Discrete Modiﬁed Projection Methods for Urysohn Integral Equations with Green’s Function Type Kernels

. In the present paper we consider discrete versions of the modiﬁed projection methods for solving a Urysohn integral equation with a kernel of the type of Green’s function. For r ≥ 0 , a space of piecewise polynomials of degree ≤ r with respect to an uniform partition is chosen to be the approximating space. We deﬁne a discrete orthogonal projection onto this space and replace the Urysohn integral operator by a Nystr¨om approximation. The order of convergence which we obtain for the discrete version indicates the choice of numerical quadrature which preserves the orders of convergence in the continuous modiﬁed projection methods. Numerical results are given for a speciﬁc example.


Introduction
Let X = L ∞ [0, 1] and consider the following nonlinear Urysohn integral equation and assume that it has a unique solution ϕ. We are interested in approximate solutions of the above equation. For r ≥ 0, let X n be a space of piecewise polynomials of degree ≤ r with respect to a uniform partition of [0, 1] with n subintervals each of length h = 1 n . Let π n be the restriction to L ∞ [0, 1] of the orthogonal projection from L 2 [0, 1] to X n . Then in the classical Galerkin method, (1.2) is approximated by ϕ G n − π n K(ϕ G n ) = π n f.
The iterated modified projection solution is defined as The following orders of convergence are proved in Grammont et al [7]: If r = 0, then whereas if r ≥ 1, then . (1.4) In practice, it is necessary to replace the integral in the definition of K by a numerical quadrature formula. Also, the orthogonal projection π n needs to be replaced by a discrete orthogonal projection Q n . This gives rise to discrete versions of the above methods. It is of interest to choose the quadrature formula appropriately so as to preserve the above orders of convergence. Our aim is to investigate the discrete versions of the modified projection and of the iterated modified projection methods.
The discrete versions of the Galerkin and the iterated Galerkin methods are considered in Atkinson-Potra [4]. They propose a numerical quadrature formula which takes into consideration the fact that the kernel κ(s, t, u) lacks smoothness when s = t and obtain the order of convergence of the discrete iterated Galerkin solution.
We follow a different approach. We choose a uniform partition with m = np, p ∈ N, subintervals. A composite quadrature formula associated with this fine partition is then used to replace the integrals in the definition of K and in the definition of the inner product. Leth = 1 m . Let z M n andz M n denote respectively the discrete modified projection solution and the discrete iterated modified projection solution. We prove the following orders of convergence: If r = 0, then Thus, if r = 0 andh 2 ≤ h 4 , that is, m ≥ n 2 , then the orders of convergence in (1.3) are preserved. If r ≥ 1 andh 2 ≤ h r+5 , then the orders of convergence in (1.4) are preserved. Note that the termh 2 in the above estimates appear because of the discretization. If the kernel is smooth, then it is possible to choose a composite quadrature formula associated with the coarse partition with n subintervals and with a precision d. Then the termh 2 is replaced by h d and an appropriate choice of d will preserve the orders of convergence in (1.3) and (1.4). However, in the case of the kernel of the type of Green's function, the error in the higher order quadrature rules also is only of the order of h 2 . Hence we need to choose a different partition for the quadrature rule which makes the proofs more involved. It is to be noted that even if m > n, the size of the system of equations that need to be solved in order to compute z M n remains n(r + 1). Note that in Grammont et al [7], the orders of convergence (1.3) and (1.4) for the (continuous) modified projection and the iterated modified projection are proved. However, the numerical results are given for the discrete versions of the modified projection and of the iterated modified projection methods. In the present paper we fill this gap and justify the numerical results of Grammont et al [7].
The paper has been arranged in the following way. In Section 2, we define a discrete orthogonal projection operator and discrete versions of the modified projection methods. In Section 3,, we consider the case of a piecewise polynomial space of degree r ≥ 1 and prove (1.6). Section 4, is devoted to the proof of (1.5) in the case of piecewise constant functions. Numerical results for illustrative purpose are given in Section 5.

Discrete modified projection method
In this section we describe the Nyström approximation of K and the discrete orthogonal projection. We then define discrete analogues of the modified pro-jection method and its iterated version.

Kernel of the type of Green's function
Let r ≥ 0 be an integer and assume that the kernel κ of the integral operator K appearing in (1.1) has the following properties.

Nyström approximation
Let m ∈ N and consider the following uniform partition of [0, 1] : where the weights w q > 0 and the nodes µ q ∈ [0, 1]. It is assumed that the quadrature rule is exact at least for polynomials of degree ≤ 2r.
A composite integration rule with respect to the partition (2.1) is then defined as We replace the integral in (1.1) by the numerical quadrature formula (2.2) and define the Nyström operator as Note that K m is twice Fréchet differentiable and Then for x, y ∈ B(ϕ, δ 0 ), denotes the operator norm.
Since the kernel lacks smoothness along s = t, we only have the following order of convergence from Atkinson-Potra [4]: If x ∈ C 2 [0, 1], then (2.5) In the Nyström method, (1.2) is approximated by For all m big enough, the above equation has a unique solution ϕ m in B(ϕ, δ 0 ) and See Atkinson [1]. We quote the following result from Krasnoselskii et al [9] for future reference: , then by the generalized Taylor's theorem, where It then follows that
Then Q n,j x, y ∆j = x, Q n,j y ∆j , Q 2 n,j = Q n,j and Q n,j Q n,i = 0 for i = j. Also, A discrete orthogonal projection Q n : C[0, 1] → X n is defined as follows: (2.12) Using the Hahn-Banach extension theorem, as in Atkinson et al [2], Q n can be extended to L ∞ [0, 1]. Then The following estimate is standard: if x ∈ C r+1 (∆ j ), then we have,

Discrete projection methods
We define below the discrete versions of various projection methods given in Section 1 by replacing the integral operator K by the Nyström operator K m and the orthogonal projection π n by the discrete orthogonal projection Q n .
Discrete Modified Projection method: Discrete Iterated Modified Projection method: 3 Piecewise polynomial approximation: r ≥ 1 In this section we consider the case r ≥ 1 and obtain orders of convergence in the discrete modified projection method and its iterated version.
Since v ∈ C r+1 [0, 1], it follows from (2.14), For j = i, * ,s ∈ C r+1 (∆ j ) and v ∈ C r+1 (∆ j ). Hence We now consider the case j = i. Note that * ,s is only continuous on [t i−1 , t i ]. Define a constant function: g i (t) = * ,s (s), t ∈ [t i−1 , t i ]. Note that Thus, Combining the above estimate with (3.2) we obtain the required result.
Proof. The proof of (3.6) is similar to that of (3.1). For s ∈ [0, 1], we write If s = t i , for some i, then using (2.14) and (3.1) we obtain If s ∈ (t i−1 , t i ), then we write Proceeding as in the proof of Proposition 1, we obtain The estimate (3.6) follows from the above two estimates. In order to prove (3.7), consider v ∈ C[0, 1]. Let s = t i for some i. Then Combining (3.8) and the above estimate, we obtain and the required result follows taking the supremum over unit ball in C[0, 1].

Error in the discrete modified projection method
As in Grammont [5], it can be shown that there is a δ 0 > 0 such that (2.16) has a unique solution z M n in B(ϕ, δ 0 ) and that In the following theorem, we obtain the order of convergence of the discrete modified projection solution.

Error in the discrete iterated modified projection method
From (2.6) and Theorem 1, we obtain We quote the following result from Kulkarni-Rakshit [11]: We obtain below orders of convergence for the three terms in (3.15).

Proposition 3. Let ϕ m be the Nyström solution. Then
Proof.
Proposition 4. Let ϕ m be the Nyström solution and z M n be the discrete modified projection solution. Then Proof. Note that for m and n big enough, ϕ m , z M n ∈ B (ϕ, δ 0 ) . By the generalized Taylor's theorem, It can be shown that We skip the details. The required result then follows from Theorem 1.
Proposition 5. Let ϕ m be the Nyström solution and z M n be the discrete modified projection solution. Then Proof. Note that

By (2.4) and (3.18),
Since by (2.3), K m (ϕ m ) ≤ C 1 , it follows that The required result follows using the estimate for z M n − ϕ ∞ from Theorem 1.
We now prove our main result about the order of convergence in the discrete iterated modified projection method.
Theorem 2. Let r ≥ 1, κ be of class G 2 (r + 1, 0) and f ∈ C r+1 [0, 1]. Let ϕ be the unique solution of (1.2) and assume that 1 is not an eigenvalue of K (ϕ). Let X n be the space of piecewise polynomials of degree ≤ r with respect to the partition (2.9) and Q n be the discrete orthogonal projection defined by (2.12). Letz M n be the discrete iterated modified projection solution defined by (2.17). Then (3.20) Proof. We have from (3.14) A composite integration rule with respect to the fine partition (2.1) is then defined as Recall from (2.10) that for f, g ∈ C(∆ j ), The discrete orthogonal projection Q n,j : C(∆ j ) → P 0,∆j is defined as follows: A discrete orthogonal projection Q n : C[0, 1] → X n is defined as The following result is crucial in obtaining improved orders of convergence.

Numerical results
For the sake of illustration, we quote some numerical results from Grammont et al [7] for the following example considered in Atkinson-Potra [3]. Consider is the solution of (5.1). In this example, r can be chosen as large as we want.

Piecewise constant functions (r = 0)
Let X n be the space of piecewise constant functions with respect to the partition (2.9) and Q n : L ∞ [0, 1] → X n be the discrete orthogonal projection defined by (4.1). The numerical quadrature is chosen to be the composite Gauss 2 rule with respect to partition (2.1) with m = n 2 subintervals. Thenh = h 2 .
In Table 1 and Table 2, the computed orders of convergence in the discrete Galerkin, discrete iterated Galerkin, discrete Modified Projection and the discrete iterated Modified Projection methods are denoted respectively by δ G , δ S , δ M and δ IM . It can be seen from Table 1 that the computed values of order of convergence match well with the theoretically predicted values in (4.5), (4.9) and (4.16).

Piecewise linear functions (r = 1)
Let X n be the space of piecewise linear polynomials w.r.t. the partition (2.9) and Q n be the discrete orthogonal projection defined by (2.12). We choose the composite Gauss 2 point rule with n 2 intervals for the Galerkin methods and the composite Gauss 2 point rule with n 3 intervals for the modified projection methods. In the latter caseh 2 = h 6 . It can be seen from Table 2 that the computed values of the orders of convergence match well with the theoretically predicted values in (3.11), (3.13) and (3.20).