the boundary of \(\tau \) with a fixed distance and \(u\in W^{k+3,\infty }(\tau )\), then
Proof
Assume that \(d\simeq 1\) satisfies \(B(y_{0},2d)\subset \tau \). Let \(\phi \in C^{\infty }(\mathfrak {R}^{n})\) satisfy \(0\le \phi \le 1\), and \(\phi =1\) in B\((y_0,d)\), \(\phi =0\) in \( \mathfrak {R}^{n}\smallsetminus B(y_0,2d)\), and \(\Vert \phi \Vert _{W^{k+3,\infty }(\mathfrak {R}^{n})}\lesssim 1\). Let \(\upsilon (y)=\phi (y)u(y)\). One observes that \(\nabla (u-\varPi _{2kh}^{2k}(R_{h}^{\Omega }u))(y_0)\) can be decomposed into
Using (3.4), we have
We turn now to the estimation of \(\nabla \varPi ^{2k}_{2kh}(R_{h}\upsilon -R_{h}^{\Omega }u)(y_0)\). We denote that \(S_{h}^{0}(B(y_{0},d))=\{v^h\in C(\Omega ):v^h|_e\in P_k,\forall e\in \overline{\mathcal T}_{h}^\Omega \} \cap H_{0}^{1}(B(y_{0},d))\). Note that \(\overline{\mathcal T}_{h}^\Omega \cap \tau =\overline{\mathcal T}_{h}\cap \tau \), we have, for any \(w\in S_{h}^{0}(B(y_{0},d))\),
Then by the arguments of Schatz et al. in [22, 23], we have
Furthermore, we get
Inserting the estimates (5.4) and (5.6) into the equality (5.3), we get the desired result (5.2). \(\square \)
Next we consider the case that k is odd. Let \(\overline{\mathcal T}_{h/2}^{\Omega }\) be a quasi-uniform partition obtained by decomposing each element of \(\overline{\mathcal T}_h^{\Omega }\) into \(2^{n}\) elements. Furthermore, we assume that \(\overline{\mathcal T}_{h/2}^{\Omega }\cap \tau =\overline{\mathcal T}_{h/2}\cap \tau \) where \(\overline{\mathcal T}_{h/2}\) is defined as in Section 4. We denote by \(R_{h/2}^{\Omega }u\) the finite element solution corresponding to the mesh \(\overline{\mathcal T}_{h/2}^{\Omega }\). We define the Richardson extrapolating function \(P(R_{h/2}^{\Omega }u)\in S_{h}(\Omega )\) by letting
Similarly to Theorem 5.1, we have the following result.
Theorem 5.2
Under the assumptions of Theorem 5.1, if \(k\ge 3\) is odd, then
Based on Theorems 5.1 and 5.2, we have the following corollary.
Corollary 5.3
Under the assumptions of Theorem 5.1, if k is even, then
and, if \(k\ge 3\) is odd, then
Proof
Recall a classical result [21]
when \(u\in W^{k+l,2}(\Omega )\) for any \(2\le l\le k\). Then (5.9) and (5.10) follows from (5.2) and (5.8), respectively.
6 Numerical examples
We consider (1.1) with \(\Omega =[0,1]^2\) and the coefficients
The problem admits the exact solution
We will validate (5.9) and (5.10) with numerical experiments. For simplicity, the underlying mesh is chosen as a uniform one which consists of equal-sized isosceles right-angled triangles.
Since the estimates (5.9) and (5.10) are only valid for an interior vertex \(y_0\), without loss of generality, we test our results in the following vertices set
Correspondingly, we define the discrete norm \(\quad \Vert v\Vert _{\infty ,h}=\max _{x_{j}\in \mathcal N_h^0}|v(x_{j})|\).
We will test our estimates for different orders \(k=2,3,4\). Note that once k and the mesh size h are given, the corresponding finite element solution \(R_h^\Omega u\) can be computed with the standard finite element method. Let \(y_0\in \mathcal N_h^0, \Delta y_{1}=(h,0)\) and \(\Delta y_{2}=(0,h)\). One observes that \(\varPi _{2kh}^{2k}R_{h}^{\Omega }u(y)\) is a polynomial of degree 2k along each direction \(y_{i},i=1,2\).
In the following, we explain how to compute \(\varPi _{2kh}^{2k}R_{h}^{\Omega }u(y_0)\) and \(\varPi _{2kh}^{2k}(PR_{h}^{\Omega })u(y_0)\).
When \(k=2\), we have
When \(k=4\), we use
to compute \(\frac{\partial \varPi _{2kh}^{2k}R_{h}^{\Omega }u(y_{0})}{\partial y_{i}}\).
When \(k=3\), we first use (5.7) to compute \(P(R_h^\Omega u)(y_0)\) and we obtain
Depicted in Tables 1, 2, 3 are our numerical ultraconvergence results corresponding to the finite element degree \(k=2,4,3\) respectively.
From Tables 1, 2, 3, we observe that the gradients of the post-processed FE solutions approximate the gradient of u with orders \(\mathcal{O}(h^4)\), \(\mathcal{O}(h^6)\) and \(\mathcal{O}(h^5)\), respectively, which validate the estimates (5.9) and (5.10). Moreover, it is interesting to find that the hidden constant is independent of the mesh size h which indicates that maybe the ‘lnh’ appeared in the right-hand side of the estimates (5.9) and (5.10) can be removed.
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Acknowledgments
The authors would like to thank both the anonymous referees for their careful reading of the paper and their valuable comments which leads to a significant improvement of the paper. The first author is supported in part by the National Natural Science Foundation of China (11671304, 11171257, 11301396), the Zhejiang Provincial Natural Science Foundation, China (No. LY15A010015). The second author is supported in part by the National Natural Science Foundation of China (11471031,91430216) and the US National Science Foundation through grant U1530401. The third author is supported in part by the National Natural Science Foundation of China through grants 11571384 and 11428103, by the Fundamental Research Funds for the Central Universities through grant 16lgjc80, and by Guangdong Provincial Natural Science Foundation of China through grant 2014A030313179.
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He, Wm., Zhang, Z. & Zou, Q. Ultraconvergence of high order FEMs for elliptic problems with variable coefficients. Numer. Math. 136, 215–248 (2017). https://doi.org/10.1007/s00211-016-0838-6
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DOI: https://doi.org/10.1007/s00211-016-0838-6