SUPERCONVERGENCE AND POSTPROCESSING OF THE CONTINUOUS GALERKIN METHOD FOR NONLINEAR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS

. We propose a novel postprocessing technique for improving the global accuracy of the continuous Galerkin (CG) method for nonlinear Volterra integro-differential equations. The key idea behind the postprocessing technique is to add a higher order Lobatto polynomial of degree 𝑘 + 1 to the CG approximation of degree 𝑘 . We first show that the CG method superconverges at the nodal points of the time partition. We further prove that the postprocessed CG approximation converges one order faster than the unprocessed CG approximation in the 𝐿 2 -, 𝐻 1 - and 𝐿 ∞ -norms. As a by-product of the postprocessed superconvergence results, we construct several a posteriori error estimators and prove that they are asymptotically exact. Numerical examples are presented to highlight the superconvergence properties of the postprocessed CG approximations and the robustness of the a posteriori error estimators.


Introduction
In this paper, we consider the nonlinear Volterra integro-differential equation (VIDE) of the form VIDEs arise widely in mathematical modelling of physical, biological, engineering and other phenomena that are governed by memory effects [9,19,26].During the past few decades, various numerical methods have been studied for VIDEs, such as Runge-Kutta methods [4,20,27,32], collocation methods [8,14,24,25,28], continuous Galerkin (CG) methods [17,18,[29][30][31] and discontinuous Galerkin (DG) methods [7,21].We refer the readers to the monographs [5,6] and the literature given therein.Among the above mentioned numerical methods, the Galerkin type methods have received considerable attention due to the (possibly) arbitrary high-order convergence.In the context of Galerkin methods, postprocessing techniques are attractive ways to improve the accuracy of an already obtained Galerkin approximation.Several postprocessing techniques have been introduced for the VIDEs.For example, defect correction methods (based on interpolation and iteration) [18,33] and Richardson extrapolation method [34] were studied for the CG approximations of the nonlinear VIDEs with smooth kernels; superconvergence extraction technique based on Lagrange interpolation was developed in [22] for the DG approximations of the linear VIDEs with smooth and non-smooth kernels.For other types of integral equations, some postprocessing techniques for Galerkin approximations were also investigated, see, e.g., [15,16] and references therein.
The aim of this paper is to propose and analyze a novel postprocessing technique to improve the accuracy of the CG method for the VIDE (1.1) with regular solutions.The key idea of our postprocessing technique is to add a higher order Lobatto polynomial of degree  + 1 to the CG approximation of degree  (see 4.3), which can be regarded as a simple correction for the CG approximation.The main contributions and features of this paper can be summarized as following: -We show that the CG method superconverges at the nodal points of the time partition with respect to the step-size.-We prove that the proposed postprocessing technique improves the convergence rates of the CG method in the  2 -,  1 -and  ∞ -norms by one order.-Based on the postprocessed superconvergence results, we construct asymptotically exact a posteriori error estimators for the CG method as the step-size decreases.-The postprocessing is local, in the sense that it can be done independently on each local time interval, which enables the design of parallel numerical algorithms.-The postprocessing is very easy to implement and can achieve global superconvergence with a small cost which only requires to compute an integral on each local time interval.
This paper is organized as follows.In Section 2, we introduce the CG scheme for the VIDE 1.1 and state the a priori error estimates.In Section 3, we prove the nodal superconvergence estimates and obtain some superclose results.In Section 4, we describe the postprocessing technique and analyze its superconvergence properties.In Section 5, we construct several a posteriori error estimators and prove that they are asymptotically accurate.In Section 6, we present some numerical experiments to validate the theoretical results.Finally, we give some concluding remarks in Section 7.

Continuous Galerkin method
Let  ℎ be a partition of [0,  ] into  time intervals We define the length of   by ℎ  =   −  −1 and the maximum step-size by ℎ = max 1≤≤ {ℎ  }.For simplicity, we assume that the time mesh is quasiuniform, i.e., there exists a positive constant   such that although the postprocessing technique proposed in this work can be applied to an arbitrary mesh.
The trial and test function spaces are given by respectively.Here,   (  ) denotes the space of polynomials of degree at most  on   and the space  −1 (  ) is defined analogously.
To describe the CG method, we define an integral operator  : ([0,  ]) → ([0,  ]) by The CG approximation of the VIDE (1.1) can be read as: find for any  ∈  −1,0 ( ℎ ).Since the CG scheme (2.1) employs different trial and test function spaces, it can be regarded as a Petrov-Galerkin scheme.The well-posedness of the CG solution defined by (2.1) has been proved in [18,29].
Due to the discontinuous character of the test space  −1,0 ( ℎ ), the CG scheme (2.1) can be decoupled into local problems on each time step, namely, if  is given on for all  ∈  −1 (  ).Here, we set the initial value  | 1 ( 0 ) =  0 .
The following a priori error estimates have been established in [18,29].

Natural superconvergence of the CG method
In this section, we show that the CG method for the VIDE (1.1) superconverges at the nodal points of the time partition.We further prove that the projection Π   (see (3.12)) is superclose to the CG solution  .

Preliminaries
Let   be the Legendre polynomial of degree  on [−1, 1].It is well-known that there holds the orthogonality where  , is the Kronecker symbol.Let   be the Lobatto polynomial of degree  on [−1, 1], namely (see, e.g., [11]) It is easy to verify that For our purpose, we also define the shifted Legendre and Lobatto polynomials on   by and for  ∈   , respectively.Combining (3.1)-(3.5),we can obtain the following properties of the shifted Legendre and Lobatto polynomials  , and  , in a straightforward way.Lemma 3.1.For the polynomials  , () and  , (), there hold Let  ℎ be a given partition of [0,  ] with  subintervals {  }  =1 .For any  ∈  1 (  ), we have  ′ ∈  2 (  ).Since the Legendre polynomials form a complete orthogonal basis of the  2 space, by using the Riesz-Fischer Theorem (see Theorem 3 in Chapter 7.3 of [23]), we can expand  ′ ∈  2 (  ) into the Fourier-Legendre series with û, = 2+1 ℎ ∫︀   ′  , d be the Fourier coefficients of  ′ , here the "=" means that the partial sums of the Fourier-Legendre series of the function  ′ converge to  ′ in the sense of the metric in  2 (  ), namely, lim Moreover, due to the generalized Parseval's identity (see Corollary of Theorem 1 in Chapter 7.3 of [23]), the Fourier-Legendre series (3.7) can be integrated term by term over the interval ( −1 , ) ⊂   , which implies that where  0, = ( −1 ),  1, = (  ) and We now define an projector    :  1 (  ) →   (  ) with  ≥ 1 by It is worth noting that the projection     has been frequently used for the superconvergence analysis of the finite element methods and finite volume methods for various partial differential equations; see, e.g., [11,12] and the references therein.In this paper, we shall also use this projection for superconvergence analysis of the CG method for VIDEs.
The constants  > 0 are independent of ℎ  .

Superconvergence at the nodes
In this section, we will prove that the CG method superconverges at the nodes of the time partition  ℎ with respect to the step-size ℎ.

Postprocessed superconvergence analysis of the CG method
In this section, we propose a postprocessing technique for the CG method.Based on the superclose results established in Theorem 3.5, we prove that the postprocessed CG solution  * is superconvergent to the exact solution .

Postprocessed superconvergence results
The main results of this section are stated in the following theorem.
Remark 4.2.According to Lemma 2.1 and Theorem 4.1, we observe that the convergence rates of the  2 -,  1 -and  ∞ -error estimates are improved by one order, namely, and Moreover, using (4.3), the fact  +1, (  ) = 0 and (3.16), we have which implies that the postprocessed CG solution  * keeps the same nodal superconvergence as the CG solution.

Asymptoticlly exact a posteriori error estimators
In this section, we construct several asymptotically exact a posteriori error estimators for the CG method based on the postprocessed superconvergence results.
In view of the definition (4.3) of the postprocessed solution  * , we define the a posteriori error estimators by (5.1) For convenience, we also define the unprocessed and postprocessed  2 -,  1 -and  ∞ -errors (5.2) In the following theorem, we show that the a posteriori error estimators   ,  = 0, 1, ∞, are asymptotically exact as ℎ → 0 under reasonable assumptions.
Remark 5.2.Proving the assumption (5. 3) remains an open issue in our situation, which is beyond the scope of the present paper.However, let us make some comments.First, for one-dimensional elliptic problems, similar estimate as (5.3) has been proved in [2].Second, in order to prove the asymptotic exactness of a posterior error estimators, such assumption has been frequently used in the existing literature (see, e.g., [1] and the reference therein).Third, numerical results show that the convergence rates as predicted by Lemma 2.1 are sharp, i.e.,   = (ℎ +1 ) with  = 0, ∞ and  1 = (ℎ  ), which also imply the lower bounds as stated in (5.3).In that sense, the assumption (5.3) is reasonable.

Numerical experiments
In this section, we present some numerical results to verify the theoretical findings.The test problems are solved by the CG method with uniform degree  on uniform or nonuniform time partitions.Let  be the CG solution and  * be the postprocessed CG solution.We denote by  :=  −  and  * :=  −  * the error functions, and by 'order' the actual rate of convergence of the unprocessed or postprocessed CG method with respect to  , where  is the number of elements in the time partition.Clearly,  ≃ 1/ℎ for the uniform and quasi-uniform time partitions.Throughout, all integrals (including the  2 -and  1 -error norms) are numerically evaluated by the Gauss-Legendre quadrature formula with  + 1-points unless otherwise specified.Let {  }  =0 be the nodes of a given time partition of [0,  ].We calculate the  ∞ -errors by where  , =  −1 + ℎ  /20 with 0 ≤  ≤ 20.
For measuring the efficiency of the a posteriori error estimators, we define the global effectivity indices by where   and   are defined by (5.1) and (5.2), respectively.
Consider the problem (6.1) with  = 1.Following [13], we use the nonuniform time partition of [0,  ] with nodal points given by where () returns a uniformly distributed random number in (0, 1).Obviously, this mesh is generated based on a random perturbation on the uniform mesh.We first consider the maximum nodal errors.In Table 1, we list the maximum nodal errors and convergence rates for  = 1, 2, 3 and 4. Here, we denote by max |(  )| the maximum absolute errors at nodal points {  }  =1 of the time partition.It can be seen that the CG method superconvergences at the nodal points with order 2, which confirms the theoretical result in Theorem 3.3.
We next consider the performance of the postprocessing technique as defined by (4.3).In Tables 2-4, we list the unprocessed and postprocessed  2 -,  1 -,  ∞ -errors and their convergence orders for  = 1, 2, 3 and 4. Clearly, the convergence of the  2 -and  ∞ -errors are improved by one order for  ≥ 2, while the convergence of the  1 -errors is improved by one order for  ≥ 1, which coincides well with Theorem 4.1.We also observe  that the the global effectivity indices  0 and  ∞ always approach 1 for  ≥ 2 and the index  1 approaches 1 for  ≥ 1, which implies that the error estimators are asymptotically exact and confirms the results in Theorem 5.1.Moreover, we find that although the convergence orders of the  2 -and  ∞ -errors for  = 1 are not improved after postprocessing, the errors are slightly improved, and the effectivity indices  0 and  ∞ are around 1.63 and 1.33, respectively, which are within a reasonable range.

Example 2
We consider the nonlinear VIDE (cf.[3]): where  (, ()) =  − ln(1 + ) − 1 (1+) 2 − () and the solution of (6.2) is () = 1 1+ • Consider the problem (6.2) with  = 10.The uniform time partitions which consist of  elements are used for this example.In Table 5, we show the numerical convergence rates of the maximum nodal errors.It can be seen that the convergence rates of order 2 are observed in Table 5, thereby confirming the theoretical result in Theorem 3.3.
In Tables 6-8, we list the unprocessed and postprocessed  2 -,  1 -,  ∞ -errors and their convergence orders, as well as the global effectivity indices for different .We observe the same superconvergence results of the postprocessed CG approximations as those presented in Example 1.Additionally, we note that, the global   effectivity indices   with  = 0, 1, ∞ always approach 1 (except the case of  = 1 for  0 ), which implies that the error estimators are asymptotically exact.
In Table 9, we list the CPU time (in seconds) for obtaining the unprocessed and postprocessed CG approximations, where CPUT and CPUT-P denote the CPU time cost of the unprocessed and postprocessed CG approximations, respectively.It can be seen that, the CPU time cost of the postprocessing process is far less than the cost of the original CG approximation, which is almost negligible.

Example 3
Although the present theory does not apply to weakly singular VIDEs with nonsmooth solutions, we expect good results for the postprocessed CG approximation when (nonuniform) graded meshes are used.Thus, we consider the linear VIDE with weakly singular kernel: We choose the right-hand side  such that the solution of (6.3) is given by () =  3 2  − .Obviously,  ∈  2− (0,  )(with arbitrary  > 0) and the second-order derivative of  is unbounded near  = 0.
Consider the problem (6.3) with  = 1.We use the ℎ-version of the composite Gauss-Legendre quadrature developed in [35] to evaluate the involved weakly singular integrals.Moreover, in order to capture the initial singularity of the solution at  = 0, we use the graded mesh with nodes given by   = (︁ )︁  , 0 ≤  ≤ .
Throughout this example, we set the grading parameter  =  + 1.It is worth pointing out that, the selection of the optimal grading parameter  has been studied for the collocation method [8] and DG method [21] for weakly singular VIDEs.However, in our situation of the CG method, it still needs further investigation.In Tables 10-12, we list the unprocessed and postprocessed  2 -,  1 -,  ∞ -errors and their convergence orders, as well as the global effectivity indices for different .From these tables, we observe the same superconvergence results of the postprocessed CG approximations as those reported in Examples 1 and 2 for smooth solutions.In addition, we see again that the global effectivity indices   with  = 0, 1, ∞ always approach 1 (except the case of  = 1 for  0 and  ∞ ).

Concluding remarks
In this paper, we propose and analyze a simple but efficient postprocessing technique for the CG method of nonlinear VIDEs with smooth kernels.We prove that the postprocessing improves the convergence of the CG method in the  2 -,  1 -and  ∞ -norms by one order for regular solutions.As a result, we construct several asymptotically exact a posteriori error estimators as the step-size approaches zero.Numerical results show that, for weakly singular VIDEs with nonsmooth solutions, after postprocessing the convergence rates of the CG method with graded meshes can be also improved by one order, but this is not covered by our theoretical results.Thus, the superconvergence analysis of the postprocessed CG method for weakly singular VIDEs will be a topic of our future research.
The a posterior error estimators developed in this paper can be used for adaptive implementation of the CG time stepping method for VIDEs, although the efficiency of the local error estimators still need further study.

Table 5 .
Example 2: maximum nodal errors and convergence orders.

Table 9 .
Example 2: CPU time (in seconds) of the unprocessed and postprocessed CG approximations.