1 Introduction

In practical applications one frequently encounters the second kind Volterra integral equations of the form

$$\begin{aligned} x(t) - \int _{0}^{t} k(t,s)x(s)\,ds= f(t), \end{aligned}$$
(1.1)

where the kernel k(., .),  the source function f are given smooth functions and x is the unknown function to be determined. Since the integral equations of the form (1.1) usually cannot be solved explicitly, so it is required to obtain approximate solutions. A computational approach to solve integral equations of the form (1.1) by high-order numerical methods such as projection methods is an essential work in scientific research. Various numerical methods for solving (1.1), such as Galerkin, collocation, Petrov–Galerkin and Nystr\(\ddot{o}\)m methods are well documented in [2,3,4,5,6, 11, 14, 16,17,18]. There have been many attempts at improving the accuracy of numerical solution of the integral equation (1.1) by various projection methods. In [3], H. Brunner discussed piecewise polynomial based collocation and iterated collocation methods and their discrete versions for linear second kind Volterra integral equations and proved that the order of convergence of iterated collocation method is twice that of the collocation method at the knots. In [5], H. Brunner and N. Yan obtained global superconvergence of order \({\mathcal {O}} (h^{r+1})\) for iterated collocation solution using the piecewise polynomial space of degree \(\le (r-1)\) to linear second kind Volterra integral equations, where h is the norm of the partition of [0, 1]. In [6], H. Brunner et al. showed that the \((n-1)\)-fold application of iterated correction technique to the iterated collocation method gives global convergence of order \({\mathcal {O}}(h^n)\). In [18], S. Zhang et al. proved that the convergence rate of the \((n-1)\) step iterated Galerkin approximation is of order \({\mathcal {O}}(h^{n+1})\), which improves the corresponding result in [6]. In [16], Wan et al. discussed spectral Galerkin method for second-kind linear Volterra integral equations and showed that the errors of the spectral approximations decay exponentially, provided that the kernel function and the source function are sufficiently smooth. In [17], Z. Xie et al. proposed spectral and psedo-spectral Jacobi–Galerkin methods to provide a rigorous error analysis in both infinity and weighted norms for second kind Volterra type integral equations. In [10], multi-projection and in [8], discrete multi-projection methods were proposed to solve Fredholm integral equations of second kind and showed that under the same assumptions as that of classical Galerkin and collocation methods, the proposed multi-Galerkin and multi-collocation methods exhibit superconvergence results over iterated Galerkin and iterated collocation methods.

In this paper, we consider the Galerkin method and its iterated version to approximate the solution of Volterra integral equations of the form (1.1) with a smooth kernel using piecewise polynomial basis functions. We prove that the approximate solutions in Galerkin method converge to the exact solution with the order \({\mathcal {O}}(h^{r}),\) whereas the iterated Galerkin solutions converge globally with the order \({\mathcal {O}}(h^{2r})\) in infinity norm. We also discuss the multi-Galerkin and iterated multi-Galerkin methods for the linear Volterra integral equation (1.1). We show that under suitable assumptions on the kernel k(., .),  right hand side function f,  and the solution x, the proposed iterated multi-Galerkin method has order of convergence \({\mathcal {O}}(h^{3r})\) in infinity norm. Thus under the same assumptions as in the Galerkin method, the proposed iterated multi-Galerkin method converges faster than Galerkin and iterated Galerkin methods. Moreover the size of the system of equations that must be solved, remains the same as in Galerkin method. We illustrate our theoretical results by numerical examples.

We organize this paper as follows. In Sect. 2, we apply the Galerkin and iterated Galerkin methods to the equation (1.1) and discuss the convergence results. In Sect. 3, we consider the multi-Galerkin method and its iterated version to obtain superconvergence results. In Sect. 4, numerical results are given to illustrate the theoretical results. Throughout this paper, we assume that c is a generic constant.

2 Galerkin method: linear second kind Volterra intergal equations with smooth kernel

Let \(\mathbb {X}=L^2[0,1]\). Consider the following Volterra integral equation of second kind

$$\begin{aligned} x(t) - \int _{0}^{t} k(t,s)x(s)\,ds= f(t),\quad t\in [0,1], \end{aligned}$$
(2.1)

where the kernel \(k(.,.) \in {{\mathcal {C}}}([0,1] \times [0,1]),\) f are given smooth functions and x is the unknown function to be approximated in the Banach space \(\mathbb {X}\).

Instead of considering the integral interval [0, t] of equation (2.1), first we will transfer the integral interval [0, t] to a fixed interval [0, 1] (see [15, 16]) and then we solve the corresponding integral equation. To do this, we consider a transformation \(s(.,.):([0,1] \times [0,1]) \rightarrow [0,1],\) by taking \(s=t \theta \),  \((t, \theta ) \in ([0,1] \times [0,1]). \) Then the Volterra integral equation (2.1) becomes

$$\begin{aligned} x(t)- \int _{0}^{1} \ell (t,s(t,\theta )) x(s(t,\theta ))\,d\theta =f(t),\quad t\in [0,1], \end{aligned}$$
(2.2)

where

$$\begin{aligned} \ell (t,s(t,\theta )) = t k(t,s(t,\theta )). \end{aligned}$$

Define

$$\begin{aligned} {\mathcal {K}} x(t) = \int _{0}^{1} \ell (t,s(t,\theta )) x(s(t,\theta ))\,d\theta ,\quad x \in \mathbb {X}. \end{aligned}$$
(2.3)

Then the above equation (2.2) can be written as

$$\begin{aligned} x - {\mathcal {K}}x= f. \end{aligned}$$
(2.4)

We assume that 1 is not an eigenvalue of the operator \({\mathcal {K}},\) i.e., the equation (2.4) has a unique solution \(x \in \mathbb {X}.\)

In the following theorem, we show that \({\mathcal {K}}\) is a compact operator on \(\mathbb {X}.\) (see [9]).

Theorem 1

Let \({\mathcal {K}} :L^2[0,1] \rightarrow L^2[0,1]\) be the linear integral operator defined by (2.3) with a kernel \( \ell (.,.) \in {{\mathcal {C}}} ([0,1] \times [0,1])\). Then \({\mathcal {K}}\) is a compact operator on \(L^2[0,1]\).

Proof

Let \(S=\{{\mathcal {K}} x: x \in B \subseteq L^2[0,1]\},\) where B denotes the closed unit ball in \(L^2[0,1].\) In order to prove that \({\mathcal {K}}\) is a compact operator, it is enough to show that S is uniformly bounded and equicontinuous.

For any \(x \in B \subseteq L^2[0,1],\) we have

$$\begin{aligned} |{\mathcal {K}} x(t)|= & {} \left| \int _{0}^{1} \ell (t,s(t,\theta )) x(s(t,\theta ))\,d\theta \right| \\\le & {} \left[ \int _0^1|\ell (t,s(t,\theta ))|^2 d\theta \right] ^{\frac{1}{2}} \left[ \int _0^1|x(s(t,\theta ))|^2 d\theta \right] ^{\frac{1}{2}} \\\le & {} \Vert \ell \Vert _{L^2} \Vert x\Vert _{L^2} \le \Vert \ell \Vert _{\infty } = M, \end{aligned}$$

where \(M= \Vert \ell \Vert _\infty =\sup \limits _{t,s(t,\theta )\in [0,1]} | \ell (t,s(t,\theta ))|<\infty .\)

This implies \(\Vert {\mathcal {K}} x\Vert _\infty =\sup \limits _{t \in [0,1]} |{\mathcal {K}}x(t)| \le M < \infty .\)

Hence

$$\begin{aligned} \Vert {\mathcal {K}}\Vert _\infty =\sup \limits _{x \in B} \Vert {\mathcal {K}} x\Vert _\infty \le M < \infty , \end{aligned}$$
(2.5)

i.e., the set S is uniformly bounded.

Now for any \(t, t' \in [0,1]\) and \(x \in L^2[0,1],\) we consider

$$\begin{aligned}&|{\mathcal {K}} x(t')-{\mathcal {K}} x(t)| \nonumber \\&\quad = \left| \int _{0}^{1} \left[ \ell (t',s(t',\theta )) x(s(t',\theta ))- \ell (t,s(t,\theta )) x(s(t,\theta ))\right] \,d\theta \right| \nonumber \\&\quad = \left| \int _{0}^{1} \left[ \ell (t',s(t',\theta )) x(s(t',\theta ))- \ell (t',s(t',\theta )) x(s(t,\theta ))\right. \right. \nonumber \\&\left. \left. \quad \quad +\, \ell (t',s(t',\theta )) x(s(t,\theta ))- \ell (t,s(t,\theta )) x(s(t,\theta ))\right] \,d\theta \right| \nonumber \\&\quad \le \left| \int _{0}^{1} \ell (t',s(t',\theta )) [x(s(t',\theta ))- x(s(t,\theta ))]\,d\theta \right| \nonumber \\&\quad \quad +\,\left| \int _{0}^{1}[ \ell (t',s(t',\theta )) - \ell (t,s(t,\theta ))] x(s(t,\theta ))]\,d\theta \right| \nonumber \\&\quad \le \Vert \ell \Vert _{L^2} \Vert x(s(t',\theta ))- x(s(t,\theta ))\Vert _{L^2} +\Vert \ell (t',s(t',\theta )) - \ell (t,s(t,\theta ))\Vert _{L^2} \Vert x\Vert _{L^2}\nonumber \\&\quad \le \Vert \ell \Vert _{\infty } \Vert x(s(t',\theta ))- x(s(t,\theta ))\Vert _{L^2} +\Vert \ell (t',s(t',\theta )) \nonumber \\&\quad \quad -\,\ell (t,s(t,\theta ))\Vert _{L^2} \Vert x\Vert _{L^2}. \end{aligned}$$
(2.6)

Let \(\epsilon >0\) be given. Since \({{\mathcal {C}}}[0,1]\) is dense in \(L^2[0,1],\) it follows (see. [13], pp - 71]) that for any \(x \in L^2[0,1],\) there exists \(v \in {{\mathcal {C}}}[0,1]\) such that \(\Vert x(.)- v(.)\Vert _{L^2} <\frac{\epsilon }{3}.\) Also, since \(v \in {{\mathcal {C}}}[0,1] \subseteq L^2[0,1],\) we can find \(\delta >0\) such that \(\Vert v (s(t',\theta )) -v (s(t,\theta ))\Vert _{L^2}\le \Vert v (s(t',\theta )) -v (s(t,\theta ))\Vert _{\infty }<\frac{\epsilon }{3},\) for all \(t', t \in [0,1],\) satisfying \(|t'-t|<\delta \). Using this we get

$$\begin{aligned} \Vert x(s(t',\theta ))-x(s(t,\theta ))\Vert _{L^2}= & {} \Vert x(s(t',\theta ))-v(s(t',\theta ))+v(s(t',\theta ))-v(s(t,\theta ))\nonumber \\&\quad +\,v(s(t,\theta ))-x(s(t,\theta ))\Vert _{L^2} \nonumber \\\le & {} \Vert x(s(t',\theta ))-v(s(t',\theta ))\Vert _{L^2} +\Vert v(s(t',\theta ))\nonumber \\&\quad -\, v(s(t,\theta ))\Vert _{L^2} +\Vert v(s(t,\theta ))-x(s(t,\theta ))\Vert _{L^2} \nonumber \\< & {} \frac{\epsilon }{3}+\frac{\epsilon }{3}+\frac{\epsilon }{3}=\epsilon , \end{aligned}$$
(2.7)

for all \(t, t' \in [0,1]\) satisfying \(|t'-t|<\delta .\)

Since \( \ell (.,.) \in {{\mathcal {C}}}([0,1]\times [0,1]) \subseteq L^2([0,1] \times [0,1]),\) we have \(\Vert \ell (t', s(t',. )) -\ell (t, s(t,. ))\Vert _{L^2} \le \Vert \ell (t',s(t',. )) -\ell (t,s(t,. ))\Vert _{\infty } \rightarrow 0\) uniformly  as \(t'\rightarrow t.\) Using this with the boundedness of \(\Vert \ell \Vert _{\infty }\) and combining the estimates (2.6) and (2.7), we see that

$$\begin{aligned} |{\mathcal {K}} x(t')-{\mathcal {K}} x(t)| \rightarrow 0,~ as~ t'\rightarrow t. \end{aligned}$$
(2.8)

Hence by Arzel–Ascoli theorem, S is a relatively compact set in \(L^2[0,1]\), i.e, \({\mathcal {K}}\) is a compact operator on \(L^2[0,1]\). \(\square \)

Let \({\mathcal {C}}^r[0,1]\) denote the space of r-times continuously differentiable functions and for any \(u\in {\mathcal {C}}^r[0,1]\), denote

$$\begin{aligned} \Vert u\Vert _{r,\infty }= \max \{\Vert u^{(j)}\Vert _\infty : 0\le j\le r\}, \end{aligned}$$

where \(u^{(j)}\) denotes the j-th derivative of u.

Next, we will apply Galerkin method to solve the equation (2.4). To do this, we consider \(\Pi _n:0=t_0< t_1< ... < t_n=1\), a partition of [0, 1] and \(h_i=\{t_{i}-t_{i-1}: 1\le i \le n\}.\) Let \(h=\max _{i} h_i\) denotes the norm of the partition. We assume that \(h \rightarrow 0\), as \(n \rightarrow \infty \). Here we let the approximating subspaces \(\mathbb {X}_n = S^{\nu }_{r,n}(\Pi _n)\), the space of all piecewise polynomials of order r (i.e., of degree \(\le r-1\)) with breakpoints at \(t_1,\cdots ,t_{n-1}\) and with \(\nu \) \((-1\le \nu \le r-2)\) continuous derivatives. Here \(\nu = 0\) corresponds to the case of continuous piecewise polynomials. If \(\nu = -1\), there is no continuity requirements at the breakpoints, in such case \(u_n \in \mathbb {X}_n\) is arbitrarily taken to be left continuous at \(t_1,\ldots ,t_{n}\) and right continuous at \(t_0\).

Orthogonal projection operator: Let the operator \({\mathcal {P}}_n:L^2[0,1] \rightarrow \mathbb {X}_n,\) be the orthogonal projection operator defined by

$$\begin{aligned} \langle {\mathcal {P}}_n u,v\rangle = \langle u, v \rangle ,\quad v\in \mathbb {X}_n, u \in \mathbb {X}, \end{aligned}$$
(2.9)

where \(\langle u, \; v \rangle = \int ^1_{0}u(t) v(t)dt.\)

We first quote the following Lemma from Chatelin ([7], Corollary 7.6, p. 328).

Lemma 1

Let \({\mathcal {P}}_n :\mathbb {X}\rightarrow \mathbb {X}_n\) be the orthogonal projection operator defined by (2.9). Then there hold

  1. (i)

    \({\mathcal {P}}_n \) is uniformly bounded in infinity norm, i.e, \(\exists \) a constant p independent of n such that \(\Vert {\mathcal {P}}_n \Vert _{\infty } \le p < \infty .\)

  2. (ii)

    \(\Vert {\mathcal {P}}_n {u}-u\Vert _{\infty } \rightarrow 0\),  as  \(n\rightarrow \infty ,~ u\in \mathbb {X}\).

  3. (iii)

    In particular, if \(u \in {{\mathcal {C}}}^{r}[0,1],\) then

    $$\begin{aligned} \Vert ({\mathcal {I}}-{\mathcal {P}}_n )u\Vert _{\infty } \le c h^{r}\Vert u\Vert _{r,\infty }, \end{aligned}$$
    (2.10)

    where c is a constant independent of n.

The Galerkin method for solving (2.4) is seeking an approximation \(x_{n} \in \mathbb {X}_n\) such that

$$\begin{aligned} x_{n} - {\mathcal {P}}_{n}{\mathcal {K}} x_{n} = {\mathcal {P}}_{n} f. \end{aligned}$$
(2.11)

In order to obtain more accurate approximate solution, we further define the iterated Galerkin solution by

$$\begin{aligned} \tilde{x}_{n} = {\mathcal {K}} x_{n} + f. \end{aligned}$$
(2.12)

Using \({\mathcal {P}}_n \tilde{x}_{n} = x_{n},\) the equation (2.12) can be written as

$$\begin{aligned} \tilde{x}_n - {\mathcal {K}} {\mathcal {P}}_n \tilde{x}_n = f. \end{aligned}$$
(2.13)

Now we discuss the existence and uniqueness of the Galerkin and iterated Galerkin solutions. To do this, let \(BL(\mathbb {X})\) denote the space of all bounded linear operators on \(\mathbb {X}\) and we recall the definition of \(\nu \)-convergence and a theorem from [1], which are useful in proving existence and convergence of approximated solutions.

Definition 1

(\({\varvec{\nu }}\)-convergence) Let \({\mathcal {T}} \in BL(\mathbb {X})\) and \(\{{\mathcal {T}}_n\}\) be a sequence in \(BL(\mathbb {X})\), then \(\{{\mathcal {T}}_n\}\) is said to be \(\nu \) convergent to \({\mathcal {T}}\) if \(\Vert {\mathcal {T}}_n\Vert \le C, \Vert ({\mathcal {T}}_n-{\mathcal {T}} ){\mathcal {T}} \Vert \rightarrow 0\) and \(\Vert ({\mathcal {T}}_n-{\mathcal {T}} ){\mathcal {T}}_n\Vert \rightarrow 0, ~as ~ n\rightarrow \infty .\)

Theorem 2

Let \(\mathbb {X}\) be a Banach space and \({\mathcal {T}}\), \({\mathcal {T}}_{n}\in BL(\mathbb {X})\). If \({\mathcal {T}}_{n}\) is norm convergent or \(\nu \)-convergent to \({\mathcal {T}}\) and \(({\mathcal {I}}-{\mathcal {T}})^{-1}\) exists and is bounded on \(\mathbb {X}\), then for sufficiently large n, \(({\mathcal {I}}-{\mathcal {T}}_{n})^{-1}\) exists and is uniformly bounded on \(\mathbb {X}\).

Theorem 3

Let \({\mathcal {P}}_n:\mathbb {X}\rightarrow \mathbb {X}_n\) be the orthogonal projection operator defined by (2.9) and let \({\mathcal {K}} :\mathbb {X}\rightarrow \mathbb {X}_n\) be an integral operator defined by (2.3). Then for sufficiently large n, \(\Vert ({\mathcal {I}}- {\mathcal {P}}_{n} {\mathcal {K}})^{-1}\Vert _\infty \) and \(\Vert ({\mathcal {I}}- {\mathcal {K}} {\mathcal {P}}_{n})^{-1}\Vert _{\infty }\) exist and are uniformly bounded in infinity norm i.e., \(\exists \) constants \(L_1, L_2 >0\) such that \(\Vert ({\mathcal {I}}-{\mathcal {P}}_{n} {\mathcal {K}})^{-1}\Vert _{\infty } \le L_1<\infty ,\) and \(\Vert ({\mathcal {I}} -{\mathcal {K}} {\mathcal {P}}_{n})^{-1}\Vert _{\infty }\le L_2<\infty .\)

Proof

Since \({\mathcal {K}}\) is a compact operator on \(\mathbb {X}\) and \({\mathcal {P}}_{n}\) pointwise converges to \({\mathcal {I}}\) on \(\mathbb {X}\), by applying Lemma 1, we have

$$\begin{aligned} \Vert ({\mathcal {I}}- {\mathcal {P}}_{n}){\mathcal {K}}\Vert _{\infty } \rightarrow 0,\quad as~n \rightarrow \infty , \end{aligned}$$
(2.14)

and

$$\begin{aligned} \Vert ({\mathcal {K}} {\mathcal {P}}_n - {\mathcal {K}}) {\mathcal {K}}\Vert _{\infty } =\Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n) {\mathcal {K}}\Vert _{\infty } \le \Vert {\mathcal {K}} \Vert _{\infty }\Vert ({\mathcal {I}}-{\mathcal {P}}_n) {\mathcal {K}}\Vert _{\infty } \rightarrow 0, ~ as~ n \rightarrow \infty . \end{aligned}$$

Since \(\Vert {\mathcal {P}}_n \Vert _{\infty } \le p < \infty ,\) we have

$$\begin{aligned}&\displaystyle \Vert {\mathcal {K}} {\mathcal {P}}_{n}\Vert _{\infty } \le p\Vert {\mathcal {K}} \Vert _{\infty }<\infty ,\\&\displaystyle \quad \Vert ({\mathcal {K}} {\mathcal {P}}_n - {\mathcal {K}}) {\mathcal {K}} {\mathcal {P}}_n\Vert _{\infty } \le p \Vert {\mathcal {K}} \Vert _{\infty }\Vert ({\mathcal {P}}_n - {\mathcal {I}}) {\mathcal {K}}\Vert _{\infty } \rightarrow 0, ~ as~ n \rightarrow \infty . \end{aligned}$$

Thus \(\Vert {\mathcal {P}}_{n}{\mathcal {K}}-{\mathcal {K}}\Vert _{\infty } \rightarrow 0\) and \( {\mathcal {K}} {\mathcal {P}}_n \overset{\nu }{\longrightarrow } {\mathcal {K}}.\)

Now by direct application of Theorem 2, it follows that, for sufficiently large n, \(({\mathcal {I}}- {\mathcal {P}}_{n} {\mathcal {K}})^{-1}\) and \(({\mathcal {I}}- {\mathcal {K}} {\mathcal {P}}_{n})^{-1}\) exist and are uniformly bounded in infinity norm i.e., \(\exists \) constants \(L_1, L_2 >0\) such that \(\Vert ({\mathcal {I}}-{\mathcal {P}}_{n} {\mathcal {K}})^{-1}\Vert _{\infty } \le L_1<\infty ,\) and \(\Vert ({\mathcal {I}} -{\mathcal {K}} {\mathcal {P}}_{n})^{-1}\Vert _{\infty }\le L_2<\infty .\) This completes the proof. \(\square \)

In the following theorem, we give the error bounds for the Galerkin solution \(x_n\) and iterated Galerkin solution \(\tilde{x}_n\).

Lemma 2

Let \({\mathcal {P}}_n: \mathbb {X}\rightarrow \mathbb {X}_n\) be the orthogonal projection operator defined by (2.9). Let \(x_n\) and \(\tilde{x}_n\) be the Galerkin and iterated Galerkin approximate solutions of the equation (2.4), respectively. Then for sufficiently large n, there hold

$$\begin{aligned} \Vert x-x_{n}\Vert _{\infty } \le L_1 \Vert ({\mathcal {I}}-{\mathcal {P}}_n)x\Vert _{\infty }, \end{aligned}$$
(2.15)

and

$$\begin{aligned} \Vert x-\tilde{x}_{n}\Vert _{\infty } \le \ L_2\Vert {\mathcal {K}}({\mathcal {I}}-{\mathcal {P}}_n)x\Vert _{\infty }. \end{aligned}$$
(2.16)

Proof

The proof of the above Lemma follows according to the analysis of ([2, 7]). \(\square \)

Now we discuss the convergence rates for the Galerkin solution \(x_{n}\) and iterated Galerkin solution \(\tilde{x}_{n}.\)

Theorem 4

Let \({\mathcal {P}}_n: \mathbb {X}\rightarrow \mathbb {X}_n\) be the orthogonal projection operator defined by (2.9). Let \(x_n\) and \(\tilde{x}_n\) be the Galerkin and iterated Galerkin approximate solutions of x,  respectively. Assume that \( \ell (.,.) \in {{\mathcal {C}}}^{r}([0, 1] \times [0,1])\) and \(x, f \in {{\mathcal {C}}}^{r}[0,1]\).Then there hold

$$\begin{aligned} \Vert x- x_{n}\Vert _{\infty }= & {} {\mathcal {O}}\left( h^{r}\right) ,\\ \Vert x- \tilde{x}_{n}\Vert _{\infty }= & {} {\mathcal {O}}\left( h^{2r}\right) . \end{aligned}$$

Proof

From estimates (2.10) and (2.15), we obtain

$$\begin{aligned} \Vert x-x_{n}\Vert _{\infty }\le L_1 \Vert ({\mathcal {I}}-{\mathcal {P}}_n)x\Vert _{\infty }={\mathcal {O}}(h^{r}). \end{aligned}$$

Using orthogonality of \({\mathcal {P}}_n \) and estimate (2.10), we have

$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x\Vert _{\infty }= & {} \sup _{t \in [0,1]}|{\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x(t)|\nonumber \\= & {} \sup _{t \in [0,1]}\left| \int _{0}^{1} \ell (t,s(t,\theta ))({\mathcal {I}}-{\mathcal {P}}_n ) x(s(t,\theta ))\,d\theta \right| \nonumber \\= & {} \sup _{t \in [0,1]}\left| \langle \ell (t,s(t,.)),({\mathcal {I}}-{\mathcal {P}}_n )x(s(t,.))\rangle \right| \nonumber \\= & {} \sup _{t \in [0,1]}\left| \langle ({\mathcal {I}}-{\mathcal {P}}_n ) \ell (t,s(t,.)),({\mathcal {I}}-{\mathcal {P}}_n )x(s(t,.))\rangle \right| \nonumber \\= & {} \sup _{t \in [0,1]}\left| \int _{0}^{1}({\mathcal {I}}-{\mathcal {P}}_n ) \ell (t,s(t,\theta ))({\mathcal {I}}-{\mathcal {P}}_n ) x(s(t,\theta ))\,d\theta \right| \nonumber \\\le & {} \Vert ({\mathcal {I}}-{\mathcal {P}}_n ) \ell \Vert _{\infty } \Vert ({\mathcal {I}}-{\mathcal {P}}_n )x\Vert _{\infty } \end{aligned}$$
(2.17)
$$\begin{aligned}\le & {} c \Vert \ell \Vert _{r,\infty } \Vert x\Vert _{r,\infty } h^{2r}. \end{aligned}$$
(2.18)

Combining the estimates (2.16) and (2.18), we have

$$\begin{aligned} \Vert x-\tilde{x}_{n} \Vert _{\infty }={\mathcal {O}}(h^{2r}). \end{aligned}$$

This completes the proof. \(\square \)

3 Superconvergence results by multi-Galerkin method

In this section, we propose multi-Galerkin (M-Galerkin) and iterated multi-Galerkin (iterated M-Galerkin) methods (see [8, 10, 12]) for solving (1.1) and obtain the superconvergence results. To do this, we define the multi-projection operator \({\mathcal {K}}_n^M\) by

$$\begin{aligned} {\mathcal {K}}_{n}^{M} := {\mathcal {P}}_n {\mathcal {K}} {\mathcal {P}}_n + {\mathcal {P}}_n {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n )+({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}} {\mathcal {P}}_n . \end{aligned}$$
(3.1)

Then the M-Galerkin method for solving the equation (2.4) is seeking an approximate solution \(x_n^{M} \in \mathbb {X}\) such that

$$\begin{aligned} x_{n}^{M}- {\mathcal {K}}_{n}^{M} x_{n}^{M} = f. \end{aligned}$$
(3.2)

The iterated M-Galerkin solution is defined by

$$\begin{aligned} \tilde{x}_{n}^{M} = {\mathcal {K}} x_{n}^{M} + f. \end{aligned}$$
(3.3)

To solve the equation (3.2), applying \({\mathcal {P}}_n\) and \(({\mathcal {I}}- {\mathcal {P}}_n)\) to the equation (3.2), gives

$$\begin{aligned} {\mathcal {P}}_n x_{n}^{M}= & {} {\mathcal {P}}_n {\mathcal {K}} {\mathcal {P}}_n x_{n}^{M}+{\mathcal {P}}_n {\mathcal {K}}({\mathcal {I}}- {\mathcal {P}}_n) x_{n}^{M} + {\mathcal {P}}_n f.\end{aligned}$$
(3.4)
$$\begin{aligned} ({\mathcal {I}}-{\mathcal {P}}_n) x_{n}^{M}= & {} ({\mathcal {I}}-{\mathcal {P}}_n) {\mathcal {K}} {\mathcal {P}}_n x_{n}^{M}+ ({\mathcal {I}}-{\mathcal {P}}_n)f. \end{aligned}$$
(3.5)
$$\begin{aligned} \Rightarrow x_{n}^{M}= & {} {\mathcal {P}}_n x_{n}^{M} + ({\mathcal {I}}-{\mathcal {P}}_n) {\mathcal {K}} {\mathcal {P}}_n x_{n}^{M}+ ({\mathcal {I}}-{\mathcal {P}}_n)f. \end{aligned}$$
(3.6)

Substituting (3.6) into (3.4), we obtain

$$\begin{aligned} {\mathcal {P}}_n x_{n}^{M}= & {} {\mathcal {P}}_n {\mathcal {K}} {\mathcal {P}}_n x_{n}^{M}+ {\mathcal {P}}_n {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n) ({\mathcal {P}}_n x_{n}^{M} + ({\mathcal {I}}-{\mathcal {P}}_n) {\mathcal {K}} {\mathcal {P}}_n x_{n}^{M}\nonumber \\&\quad \quad + ({\mathcal {I}}-{\mathcal {P}}_n)f) +{\mathcal {P}}_n f \nonumber \\&\Rightarrow {\mathcal {P}}_n x_{n}^{M}-({\mathcal {P}}_n {\mathcal {K}} +\,{\mathcal {P}}_n {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n){\mathcal {K}}){\mathcal {P}}_n x_{n}^{M} ={\mathcal {P}}_n {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n) f \nonumber \\&\quad \quad +\,{\mathcal {P}}_n f. \end{aligned}$$
(3.7)

Let \({\mathcal {W}}_n^M={\mathcal {P}}_n x_{n}^{M},\) then we can seek \({\mathcal {W}}_n^M \in \mathbb {X}_n\) from the equation

$$\begin{aligned} \left( {\mathcal {I}} - Q_n {\mathcal {K}} \right) {\mathcal {W}}_n^M = Q_n f, \end{aligned}$$
(3.8)

where \(Q_n=({\mathcal {P}}_n +{\mathcal {P}}_n {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n))\), and from equation (3.6), we obtain

$$\begin{aligned} x_n^M ={\mathcal {W}}_n^M + ({\mathcal {I}}-{\mathcal {P}}_n) ({\mathcal {K}} {\mathcal {W}}_n^M+ f). \end{aligned}$$
(3.9)

Note that from equation (3.8) the size of the system of equations to be solved in multi-Galerkin method remains the same as the Galerkin method.

Also note that

$$\begin{aligned} {\mathcal {K}}-{\mathcal {K}}_{n}^{M} = ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ). \end{aligned}$$
(3.10)

Next we discuss the existence and uniqueness of \(x_{n}^{M}\). To do this, we first prove the invertibility of \(\Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}\Vert _{\infty }.\)

Theorem 5

For sufficiently large n, \(\Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}\Vert _\infty \) exists and is uniformly bounded on \(\mathbb {X}\) i.e., \(\exists \) a constant \(L>0\) such that \(\Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}\Vert _{\infty } \le L<\infty \).

Proof

Using Lemma 1 and estimates (2.14) and (3.10), we have

$$\begin{aligned} \Vert {\mathcal {K}}-{\mathcal {K}}_{n}^{M}\Vert _{\infty }= & {} \Vert ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n )\Vert _{\infty } \le \Vert ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}}\Vert _{\infty }~(1+\Vert {\mathcal {P}}_n \Vert _{\infty }) \\\le & {} (1+p)\Vert ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}}\Vert _{\infty } \rightarrow 0,~~ as~~ n \rightarrow \infty . \end{aligned}$$

This implies \({\mathcal {K}}_{n}^{M}\) is norm convergent to \({\mathcal {K}}\). Hence by direct application of Theorem 2, it follows that \(\Vert ({\mathcal {I}}- {\mathcal {K}}_{n}^{M})^{-1}\Vert _\infty \) exists and is uniformly bounded on \(\mathbb {X},\) i.e., for n large enough, \(\exists \) a constant \(L>0,\) such that \(\Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}\Vert _\infty \le L<\infty \). This completes the proof. \(\square \)

In the following lemma, we give some error estimates which we need to obtain the superconvergence rates.

Lemma 3

Let \(x \in {{\mathcal {C}}}^{r}[0,1] \) and \( \ell (.,.) \in {{\mathcal {C}}}^{r}([0, 1] \times [0,1])\), then the following estimates hold

$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) \Vert _{\infty }\le & {} ch^{r} \Vert \ell \Vert _{r,\infty }, \end{aligned}$$
(3.11)
$$\begin{aligned} \Vert ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x)\Vert _{\infty }\le & {} ch^{2r} \Vert \ell \Vert _{r,\infty } \Vert x\Vert _{r,\infty }, \end{aligned}$$
(3.12)
$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x \Vert _{\infty }\le & {} ch^{3r} \Vert \ell \Vert _{r,\infty }^{2} \Vert x\Vert _{r,\infty }, \end{aligned}$$
(3.13)
$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) \Vert _{\infty }\le & {} c h^{2r} \Vert \ell \Vert _{r,\infty }^{2}, \end{aligned}$$
(3.14)

where c is constant independent of n.

Proof

From estimate (2.17) and Lemma 1, it follows that

$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x\Vert _{\infty }\le & {} \Vert ({\mathcal {I}}-{\mathcal {P}}_n ) \ell \Vert _{\infty } \Vert ({\mathcal {I}}-{\mathcal {P}}_n )x\Vert _{\infty }\le ch^{r} \Vert \ell \Vert _{r,\infty }(1+\Vert {\mathcal {P}}_n \Vert _{\infty }) \Vert x\Vert _{\infty }\\\le & {} c(1+ p)h^{r} \Vert \ell \Vert _{r,\infty } \Vert x\Vert _{\infty }. \end{aligned}$$

This implies

$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) \Vert _{\infty } \le ch^{r} \Vert \ell \Vert _{r,\infty }, \end{aligned}$$

where c is a constant independent of n. This completes the proof of (3.11).

Again using Lemma 1 and estimate (2.18), we have

$$\begin{aligned} \Vert ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x\Vert _{\infty }\le & {} (1+p) \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x\Vert _{\infty }\nonumber \\\le & {} c h^{2r} \Vert \ell \Vert _{r,\infty } \Vert x\Vert _{r,\infty }. \end{aligned}$$

This proves (3.12).

Now using the estimates (2.18) and (3.11), we obtain

$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x \Vert _{\infty }\le & {} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n )\Vert _{\infty } \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) x\Vert _{\infty }\nonumber \\\le & {} ch^{3r} \Vert \ell \Vert _{r,\infty }^{2} \Vert x\Vert _{r,\infty }. \end{aligned}$$

This completes the proof of (3.13).

Similarly from estimate (3.11), it follows that

$$\begin{aligned} \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n )\Vert _{\infty }\le & {} \Vert {\mathcal {K}} ({\mathcal {I}}- {\mathcal {P}}_n )\Vert _{\infty } \Vert {\mathcal {K}} ({\mathcal {I}}- {\mathcal {P}}_n )\Vert _{\infty }\nonumber \\\le & {} ch^{2r} \Vert \ell \Vert _{r,\infty }^{2}, \end{aligned}$$

which proves (3.14). This completes the proof. \(\square \)

In the following theorem, we give the superconvergence results in M-Galerkin and iterated M-Galerkin methods.

Theorem 6

Let \({\mathcal {P}}_n : \mathbb {X}\rightarrow \mathbb {X}_n\) be the orthogonal projection operator defined by (2.9). Let \(x_{n}^{M}\) and \(\tilde{x}_{n}^{M}\) be the M-Galerkin and iterated M-Galerkin approximate solutions of the equation (2.4), respectively. Assume \( \ell (.,.) \in {{\mathcal {C}}}^{r}([0, 1] \times [0,1])\) and \(x, f \in {{\mathcal {C}}}^{r}[0,1]\), then there hold

$$\begin{aligned} \Vert x- x_{n}^{M}\Vert _{\infty }= & {} {\mathcal {O}}(h^{2r}), \end{aligned}$$
(3.15)
$$\begin{aligned} \Vert x- \tilde{x}_{n}^{M}\Vert _{\infty }= & {} {\mathcal {O}}(h^{3r}). \end{aligned}$$
(3.16)

Proof

Using Theorem 5, equations (2.4), (3.2), (3.10) and estimate (3.12), we have

$$\begin{aligned} \Vert x-x_{n}^{M}\Vert _{\infty }= & {} \Vert ({\mathcal {I}}-{\mathcal {K}})^{-1}f-({\mathcal {I}}- {\mathcal {K}}_{n}^{M})^{-1}f\Vert _\infty \\= & {} \Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M} )^{-1}({\mathcal {K}}-{\mathcal {K}}_{n}^{M})({\mathcal {I}}-{\mathcal {K}})^{-1}f \Vert _\infty \nonumber \\= & {} \Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}({\mathcal {K}}- {\mathcal {K}}_{n}^{M})x\Vert _\infty \nonumber \\\le & {} \Vert ({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}\Vert _\infty \Vert ({\mathcal {K}}- {\mathcal {K}}_{n}^{M})x\Vert _\infty \nonumber \\\le & {} L \Vert ({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}}({\mathcal {I}}- {\mathcal {P}}_n )x\Vert _{\infty }\nonumber \\\le & {} L c h^{2r} \Vert \ell \Vert _{r,\infty } \Vert x\Vert _{r,\infty }={\mathcal {O}}(h^{2r}), \end{aligned}$$

which proves (3.15).

From the equations (2.4) and (3.3), we have

$$\begin{aligned} x-\tilde{x}_{n}^{M}= & {} {\mathcal {K}} (x-x_{n}^{M})\\= & {} {\mathcal {K}}({\mathcal {I}}-{\mathcal {K}})^{-1}({\mathcal {K}}-{\mathcal {K}}_{n}^{M})({\mathcal {I}}-{\mathcal {K}}_{n}^{M})^{-1}f\\= & {} ({\mathcal {I}}-{\mathcal {K}})^{-1}{\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ) (x+x_{n}^{M}-x)\\= & {} ({\mathcal {I}}-{\mathcal {K}} )^{-1}[{\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}}({\mathcal {I}}-{\mathcal {P}}_n )x - {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}}({\mathcal {I}}-{\mathcal {P}}_n ) (x-x_{n}^{M})]. \end{aligned}$$

Hence using estimates (3.13)–(3.15), we obtain

$$\begin{aligned} \Vert x-\tilde{x}_{n}^{M}\Vert _{\infty }\le & {} \Vert ({\mathcal {I}}-{\mathcal {K}} )^{-1}\Vert _{\infty } \left[ \Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}}({\mathcal {I}}-{\mathcal {P}}_n )x\Vert _{\infty }\right. \\&\left. \quad +\,\Vert {\mathcal {K}} ({\mathcal {I}}-{\mathcal {P}}_n ){\mathcal {K}}({\mathcal {I}}-{\mathcal {P}}_n )\Vert _{\infty } \Vert x-x_{n}^{M}\Vert _{\infty }\right] \\= & {} {\mathcal {O}}(h^{3r}). \end{aligned}$$

This completes the proof. \(\square \)

Remark 1

The above results should be compared with following known bounds in ([18], Theorem 3.1) and in ([4,5,6])

$$\begin{aligned} \Vert x-x_n\Vert _{\infty }= & {} {\mathcal {O}}(h^{r}),\\ \Vert x-\tilde{x}_n\Vert _{\infty }= & {} {\mathcal {O}}\left( h^{r+1}\right) , \end{aligned}$$

where \(x_n\) and \(\tilde{x}_n\) are the approximate and iterated approximate solutions in Galerkin and collocation methods.

Remark 2

From Theorems 4 and 6, we see that the iterated multi-Galerkin method has order of convergence \({\mathcal {O}}(h^{3r}),\) whereas the order of convergence of Galerkin and iterated-Galerkin methods are of the order \({\mathcal {O}}(h^{r})\) and \({\mathcal {O}}(h^{2r}),\) respectively. This shows that the iterated multi-Galerkin method improves over both Galerkin and iterated-Galerkin methods.

4 Numerical results

In this section, we present the numerical results. For that we take the piecewise polynomials as the basis functions for the subspace \(\mathbb {X}_n\). We present the errors of the approximate and iterated approximate solutions under the piecewise polynomial based Galerkin and M-Galerkin methods in infinity norm. We denote the Galerkin, iterated Galerkin, M-Galerkin and iterated M-Galerkin solutions by \(x_n \), \(\tilde{x}_n \), \(x_n^{M}\) and \(\tilde{x}_n^{M},\) respectively. Also we denote \(\Vert x - x_n \Vert _{\infty } = {\mathcal {O}} (h^{\alpha })\), \(\Vert x - \tilde{x}_n \Vert _{\infty } = {\mathcal {O}} (h^{a})\), \(\Vert x - x_n^{M}\Vert _{\infty } = {\mathcal {O}} (h^{\gamma })\), \(\Vert x - \tilde{x}_n^{M}\Vert _{\infty } = {\mathcal {O}} (h^{c})\). The numerical tests were performed on a PC Intel(R) Core (TM) i5-3470 CPU @ 3.20GHz Processor, 4.00GB RAM and 32-bit operating system on Matlab (R2012b).

Table 1 Galerkin and iterated Galerkin methods
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Table 2 M-Galerkin and iterated M-Galerkin methods
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Consider the uniform partition of [0, 1]:

$$\begin{aligned} 0=t_0<t_1< t_2<...< t_{n}=1 \end{aligned}$$

where \(t_i=\frac{i}{n},~i=0,1,2,..., n.\)

We choose the approximating subspaces as the space of piecewise constant functions \((r=1)\), which has dimension n. Then for \(r=1\), the expected orders of convergence are \(\alpha =1\), \(\gamma =2\), \(a=2\) and \(c=3,\) which are calculated in the following Tables [1–4] of Example 1 and Example 2. In Tables 1 and 3, we present the errors in Galerkin and iterated Galerkin methods. The errors in M-Galerkin and iterated M-Galerkin methods are given in Tables 2 and 4.

Example 1

Consider the following Volterra integral equation of second kind

$$\begin{aligned} x(t) =f(t) + \int _{0}^{t} k(t,s)x(s)\,ds,~~ t\in [0,1]. \end{aligned}$$

with the kernel function \(k(t,s)=2 \cos (t-s)\) and the function \(f(t)=\sin t\) and the exact solution is given by \(x(t)=t e^t.\)

The transformed integral equation is

$$\begin{aligned} x(t)- \int _{0}^{1} \ell (t,s(t,\theta )) x(s(t,\theta ))\,d\theta =f(t),\quad t\in [0,1], \end{aligned}$$

where

$$\begin{aligned} \ell (t,s(t,\theta )) =2 t \cos (t-t \theta ),\quad t, \theta \in [0,1]. \end{aligned}$$

Example 2

Consider the following Volterra integral equation of second kind

$$\begin{aligned} x(t) =f(t) + \int _{0}^{t} k(t,s)x(s)\,ds,~~ t\in [0,1], \end{aligned}$$

with the kernel function \(k(t,s)=ts\), \(f(t)=t^5-\frac{t^8}{7};\) and its exact solution is given by \(x(t)=t^5.\)

The transformed integral equation is

$$\begin{aligned} x(t)- \int _{0}^{1} \ell (t,s(t,\theta )) x(s(t,\theta ))\,d\theta =f(t),~ t\in [0,1], \end{aligned}$$

where

$$\begin{aligned} \ell (t,s(t,\theta )) = t^3 \theta , ~~t, \theta \in [0,1]. \end{aligned}$$
Table 3 Galerkin and iterated Galerkin methods
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Table 4 M-Galerkin and iterated M-Galerkin methods
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From Table 1 of Example 1 and Table 3 of Example 2, we see that the iterated approximate solutions give better convergence rates than the approximate solutions in Galerkin method. From Table 2 of Example 1 and Table 4 of Example 2, we also see that the iterated M-Galerkin method gives better convergence rates than standard Galerkin and iterated Galerkin methods. Note that the size of the system of equations that must be solved, remains the same as in Galerkin method.