Abstract
It is well known that for the second-kind Volterra integral equations (VIEs) with weakly singular kernel, if we use piecewise polynomial collocation methods of degree m to solve it numerically, due to the weak singularity of the solution at the initial time \(t = 0\), only \(1 - \alpha \) global convergence order can be obtained on uniform meshes, comparing with m global convergence order for VIEs with smooth kernel. However, in this paper, we will see that at mesh points, the convergence order can be improved, and it is better and better as n increasing. In particular, 1 order can be recovered for \(m = 1\) at the endpoint. Some superconvergence results are obtained for iterated collocation methods, and a representative numerical example is presented to illustrate the obtained theoretical results.
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1 Introducation
We consider the following second-kind Volterra integral equation (VIE) with weakly singular kernel:
where g and K are continuous functions on their respective domains, and \(K(t, t) \ne 0\) for \(t \in I\). In [1], it is shown that on uniform meshes, the convergence order of piecewise polynomial collocation methods is only \(1 - \alpha \). In order to improve the convergence order, graded meshes are employed to overcome the lower regularity at the initial time \(t = 0\). However, in [2], it is said that “the commonly used graded meshes may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors”, and in order to avoid this problem, a kind of hybrid collocation methods is presented, but the original singularity has to be considered for carefully designing the mesh.
In this paper, at the mesh point \(t_n\), a fine error estimation with order \(t_n^{ - \alpha } h^{2 - \alpha } + t_n^{1 - m - \alpha } h^m\) for piecewise polynomial collocation methods on uniform meshes is obtained, where m is the degree of the piecewise polynomial. In particular, at the endpoint, the convergence order is \(\min \{ 2 - \alpha , m \}\); for \(m = 1\) and \(\alpha \le 0.5\), at the collocation point, the convergence order is always 1, which is not affected by the initial singularity. In order to improve the convergence order, the general iterated collocation methods are presented for \(m = 1\), and it is shown that for the k-th iterated collocation method, the convergence order is \(t_n^{k - 1 - k \alpha }h^{2 - \alpha }\) at the mesh point \(t_n\).
The outline of this paper is as follows. In Sect. 2, the classical piecewise polynomial collocation method on uniform meshes is recalled. In Sect. 3, fine error estimations at mesh points for VIEs with \(m = 1\) and \(K(t, s) \equiv 1\) are investigated, and the error estimations for \(m \ge 2\) and general kernels are given in Sect. 4. The iterated collocation methods and the convergence are analyzed in Sect. 5. A typical numerical example is given to illustrate the obtained theoretical results in Sect. 6.
2 Collocation Methods on Uniform Meshes
Let \(N \ge 2\) be a positive integer, and \(I_h := \left\{ t_{n} := n h: \; n = 0,1, \ldots , N \;\, \left( t_{N} := T \right) \right\} \) be a given mesh on \(I = [0, T]\), with \(\sigma _n := \left( t_n, t_{n + 1} \right] \) and mesh diameter \(h := T/N\).
We seek a collocation solution \(u_h\) for (1) in the piecewise polynomial collocation space
where \(\pi _m\) denotes the space of all (real) polynomials of degree not exceeding m. For a prescribed set of collocation points
\(u_h\) is defined by the collocation equation
In [1], it is shown that the collocation solution \(u_h\) converges to the exact solution u, with order \(1 - \alpha \), i.e.,
In this paper, we will show that at the mesh points, especially at the endpoint, a better convergence result can be expected.
The following lemma, coming from [1, Lemma 6.2.10], is useful.
Lemma 1
Let \(I_h\) be a uniform mesh on \(I = [0, T]\). If \(\{ c_i \}\) satisfy \(0 \le c_1< \cdots < c_m \le 1\), then, for \(0 \le l < n \le N - 1\) and \(\nu \in \mathbb {N}_0\),
where \(\gamma (\alpha ) := \frac{2^{\alpha }}{1-\alpha }\).
3 Fine Error Estimations for \(m = 1\) and Constant Kernels at Mesh Points
In order to obtain the first insight, in this section, we assume that \(m = 1\) and \(K(t, s) \equiv 1\).
Let \(e_h := u - u_h\). On the first mesh interval \([ t_0, t_1 ] = [ 0, h ]\), by [1, Theorem 6.2.9], we know that there exists a constant \(C_1\), which is independent of h and N, such that
For \(1 \le n \le N - 1\), the collocation error on \(\left( t_{n}, t_{n + 1} \right] \) has the local Lagrange representation
where \(\varepsilon _{n, 1} := e_h( t_{n, 1} ), t_{n, 1} := t_n + c_1 h\) and
By [2] (see also [1, Theorem 6.1.6]), there exists a constant \(C_2\), such that
i.e.,
where
and
defined as in [5].
Therefore, for \(1 \le n \le N - 1\), (8) can be written as
Let \(\varvec{\varepsilon }_{n} := \left( \begin{array}{c} \varepsilon _{1, 1} \\ \varepsilon _{2, 1} \\ \varepsilon _{3, 1} \\ \vdots \\ \varepsilon _{n, 1} \\ \end{array}\right) \) and \(\mathbf {r}_{n}(\alpha ):=\left( \begin{array}{c} r_{1}(\alpha ) \\ r_{2}(\alpha ) \\ r_{3}(\alpha ) \\ \vdots \\ r_{n}(\alpha ) \\ \end{array}\right) \). Then
where \(\mathbf {I}_{n}\) denotes the identity in \(L(\mathbb {R}^{n})\) and \(\mathbf {T}_{n}\) is the lower triangular Toeplitz matrix (see [5]).
It is easy to prove the following lemma.
Lemma 2
Let \(r \in \mathbb {N}\), and
be a lower triangular matrix with \(a_{i, i} \ne 0 \ (i = 1, 2, \ldots , r)\). Then \(\mathbf {A}\) is invertible, and the inverse matrix \(\mathbf {A}^{- 1}\) is also a lower triangular matrix, with the elements
Denote the inverse of the matrix \(\mathbf {I}_{n} - h^{1 - \alpha } \mathbf {T}_{n}\) as \(\mathbf {B}_{n}\) with the element \(b_{i, j}\), and \(\bar{a}_{0} = \bar{a}_{0}(c_1; \alpha ) := 1 - h^{1 - \alpha } a_{0}(c_1; \alpha )\). By Lemma 2, we easily obtain the following corollary.
Corollary 1
For \(1 \le n \le N - 1\), the matrix \(\mathbf {I}_{n} - h^{1 - \alpha } \mathbf {T}_{n}\) is invertible for sufficiently small h, and \(\mathbf {B}_{n} = \left( \mathbf {I}_{n} - h^{1 - \alpha } \mathbf {T}_{n} \right) ^{- 1}\) is also a lower triangular matrix, with the elements
Lemma 3
For \(1\le n\le N\),
and
Proof
The first part follows from [4, Lemma 5.6], and the second part follows from [3, Lemma 6.1]. \(\square \)
Lemma 4
For \(1 \le n \le N - 1, 1 \le i \le n, 1 \le k \le n - i\), \(b_{i + k, k}\) has the same value as \(b_{i + 1, 1}\), which is independent of k, i.e.,
In addition, there exists a constant \(C_3\), which is independent of h and N, such that
Proof
We use the argument of the mathematical induction. First, for \(i = 1, 1 \le k \le n - i\),
so the values of \(b_{1 + k, k}\) are same, i.e., \(b_{1 + k, k} = b_{2, 1}\).
We assume that for \(1 \le i \le n - 1, 1 \le k \le n - i\) the values of \(b_{i + k, k}\) are same, which implies \(b_{i + k, k} = b_{i + 1, 1}\). Then by Corollary 1,
which is independent of k with \(1 \le k \le n - (i + 1)\), i.e., the values of \(b_{i + 1 + k, k}\) are same, and \(b_{i + 1 + k, k} = b_{i + 2, 1}\). The proof of the first part is complete.
In addition, for sufficiently small h, there exists a constant \(D_0\), which is independent of h and N, such that
So by Corollary 1 and Lemma 1, we have,
By the discrete Gronwall inequality (see [1, Theorem 6.1.19]), we know that there exists a constant \(C_3\), which is independent of h and N, such that
\(\square \)
Theorem 1
Assume that \(g \in C^{1}(I), K \in C^{1}(D)\). Let u and \(u_h \in S_{0}^{(- 1)}(I_h)\) be the exact solution and the collocation solution defined by the collocation Eq. (3), respectively, for the second-kind Volterra integral Eq. (1). Then for sufficiently small h,
where C is a constant independent of h and N.
In particular, there exist constants \(\hat{C}\) and \(\bar{C}\), independent of h and N, such that at the collocation points,
and at the endpoint,
Proof
First, by (5), (7), (9), Lemmas 1 and 3, there exists a constant \(C_4\), such that
Next, by (10), Lemmas 3 and 4, we have
In particular, by (6) and (7), there exists a constant C, such that for \(1 \le n \le N - 1\),
Further, at \(t = t_N = T\), for \(N \ge 2\),
\(\square \)
Corollary 2
If \(\alpha \le 0.5\), the order of the error at the collocation points is always 1; i.e.
4 Fine Error Estimations for \(m \ge 1\) and General Kernels at Mesh Points
Let \(e_h := u - u_h\). On the first mesh interval \([t_0, t_1] = [0, h]\), by [1, Theorem 6.2.9], we know that there exists a constant \(M_1\), such that
For \(1 \le n \le N - 1\), the collocation error on \(\left( t_{n}, t_{n + 1} \right] \) has the local Lagrange representation
where \(\varepsilon _{n, j} := e_h(t_{n, j})\), \(t_{n, j} := t_n + c_j h\) and
By [2] (see also [1, Theorem 6.1.6]), there exists a constant \(M_2\), such that
where
For \(1 \le n \le N - 1\) and \(1 \le l \le n - 1\), denote
Then
and
Denote
Then the coefficient matrix can be written as \(\mathbf {I}_{m n} - h^{1 - \alpha } \bar{\mathbf {T}}_{m n}\).
It is easy to prove the following lemma.
Lemma 5
Let \(r \in \mathbb {N}\) and \(\mathbf {D}_{p, q} \ (1 \le q \le p \le r)\) be square matrices, and
be a lower triangular block matrix with invertible \(\mathbf {D}_{p, p} \ (i = 1, 2, \ldots , r)\). Then \(\mathbf {D}\) is invertible, and the inverse matrix \(\mathbf {D}^{- 1}\) is also a lower triangular block matrix, with the elements
Denote the inverse of the matrix \(\mathbf {I}_{m n} - h^{1 - \alpha } \bar{\mathbf {T}}_{m n}\) as \(\bar{\mathbf {B}}_{m n}\) with the element \(\mathbf {B}_{i, j}\), and \(\bar{\mathbf {A}}_{i, i} = \bar{\mathbf {A}}_{i, i}(\alpha ) := \mathbf {I}_m - h^{1-\alpha } \mathbf {A}_{i, i}\). By Lemma 5, we easily obtain the following corollary.
Corollary 3
The matrix \(\mathbf {I}_{m n} - h^{1 - \alpha } \bar{\mathbf {T}}_{m n}\) is invertible for sufficiently small h, and the inverse matrix \(\bar{\mathbf {B}}_{m n} = \left( I_{m n} - h^{1 - \alpha } \bar{\mathbf {T}}_{m n} \right) ^{- 1}\) is also a lower triangular block matrix, with the elements
For \(1 \le n \le N - 1, 1 \le p \le n, 1 \le k \le n - p\), it is easy to see that for non-constant kernel K(t, s), the values of \(\mathbf {B}_{p + k, k}\) are usually different, which is different from the constant kernel case (see Lemma 4). But the estimation for \(\mathbf {B}_{p + k, k}\) still holds, which is described in the following lemma.
Lemma 6
Assume that \(K \in C(D)\), where \(D := \left\{ (t, s): 0 \le s \le t \le T \right\} \). Then for \(1 \le n \le N - 1, 1 \le p \le n, 1 \le k \le n - p\), there exists a constant \(M_3\), which is independent of h and N, such that
Proof
Denote \(\bar{K} := \max \limits _{(t, s) \in D} \left| K(t, s) \right| \) and \(\bar{L} := \max \limits _{1 \le j \le m, s \in [0,1]} \left| L_j(s) \right| \). Then by Lemma 1, we know that
For sufficiently small h, \(\bar{\mathbf {A}}_{p, p}^{- 1}\) is uniformly bounded, which implies that there exists a constant \(\bar{D}_0\), which is independent of h and N, such that
So by Corollary 3 and Lemma 1, we have
By the discrete Gronwall inequality (see [1, Theorem 6.1.19]), we know that there exists a constant \(M_3\), which is independent of h and N, such that
Lemma 7
For \(1\le n\le N\), \(m \ge 2\) and \(0< \alpha < 1\),
where \(\bar{\gamma }(\alpha ) := 2 ^{\alpha } \left( 1 +\frac{1}{m - 2 + \alpha }\right) + \frac{2^{m - 2 (1 - \alpha )}}{1 - \alpha }\).
Proof
By
and together with Lemma 3, we obtain
\(\square \)
Theorem 2
Assume that \(g \in C^{m}(I), K \in C^{m}(D)\), and \(u_h \in S_{m - 1}^{(- 1)}(I_h)\) is the collocation solution for the second-kind Volterra integral Eq. (1) defined by the collocation Eq. (3). Then for sufficiently small h,
where M is a constant independent of h and N.
In particular, there exist constants \(\hat{M}\) and \(\bar{M}\), independent of h and N, such that at the collocation points,
and at the endpoint,
Proof
We divide into the following two cases.
Case I:\(m = 1\).
First, by (12), (14), (15), Lemmas 1 and 3, there exists a constant \(\hat{M}_4\), which is independent of h and N, such that
Similar to the case of \(m = 1\) and constant kernels in Sect. 3, it is easy to obtain that there exist constants \(\hat{M}_5\) and \(\hat{M}_6\), such that
and
In particular, at \(t = t_N = T\), for \(N \ge 2\),
which completes the proof.
Case II:\(m > 1\).
First, by (12), (14), (15), Lemmas 1 and 7, there exists a constant \(M_4\), which is independent of h and N, such that
Next, by (17), Lemmas 3 and 6, we have
By (13) and (14), there exists a constant \(M_6\), such that
In particular,
which completes the proof. \(\square \)
Corollary 4
For the general kernel, if \(m = 1\) and \(\alpha \le 0.5\), the order of the error at the collocation points is always 1; i.e.
5 Iterated Collocation Methods for \(m = 1\)
In the following, we investigate the iterated collocation methods for \(m = 1\) to obtain some further superconvergence results.
5.1 The First Iterated Collocation Method
Let
be the first iterated collocation method. It is obvious that
Let
with \(\delta _h(t) = 0\) whenever \(t \in X_h\). Then
At \(t = t_n + v h\), by Lemmas 1, 3, and Theorem 2, there exists a constant \(\tilde{E}_0\), such that
By (13) and (14), for \(1 \le n \le N - 1\), and \(t \in \left( t_{n}, t_{n + 1} \right] \), there exists a constant \(E_1\), such that
Similarly, there exists a constant \(E_2\), such that
In addition,
therefore,
and by Lemmas 1, 3 and 7, there exist constants \(\tilde{E}_1\) and \(\tilde{E}_2\), such that
and
where \(\bar{K}_j := \max \limits _{0 \le s \le t \le T} \sum \limits _{i = 0}^j \left| \frac{\partial ^j K(t, s)}{\partial t^i \partial s^{j - i}}\right| \ (j \in \mathbb {N})\).
Denote \(e^{it, 1}_h := u - u^{it, 1}_h\). Then by [1, Theorem 6.1.2],
where \(R_{\alpha } (t, s) := \left( t - s \right) ^{- \alpha } Q(t, s; \alpha )\), \(Q(t, s; \alpha ) := \sum \limits _{n = 1} ^{\infty } \left( t - s \right) ^{(n - 1) (1 - \alpha )} \Phi _n (t, s; \alpha )\), and the functions \(\Phi _n\) are defined recursively by
\(( n \ge 2)\), with \(\Phi _1 (t, s; \alpha ) := K(t, s)\) and \(\Phi _n (\cdot , \cdot ; \alpha ) \in C(D)\).
Therefore, at the first interval \([0, t_1]\), there exists a constant \(E_3\), such that
where \(\bar{Q} := \max \limits _{0 \le s \le t \le T, 0< \alpha < 1} \left| Q(t, s; \alpha ) \right| \).
For \( 1 \le n \le N - 1\),
Since
where \(\xi _l \in (0, 1)\), so if the orthogonality condition \(\int _0^1 (s - c_1) \ d s =0\) holds, by the proof of [1, Theorem 6.2.13], there exists a constant \(C_1^{it}\), such that
Therefore, we have proved the following theorem.
Theorem 3
Assume that \(g \in C^{2}(I), K \in C^{4}(D)\), and \(u_h \in S_{0}^{(- 1)}(I_h)\) is the collocation solution for the second-kind Volterra integral Eq. (1) defined by the collocation Eq. (3), with the corresponding first iterated collocation solution \(u_h^{it, 1}\). The collocation parameter satisfies
Then for sufficiently small h,
where \(C_1^{it}\) is a constant independent of h and N.
In particular, there exists a constant \(\bar{C}_1^{it}\), which is independent of h and N, such that
5.2 The Second Iterated Collocation Method
Let
be the second iterated collocation method.
Denote \(e^{it, 2}_h := u - u^{it, 2}_h\). Then
where \(\tilde{Q}(t, s; \alpha ) := \int _0^1 \left( 1 - x \right) ^{- \alpha } x^{- \alpha } K(t, s + x (t - s)) Q(s + x (t - s), s; \alpha )\ d x\).
Therefore, at the first interval \([ 0, t_1 ]\), there exists a constant \(\hat{E}_3\), such that
For \( 1 \le n \le N - 1\),
since
where \(\xi '_l \in (0, 1)\), so if the orthogonality condition \(\int _0^1 (s - c_1) \ d s =0\) holds, by the proof of [1, Theorem 6.2.13], there exists a constant \(C^{it}_2\), such that
Therefore, we have proved the following theorem.
Theorem 4
Assume that \(g \in C^{2}(I), K \in C^{4}(D)\), and \(u_h \in S_{0}^{(- 1)}(I_h)\) is the collocation solution for the second-kind Volterra integral Eq. (1) defined by the collocation Eq. (3), with the corresponding second iterated collocation solution \(u_h^{it2}\). The collocation parameter satisfies
Then for sufficiently small h,
where \(C^{it}_2\) is a constant independent of h and N.
In particular, there exists a constant \(\bar{C}^{it}_2\), which is independent of h and N, such that
Corollary 5
If \(\alpha \le 0.5\), the order of the error for the second iterated collocation solution is always \(2 - \alpha \); i.e.
5.3 The k-th Iterated Collocation Method
Let
be the k-th iterated collocation method.
Similarly, we have the following theorem.
Theorem 5
Assume that \(g \in C^{2}(I), K \in C^{4}(D)\), and \(u_h \in S_{0}^{( - 1)}(I_h)\) is the collocation solution for the second-kind Volterra integral Eq. (1) defined by the collocation Eq. (3), with the corresponding k-th iterated collocation solution \(u_h^{it, k}\). The collocation parameter satisfies
Then for sufficiently small h,
where \(C^{it}_k\) is a constant independent of h and N.
In particular, there exists a constant \(\bar{C}^{it}_k\), which is independent of h and N, such that
Corollary 6
If \(\alpha \le \frac{k - 1}{k}\), the order of the error for the k-iterated collocation solution is always \(2 - \alpha \); i.e.
6 Numerical Results
Example 1
In (1) let \(T = 1\), \(K(t, s) = \frac{1}{10 \Gamma (1 - \alpha )}\) and \(g(t) = 1\) such that the exact solution \(u(t) = E_{1 - \alpha , 1}(\frac{t^{1 - \alpha }}{10})\), where the Mittag-Leffler function \(E_{\mu , \theta }\) is defined by
In Tables 1, 2, 3, 4, 5, 6 and 7, we take \(m = 1\) for \(\alpha = 0.3, 0.5, 0.7\), respectively. From these tables, we observe that the numerical results agree with our theoretical analysis.
At the mesh points, in Tables 1, 3 and 6, we observe that the order is \(\min \{ 2 ( 1 - \alpha ), 1\}\) for \(c_1 = 1\). The reason is that for this case, the mesh point \(t_n = t_{n - 1} + c_1 h\) is also a collocation point. In Tables 8 and 10, the similar phenomena appear for Rauda IIA , \((\frac{1}{2}, 1)\) for \(m = 2\), and Rauda IIA , \((\frac{1}{3}, \frac{1}{2}, 1)\) for \(m = 3\). At collocation points, in Table 5, we observe that the order for \(\alpha = 0.5\) and \(m = 1\) is 1.
In Tables 8, 9, 10 and11, we take \(\alpha = 0.5\) and \(m=2, 3\), respectively. From these tables, we observe that the numerical results also agree with our theoretical analysis.
In Tables 12, 13, 14, 15, 16 and 17, we take \(m = 1, c_1 = 0.5\) and \(\alpha = 0.3, 0.5, 0.7\), respectively, for the first, second and third iterated collocation methods. From these tables, we see that the numerical results are again consistent with our theoretical analysis.
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The work of the author Hui Liang was supported by the National Nature Science Foundation of China (No. 11771128, 11101130), Fundamental Research Project of Shenzhen (JCYJ20190806143201649), Project (HIT.NSRIF.2020056) Supported by Natural Scientific Research Innovation Foundation in Harbin Institute of Technology, and Research start-up fund Foundation in Harbin Institute of Technology (20190019). The work of the author Hermann Brunner was supported by the Hong Kong Research Grants Council GRF Grants HKBU 200113 and 12300014.
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Liang, H., Brunner, H. The Fine Error Estimation of Collocation Methods on Uniform Meshes for Weakly Singular Volterra Integral Equations. J Sci Comput 84, 12 (2020). https://doi.org/10.1007/s10915-020-01266-1
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DOI: https://doi.org/10.1007/s10915-020-01266-1
Keywords
- Volterra integral equations
- Weakly singular kernels
- Collocation methods
- Uniform meshes
- Convergence
- Mesh points
- Endpoint