Singularity Preserving Galerkin Method For

Hammerstein Equations With Logarithmic Kernel

Hideaki Kaneko^{∗} Richard D. Noren Department of Mathematics and Statistics

Old Dominion University Norfolk, Virginia 23529-0077, USA and Peter A. Padilla NASA Langley Research Center MS-130 Hampton, Virginia 23681-0001, USA

∗This author is partially supported by NASA under grant NCC1-213

SINGULARITY PRESERVING GALERKIN METHOD FOR HAMMERSTEIN EQUATIONS WITH LOGARITHMIC KERNEL

H. Kaneko, R. D. Noren and P. A. Padilla

Abstract In a recent paper [?], Y. Cao and Y. Xu established the Galerkin method for weakly singular Fredholm integral equations that preserves the singularity of the solution. Their Galerkin method provides a numerical solution that is a linear combination of a certain class of basis functions which includes elements that reflect the singularity of the solution. The purpose of this paper is to extend the result of Cao and Xu and to establish singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. The iterated singularity preserving Galerkin method is also discussed.

Key words: Singularity preserving Galerkin method, Hammertein equations with logarithmic kernel, Iterated singularity preserving Galerkin method. Mathematics Subject Classification (1990): 65B05, 45L10.

1 Introduction

In this paper, we are concerned with the problem of approximating the solutions of weakly singular Hammerstein equations with logarithmic kernel by the Galerkin method that preserves the singularity of the exact solution. Namely we establish a method that generates an approximate solution in terms of a collection of basis functions some of which are comprised of singular elements that reflect the characteristics of the singularity of the exact solution. The idea of the method originates in the recent paper by Cao and Xu [?]. Cao and Xu studied the special nature of the singularities that are pertinent to the solutions of the weakly singular Fredholm equations of the second kind. Let C[0, 1] denote the space of all continuous functions defined on [0, 1]. The weakly singular Fredholm integral equations of the second kind can be described as

y(s) − ∫_{0}^{1} g_{α}(|s − t|)m(s, t)y(t)dt = f(s),  0 ≤ s ≤ 1  (1.1)

where f ∈ C[0, 1], m is sufficiently smooth and

g_{α}(|s − t|) = \{ \begin{matrix}|s − t|^{α}, & −1 < α < 0, \\ log |s − t|, & α = 0, \end{matrix}  (1.2)

where y is, of course, the function to be determined. It is well documented (see, e.g. [?],[?],[?],[?]) that the solutions of the equations described in (1.1) exhibit, in general, mild singularities even in the case of a smooth forcing term f. Here by “mild” singularities, we mean singularities in derivatives. The papers of Richter [?] and Graham [?] contain singularity expansions of the solutions of equation (1.1) in the case of m(s, t) ≡ 1. The results of Graham were recently generalized by Cao and Xu for equation (1.1). Information concerning the type of singularities that solutions have is useful when solving equation (1.1) numerically. In order to approximate functions with mild singularities, many investigators utilized the important theorem of Rice [?] that gives an optimal order of approximation to such functions. Rice’s theorem is that of nonlinear approximation by splines using variable knots. Based upon this idea of approximating the solutions of equation (1.1) by splines defined on nonuniform knots, the collocation method, the Galerkin method and the product-integration method were established for equation (1.1) by Vainikko and Uba [?], by Graham [?] and by Schneider [?] respectively. A modified collocation method was introduced in [?] which also uses the idea of Rice. Recently there has been some considerable interest in the study of the following weakly singular Hammerstein equation:

y(s) − ∫_{0}^{1} g_{α}(|s − t|)m(s, t)ψ(t, y(t))dt = f(s),  0 ≤ s ≤ 1  (1.3)

where f, m and g_{α} are defined as in (1.1) and (1.2) and ψ is a known function. It is well known that an equation similar to (1.3) arises naturally as a reformulation to an integral equation of a class of two dimensional Dirichlet problems with certain nonlinear boundary conditions, -e.g., [?]. A study on the regularities of the solution y of equation (1.3) is reported in [?], extending the results of [?]. Subsequently, Kaneko, Noren and Xu used the regularity results to establish the collocation method for weakly singular Hammerstein equations in [?]. The approximate solutions provided by these methods for equations (1.1) and (1.3) are in the form of piecewise polynomials that are not always satisfactory as a tool for approximating functions with singularities. This observation is quite evident in the areas of finite element analysis. Hughes and Akin [?] list several problems (e.g. ‘upwind’ finite elements for treating convection operators [?],[?],[?]; boundarylayer elements [?] etc.) in which the finite element shape functions are constructed to include polynomials as well as singular functions. Singular shape functions are introduced to the set of basis functions through asymptotic analysis on the solution of the problem that is being considered. It should be pointed out that the analysis involved in the aforementioned papers on the finite element method is centered around the collocation method. The problems such as the choice for the extra collocation points for singular basis elements or the rate of convergence are not addressed in these papers. It should be pointed out that the location of additional collocation points for singular basis elements is critical in detemining the rate of convergence of numerical solutions. A detailed discussion on this subject can be found in [?]. A singularity preserving collocation method, because of the reasons mentioned above, seems to be more difficult to establish.

In this paper, a singularity expansion for the solution of equation (1.3) with logarithmic kernel is given. This extends the results in [?] and [?]. Only the logarithmic kernel is considered here. It is a routine matter, however, to establish, following the argument of Section 2, a singularity expansion for the solution of equation (1.3) with algebraic singularity. The details are left to the reader. The paper is organized as follows: in Section 2, we study the regularity property of the soultion of (1.3) and establish its singularity expansion. The results obtained there generalize the results of [?] and [?]. The singularity expansion is then utilized in Section 3 to achieve the singularity preserving Galerkin method for equation (1.3). Finally, in Section 4, the iterated singularity preserving Galerkin method is discussed, extending some results from [?]. Examples are given in Section 5.

2 Singularity Expansion for Weakly Singular Hammerstein Equa-

tions

In this section, we consider the following Hammerstein equation with logarithmic singularity,

y(s) − ∫_{0}^{1} log |s − t|m(s, t)ψ(t, y(t))dt = f(s),  0 ≤ s ≤ 1  (2.1)

(see (1.3) also). We let

KΨy(s) ≡ ∫_{0}^{1} log |s − t|m(s, t)ψ(t, y(t))dt.  (2.2)

Then equation (2.1) can be written in operator form as

y − KΨy = f.  (2.3)

Let H^{n} denote the Sobolev space, H^{n}[0, 1] = {w : w^{(n)} ∈ L_{2}[0, 1]}, equipped with the norm ∥w∥_{H^{n}} = (∑_{i=0}^{n} ∥w^{(i)}∥_{2}^{2})^{1/2} where w^{(i)} describes the ith generalized derivative of w. We also let W = W_{n} be the linear space spanned by the functions s^{i} log^{j} s, (1 − s)^{i} log^{j}(1 − s); i, j = 1, 2, ..., n − 1. Throughout this paper, we assume the following conditions:

m ∈ C^{2n}([0, 1] × [0, 1]), n ≥ 1, m ∈ C^{1}([0, 1] × [0, 1]), n = 0.  (2.4)
ψ ∈ C^{2n+1}(R × R)  (2.5)

f ∈ W ⊕ H^{n}.  (2.6)

We define

Ky(s) ≡ ∫_{0}^{1} log |s − t|m(s, t)y(t)dt.  (2.7)

First we quote the following result (lemma 4.4(2)) from [?].

Lemma 2.1 Let u_{1}(s) = s^{p} log^{q} s, and u_{2}(s) = (1 − s)^{p} log^{q}(1 − s), for some integers p, q ≥ 1 and let f ∈ H^{n−1}. Assume that m ∈ C^{n+1}([0, 1] × [0, 1]). Then,

(Kf)(s) = ∑_{j=1}^{n−1} [c_{j}s^{j} log s + d_{j}(1 − s)^{j} log(1 − s)] + v_{n}(s),

(Ku_{1})(s) = ∑_{j=p+1}^{n−1} ∑_{i=1}^{q+1} c_{ij}s^{j}(log s)^{i} + ∑_{j=q+1}^{n−1} d_{j}(1 − s)^{j} log(1 − s) + v_{n}(s),

and

(Ku_{2})(s) = ∑_{j=p+1}^{n−1} ∑_{i=1}^{q+1} c_{ij}(1 − s)^{j}(log(1 − s))^{i} + ∑_{j=q+1}^{n−1} d_{j}s^{j} log s + v_{n}(s).

Lemma 2.2 If u_{1}(s) = s^{p} log^{q} s, u_{2}(s) = (1 − s)^{r} log^{l}(1 − s), for some integers p, q, r, l ≥ 1 are integers, then u_{1}u_{2} ∈ W ⊕ H^{n}.

Proof: Expand u_{1} in series about s = 1 and u_{2} about s = 0 :

\begin{aligned}u_{1}(s) & = ∑_{i=0}^{n−1} b_{i}(1 − s)^{i} + f_{1}(s),  u_{2}(s) = ∑_{i=0}^{n−1} a_{i}s^{i} + f_{2}(s), \\ & ≡ P_{1}(s) + f_{1}(s)  ≡ P_{2}(s) + f_{2}(s) \end{aligned}

where f_{1}^{(k)}(s) = O((1 − s)^{n−k}) near s = 1, f_{1} is analytic at s = 1, and f_{1}^{(k)} ∼ u_{1}^{(k)}(s) − P_{1}^{(k)}(0) as s → 0+; f_{2}^{(k)}(s) = O(s^{n−k}) near s = 0, f_{2} is analytic at s = 0, and f_{2}^{(k)}(s) ∼ u_{2}^{(k)}(s) − P_{2}^{(k)}(1) as s → 1^{−}. Now u_{1}u_{2} = P_{1}P_{2} + P_{1}f_{2} + P_{2}f_{1} + f_{1}f_{2}. Clearly P_{1}P_{2} is in H^{n}. For f_{1}f_{2}, we have

\frac{d^{n}}{ds^{n}} (f_{1}(s)f_{2}(s)) = ∑_{i=0}^{n} \binom{n}{i}f_{1}^{(i)}(s)f_{2}^{(n−i)}(s).

Each term f_{1}^{(i)}(s)f_{2}^{(n−i)}(s), i = 0, 1, ..., n satisfies

f_{1}^{(i)}(s)f_{2}^{(n−i)}(s) = O(f_{1}^{(i)}(s)t^{i}) = O([u_{1}^{(i)}(s) − P_{1}^{(i)}(0)]s^{i}) → 0

as s → 0+. Similarly

f_{1}^{(i)}(s)f_{2}^{(n−i)}(s) → 0 as s → 1^{−}. Thus f_{1}f_{2} ∈ C^{n} ⊆ H^{n}. For f_{1}P_{2} we have f_{1}(s)P_{2}(s) = (u_{1}(s) − P_{1}(s))P_{2}(s) = u_{1}(s)P_{2}(s) − P_{1}(s)P_{2}(s). Since P_{2} is a polynomial, u_{1} ∈ W, it is easy to see that u_{1}P_{2} ∈ W ⊕ H^{n} (see [[?], (4.7)]). So f_{1}P_{2} ∈ H^{n}. Similarly f_{2}P_{1} ∈ W ⊕ H^{n}, and Lemma 2 has been verified. □

Lemma 2.3 A product of an H^{n} function with a function in W is in H^{n} ⊕ W.

Proof: Let g ∈ H^{n} and let u_{1} and u_{2} be defined as before prior to Lemma 1. For gu_{1} we write

\begin{aligned}u_{1}(s)g(s) & = ∑_{i=0}^{n−1} \frac{g^{(i)}(0)}{i!}s^{i+p} log^{q} s + \frac{s^{p} log^{q} s}{(n−1)!} ∫_{0}^{s} g^{(n)}(σ)(s − σ)^{n−1}dσ \\ & ≡ T_{1} + T_{2}. \end{aligned}

Since T_{1} ∈ W ⊕ H^{n}, we turn to T_{2} and write

\frac{d^{n}T_{2}}{ds^{n}} = \frac{1}{(n−1)!} ∑_{k=0}^{n} \binom{n}{k} \frac{d^{k}}{ds^{k}} [s^{p} log^{q} s]\frac{d^{n−k}}{ds^{n−k}} [∫_{0}^{s} g^{(n)}(σ)(s − σ)^{n−1}dσ]
= \frac{1}{(n−1)!} ∑_{k=1}^{n} \binom{n}{k} \frac{d^{k}}{ds^{k}} [s^{p} log^{q} s][(n − 1)...k] ∫_{0}^{s} g^{(n)}(σ)(s − σ)^{k−1}dσ
+s^{p} log^{q} sg^{(n)}(s).

But s^{p} log^{q} s ∈ L^{∞}, g^{(n)} ∈ L_{2}[0, 1] so (s^{p} log^{q} s)g^{(n)}(s) ∈ L^{2}.

For the terms

b_{n}(s) ≡ \frac{d^{k}}{ds^{k}} [s^{p} log^{q} s] ∫_{0}^{s} g^{(n)}(σ)(s − σ)^{k−1}dσ

we have, for some constant M and nonegative integer α

\begin{aligned}| b_{n}(s) | & ≤ M \frac{(− log s)^{α}}{s^{k−1}} ∫_{0}^{s} | g^{(n)}(σ) | s^{k−1}dσ \\ & = Ms(− log s)^{α} \frac{1}{s} ∫_{0}^{s} | g^{(n)}(σ) | dσ. \end{aligned}

But g^{(n)} ∈ L_{2}[0, 1], so by Hardy’s inequality [?] (p. 72) \frac{1}{s} ∫_{0}^{s} | g^{(n)}(σ) | dσ ∈ L_{2}[0, 1]. Since s(− log s)^{α} ∈ L^{∞} it follows that b_{n} ∈ L_{2}[0, 1]. Hence \frac{d^{n}T_{2}}{ds^{n}} ∈ L_{2}[0, 1], or T_{2} ∈ H^{n}, This proves gu_{1} ∈ W ⊕ H^{n}.

The case for gu_{2} ∈ W ⊕ H^{n} is similar. □

Finally we need the following:

Lemma 2.4 The operator KΨ maps W ⊕ H^{n} into W ⊕ H^{n+1}.

Proof: Let y = w + h, w ∈ W, h ∈ H^{n}. We use Taylor’s theorem in the form

ψ(t, x) = ∑_{k=0}^{n} \frac{1}{k!}ψ^{(0,k)}(t, a)(x − a)^{k} + \frac{1}{n!} ∫_{a}^{x}(x − σ)^{n}ψ^{(0,n+1)}(t, σ)dσ.  (2.8)

Letting x = y(s) and a = h(s) allows us to write

\begin{aligned}(KΨ)(y)(t) & = ∑_{k=0}^{n} \frac{1}{k!} ∫_{0}^{1} log | t − s | m(t, s)ψ^{(0,k)}(s, h(s))w(s)^{k}ds \\ & +\frac{1}{n!} ∫_{0}^{1} log | t − s | m(t, s) ∫_{h(s)}^{y(s)} ψ^{(0,n+1)}(s, σ)(y(s) − σ)^{n}dσds \\ & ≡ ∑_{k=0}^{n} \frac{1}{k!}A_{k}(t) + \frac{1}{n!}B(t). \end{aligned}  (2.9)

By (3), ψ^{(0,k)}(s, h(s)) ∈ H^{n}, k = 0, 1, ..., n, and by expanding w(s)^{k} with the multinomial expansion, it is clear that w(s)^{k} is a sum of terms in W as well as terms of the form as^{p} log^{q} s(1− s)^{r} log^{u}(1 − s), p, q, r, u ≥ 1 are integers. The constant, a, depends on p, q, r and u. Since ψ^{(0,k)}(h(s)) ∈ H^{n} and w(s)^{k} ∈ W ⊕ H^{n}, k = 0, 1, ..., n, it follows from Lemma 3 that

ψ^{(0,k)}(s, h(s))w(s)^{k} ∈ W ⊕ H^{n}.  (2.10)

By Lemma 1 and (2.10), we have

A_{k} ∈ W ⊕ H^{n+1}.  (2.11)

For B(t), if we prove that

F (s) ≡ ∫_{h(s)}^{y(s)} ψ^{(0,n+1)}(s, σ)(y(s) − σ)^{n}dσ ∈ W ⊕ H^{n},  (2.12)

then, also by Lemma 1, B(t) = K[F](t) will be in W ⊕ H^{n+1}. This will complete the proof of this lemma. First of all, suppose n ≥ 1. We write

F′(s) = −ψ^{(0,n+1)}(s, h(s))w(s)^{n}h′(s).

Since h ∈ H^{n}, ψ ∈ C^{2n+1}, ψ^{(0,n+1)}(s, h(s)) ∈ H^{n}. By Lemmas 2 and 3, −ψ^{(0,n+1)}(s, h(s))w(s)^{n} ∈ H^{n} ⊕ W. Since h′ ∈ H^{n−1}, it follows that −ψ^{(0,n+1)}(s, h(s))w(s)^{n}h′(s) ∈ H^{n−1} ⊕W (Lemma 2). Since F′ ∈ H^{n−1} ⊕ W it is clear that F ∈ H^{n} ⊕ W. Second, let n = 0. Then F (s) = ∫_{h(s)}^{y(s)} ψ′(s, σ)dσ = ψ(s, y(s)) − ψ(s, h(s)) ∈ L_{2}[0, 1] ⊆ W ⊕ H^{0}. Thus

F ∈ W ⊕ H^{n}.  (2.14)

By (2.9), (2.11) and (2.12), it follows that KΨ maps W ⊕ H^{n} into W ⊕ H^{n+1}. □ Using the lemmas which we proved above, we obtain the following main result of this section.

Theorem 2.5 Suppose the conditions (2.4)-(2.6) hold and y is an isolated solution of (2.1). Then there are constants a_{ij} and b_{ij}, for i, j = 1, 2, ..., n − 1, and there is a function v_{n} in H^{n} such that

y(t) = ∑_{i=1}^{n−1} ∑_{j=1}^{n−1}[a_{ij}t^{i} log^{j} t + b_{ij}(1 − t)^{i} log^{j}(1 − t)] + v_{n}(t).  (2.13)

Proof: For n = 0, this follows from Lemma 4 with n = 0. Assume that the result holds for n = k, that is, if f ∈ H^{k} ⊕ W_{k}, then (2.13) holds with n = k. Say y = w_{k} + v_{k}, where v_{k} ∈ H^{k}, w_{k} = ∑_{i=1}^{k−1} ∑_{j=1}^{k−1}[a_{ij}t^{i} log^{j} t + b_{ij}(1 − t)^{i} log^{j}(1 − t)].

Now consider the case n = k + 1 and suppose f ∈ H^{k+1} ⊕ W_{k+1}.

Since y = w_{k} +v_{k} we write y = KΨy +f = KΨ(w_{k} +v_{k})+f. From Lemma 1, KΨ(w_{k} +v_{k}) ∈ W_{k+1} ⊕ H^{k+1}. The proof is complete. □

3 Singularity Preserving Galerkin Method

In this section, we establish the singularity preserving Galerkin method for equation (2.1). First we recall the definition of the space of spline functions of order n. Define the partition of [0, 1] as

Δ : 0 = t_{0} < t_{1} < ... < t_{k} < t_{k+1} = 1.

Let

h = max_{1≤i≤k+1}(t_{i} − t_{i−1}),

and assume h → 0 as k → ∞. Denote by Π_{n} the set of polynomials of degree n − 1. Then the space of splines of order n with knots t_{i}’s of multiplicity n − 1 − ν is defined as

S_{h}^{n} = S_{h}^{n,ν}(Δ) = {s ∈ C^{ν}[0, 1] : s|_{I_{i}} ∈ Π_{n}, },

where 0 ≤ ν ≤ n − 1 and I_{i} = (t_{i−1}, t_{i}) for i = 1, 2, ..., k + 1. It is well known that the dimension of S_{h}^{n} is d = n(k + 1) − k(1 + ν). S_{h}^{n} is spanned by a basis consisting of B-splines {B_{i}}_{i=1}^{d}. We let

V_{h}^{n} ≡ W ⊕ S_{h}^{n}  (3.1)

and denote the orthogonal projection of L_{2}[0, 1] onto V_{h}^{n} by P_{h}^{G}. The singularity preserving Galerkin method for approximating the solution of equation (2.3) requires the solution y_{h} ∈ V_{h}^{n} to satisfy the following equation:

y_{h} − P_{h}^{G}KΨy_{h} = P_{h}^{G}f.  (3.2)

More specifically, we need to find y_{h} in the form

y_{h}(s) = ∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s + ∑_{i,j=1}^{n−1} β_{ij}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}(s)  (3.3)

where {α_{ij}, β_{ij}}_{i,j=1}^{n−1} and {γ_{i}}_{i=1}^{d} are found by solving the following system of nonlinear equations:

∑_{i,j=1}^{n−1} α_{ij}(s^{i} log^{j} s, s^{p} log^{q} s) + ∑_{i,j=1}^{n−1} β_{ij}((1 − s)^{i} log^{j}(1 − s), s^{p} log^{q} s) + ∑_{i=1}^{d} γ_{i}(B_{i}, s^{p} log^{q} s)
−(KΨ(∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s + ∑_{i,j=1}^{n−1} β_{ij}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}), s^{p} log^{q} s)
= (f, s^{p} log^{q} s) p, q = 1, 2, …, n − 1
∑_{i,j=1}^{n−1} α_{ij}(s^{i} log^{j} s, (1 − s)^{p} log^{q}(1 − s)) + ∑_{i,j=1}^{n−1} β_{ij}((1 − s)^{i} log^{j}(1 − s), (1 − s)^{p} log^{q}(1 − s))
+ ∑_{i=1}^{d} γ_{i}(B_{i}, (1 − s)^{p} log^{q}(1 − s))
−(KΨ(∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s + ∑_{i,j=1}^{n−1} β_{ij}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}),
(1 − s)^{p} log^{q}(1 − s)) = (f, (1 − s)^{p} log^{q}(1 − s)) p, q = 1, 2, …, n − 1
∑_{i,j=1}^{n−1} α_{ij}(s^{i} log^{j} s, B_{p}) + ∑_{i,j=1}^{n−1} β_{ij}((1 − s)^{i} log^{j}(1 − s), B_{p}) + ∑_{i=1}^{d} γ_{i}(B_{i}, B_{p})
−(KΨ(∑_{i,j=1}^{n−1} α_{ij}s^{i} log^{j} s + ∑_{i,j=1}^{n−1} b_{i,j}(1 − s)^{i} log^{j}(1 − s) + ∑_{i=1}^{d} γ_{i}B_{i}), B_{p})
= (f, B_{p}) p = 1, 2, …, d

where (·, ·) denotes the usual inner product defined on L_{2}[0, 1]. Now let P_{h} be the orthogonal projection of L_{2}[0, 1] onto S_{h}^{n}. Then we have

P_{h}v → v  as h → 0  for all v ∈ L_{2}[0, 1].  (3.4)

It is well known (e.g. [?]) that if g ∈ H^{n}, n ≥ 0, then for each h > 0, there exists ϕ_{h} ∈ S_{h}^{n} such that

∥g − ϕ_{h}∥_{2} ≤ Ch^{n}∥g∥_{H^{n}},  (3.5)

where C > 0 is a constant independent of h. By virtue of the fact that P_{h}u is the best L_{2} approximation of u from S_{h}^{n}, we see immediately that

∥P_{h}u − u∥_{2} ≤ ∥u − ϕ_{h}∥_{2} ≤ Ch^{n}∥u∥_{H^{n}}, for all u ∈ H^{n}.  (3.6)

The following lemma from [?] is useful in the sequel.

Lemma 3.1 Let X be a Banach space. Suppose that U_{1} and U_{2} are two subspaces of X with U_{1} ⊆ U_{2}. Assume that P_{1} : X → U_{1} and P_{2} : X → U_{2} are linear operators. If P_{2} is a projection, then

∥x − P_{2}x∥_{X} ≤ (1 + ∥P_{2}∥_{X})∥x − P_{1}x∥_{X} for all x ∈ X.

For convenience, we introduce operators T̂ and T_{h} by letting

T̂y ≡ f + KΨy  (3.7)

and

T_{h}y_{h} ≡ P_{h}^{G}f + P_{h}^{G}KΨy_{h}  (3.8)

so that equations (2.1) and (3.2) can be written respectively as y = T̂y and y_{h} = T_{h}y_{h}. The following theorem guarantees the existence of a solution of the singularity preserving Galerkin method (3.2) and describes the accuracy of its approximation.

Theorem 3.2 Let y ∈ L_{2}[0, 1] be an isolated solution of equation (2.1). Assume that 1 is not an eigenvalue of the linear operator (KΨ)′(y), where (KΨ)′(y) denotes the Fréchet derivative of KΨ at y. Then the singularity preserving Galerkin approximation equation (3.2) has a unique solution y_{h} such that ∥y − y_{h}∥_{2} < δ for some δ > 0 and for all 0 < h < h_{0} for some h_{0} > 0. Moreover, there exists a constant 0 < q < 1, independent of h, such that

\frac{α_{h}}{1 + q} ≤ ∥y − y_{h}∥_{2} ≤ \frac{α_{h}}{1 − q} ,  (3.9)

where α_{h} ≡ ∥(I − T′_{h}(y))^{−1}(T_{h}(y) − T̂(y))∥_{2}. Finally, if y = w + v with w ∈ W and v ∈ H^{n}, then

∥y − y_{h}∥_{2} ≤ Ch^{n}∥v∥_{H^{n}},  whenever 0 < h < h_{0},  (3.10)

where C > 0 is a constant independent of h.

Proof: The existence of a unique solution y_{h} of equation (3.2) in the disk of radius δ about y and the inequalities in (3.7) can be proved using Theorem 2 of Vainikko [?]. A detailed discussion on this application can be found in [?]. To get (3.10), first we note from Lemma 3.1, for v ∈ L_{2}[0, 1],

∥P_{h}^{G}v − v∥_{2} ≤ (1 + ∥P_{h}^{G}∥_{2})∥P_{h}v − v∥_{2}.  (3.11)

By assumption, (I − (KΨ)′(y))^{−1} exists. By (3.4), Lemma 3.1 and since (KΨ)′(y) is a compact linear operator, ∥P_{n}^{G}(KΨ)′(y) − (KΨ)′(y)∥_{2} → 0 as n → ∞. Hence (I − P_{n}^{G}(KΨ)′(y))^{−1} = (I − T′_{h}(y))^{−1} exists and uniformly bounded in ∥ · ∥_{2} norm. Now, from (3.9),

\begin{aligned}∥y − y_{h}∥_{2} & ≤ \frac{α_{h}}{1−q} \\ & = \frac{1}{1−q} ∥(I − T′_{h}(y))^{−1}(T_{h}(y) − T̂(y))∥_{2} \\ & ≤ C∥P_{h}^{G}KΨy − KΨy + P_{h}^{G}f − f∥_{2} \\ & = C∥P_{h}^{G}y − y∥_{2}. \end{aligned}  (3.12)

where C is independent of h. Using the uniform boundedness of {P_{h}^{G}}, (3.6), (3.11) and (3.12), we obtain

∥y − y_{h}∥_{2} ≤ Ch^{n}∥v∥_{H^{n}}.

□

4 The Iterated Singularity Preserving Galerkin Method

In this section, the superconvergence of the iterated singularity preserving Galerkin method is discussed. We remind the reader that the conditions (2.4), (2.5) and (2.6) are still in effect. The discussion of this section depends heavily upon the recent paper by Kaneko and Xu [?] so that only the points of distinct differences are explained.

Let y_{0} be an isolated solution of (2.1). Assume that y_{h} is the unique solution of (3.2) in the sphere ∥y_{0} − y∥_{2} ≤ δ, for some δ > 0. Define

y_{n}^{I} = f + KΨy_{h}.  (4.1)

Applying P_{h}^{G} to both sides of (4.1), we obtain

P_{h}^{G}y_{h}^{I} = P_{h}^{G}f + P_{h}^{G}KΨy_{h}.  (4.2)

Comparing (4.2) with (3.2),

P_{h}^{G}y_{h}^{I} = y_{h}.  (4.3)

Substitution of (4.3) into (4.1) yields that y_{h}^{I} satisfies the following Hammerstein equation,

y_{h}^{I} = f + KΨP_{h}^{G}y_{h}^{I}.  (4.4)

In order to analyse the order of convergence of the iterated singularity preserving Galerkin method for Hammerstein equations with logarithmic singular kernel, we need the following. For any ϵ ∈ R, let [0, 1]_{ϵ} = {t ∈ [0, 1] : t + ϵ ∈ [0, 1]}. Let Δ_{h} denote the forward difference operator with step size h. For α > 0 and 1 ≤ p ≤ ∞, we define the Nikol’skii space N_{p}^{α}[0, 1] by

N_{p}^{α}[0, 1] = \{x ∈ L_{p}[0, 1] : |x|_{α,p} := sup_{h≠0} \frac{1}{|h|^{α_{0}}} ∥Δ_{h}^{2}x^{[α]}∥_{L_{p}[0,1]_{2h}} < ∞\},  (4.5)

where [α] is an integer and 0 < α_{0} ≤ 1 are chosen so that α = [α] + α_{0}. N_{p}^{α}[0, 1] is a Banach space with the norm ∥x∥_{α,p} = ∥x∥_{p} + |x|_{α,p}. We remark that the function t^{α−1} is in N_{1}^{α}[0, 1] but is not in N_{1}^{β}[0, 1], for any β > α, and log t ∈ N_{1}^{1}[0, 1], hence it is in N_{1}^{α} for all 0 < α < 1.

The theorem of Kaneko and Xu [14, Theorem 3.3], with only very minor modification can be written in the following form.

Theorem 4.1 Let y_{0} ∈ C[0, 1] be an isolated solution of equation (2.1) and y_{h} be the unique solution of (2.5) in the sphere B(y_{0}, δ). Let y_{h}^{I} be defined by the iterated scheme (4.1). Assume that 1 is not an eigenvalue of (KΨ)′(y_{0}). Then, for all 1 ≤ p ≤ ∞,

∥y_{0} − y_{h}^{I}∥_{∞} ≤ C {∥y_{0} − P_{h}^{G}y_{0}∥_{∞}^{2} + sup_{0≤t≤1} inf_{u∈V_{h}^{n}} ∥k(t, .)ψ^{(0,1)}(., y_{0}(.)) − u∥_{q}∥y_{0} − P_{h}^{G}y_{0}∥_{p}} ,

where 1/p + 1/q = 1 and C is a constant independent of n.

As a corollary, we obtain the main result of the section. First, we introduce some notations. Applying the mean-value theorem to ψ(s, y) to get

ψ(s, y) = ψ(s, y_{0}) + ψ^{(0,1)}(s, y_{0} + θ(y − y_{0}))

where θ ≡ θ(s, y_{0}, y) with 0 < θ < 1 and ψ^{(0,1)} denotes the partial derivative of ψ with respect to the second variable. Also

k(s, t) ≡ log(|s − t|)m(s, t)

and

g(s, t, y_{0}, y, θ) ≡ k(s, t)ψ^{(0,1)}(s, y_{0} + θ(y − y_{0})).

Theorem 4.2 Assume the hypotheses of the previous theorem. Assume also that (2.4)-(2.6) hold. Then

∥y_{0} − y_{n}^{I}∥_{∞} = O(h^{2n}) + O(h^{n+1}) = O(h^{n+1}).

Proof: First of all, for each u ∈ V_{h}^{n},

∥y_{0} − P_{h}^{G}y_{0}∥_{∞} ≤ ∥y_{0} − u∥_{∞} + ∥P_{h}^{G}u − P_{h}^{G}y_{0}∥_{∞} ≤ (1 + P)∥y_{0} − u∥_{∞},  (4.6)

where P ≡ sup_{h>0} P_{h}^{G} < ∞. Since y_{0} = w + v for some w ∈ W and v ∈ H^{n}, we let u = w + u^{∗}, where u^{∗} ∈ S_{h}^{n}. We obtain ∥y_{0} − u∥_{∞} = ∥v − u^{∗}∥_{∞}. With (4.6) this yields

∥y_{0} − P_{h}^{G}y_{0}∥_{∞} ≤ (1 + P) inf_{u^{∗}∈S_{h}^{n}} ∥v − u^{∗}∥_{∞} ≤ Ch^{n}.  (4.7)

The last inequality follows from (3.5). Secondly, by [?], [Theorem 4 (i)], there exists v_{t} ∈ S_{h}^{n} such that ∥k_{t} − v_{t}∥_{1} = O(h). Since ν ≥ 1, S_{h}^{n} = S_{h}^{n,ν} ⊆ H^{1}, so v_{t} ∈ H^{1}.

Since y_{0} ∈ W ⊕ H^{n} it follows that ψ^{(0,1)}(·, y_{0}(·)) ∈ W ⊕ H^{n−1}, by expanding ψ^{(0,1)}(·, y_{0}(·)) in Taylor series centered at v (recall y_{0} = w +v, v ∈ H^{n}) and using (2.10) and (2.12). Consequently, v_{t}(·)ψ^{(0,1)}(·, y_{0}(·)) ∈ W ⊕ H^{n−1}. Say v_{t}(·)ψ^{(0,1)}(·, y_{0}(·)) = a_{t} + b_{t}, where a_{t} ∈ W and b_{t} ∈ H^{n−1}. Now there exists u_{t} ∈ S_{h}^{n} such that ∥u_{t} − b_{t}∥_{1} = O(h^{n−1}) and

∥g_{t} − u_{t} − a_{t}∥_{1} ≤ ∥k_{t} − v_{t}∥_{1}∥ψ^{(0,1)}(·, y_{0}(·))∥_{∞} + ∥v_{t}(·)ψ^{(0,1)}(·, y_{0}(·)) − u_{t} − a_{t}∥.
= O(h) + O(h^{n−1}) = O(h),

provided n ≥ 2. Now we apply Theorem 4.1 to get

∥y_{0} − y_{h}^{I}∥_{∞} = O(h^{2n}) + O(h^{n+1}) = O(h^{n+1}).

5 Numerical Example

Let m(s, t) = 1, g(|s − t|) = log(|s − t|) and ψ(s, t) = cos(s + t) in equation (1.3). We assume f in such a way that x(t) = sin t + t log t is the solution. Using splines of order 2 we approximate the solution of the Hammerstein equation with

y_{0}(t) = ∑_{i=1}^{d} γ_{i}B_{i}

and

y_{1}(t) = ∑_{i=1}^{d} γ_{i}B_{i} + αt log t + β(1 − t) log(1 − t)  (5.1)

y_{0} represents the numerical solution that uses only the slpine basis elements whereas y_{1} represents the current scheme. y_{0} is computed for comparison. Notice that the convergence rate for y_{0} is lower due to the logarithmic singularity in the kernel and due to the use of the uniform partition of [0, 1]. The use of nonuniform partition to obtain the optimal rate of convergence of numerical solution was recently established in [?] for the Galerkin method. It should be pointed out that, as the number of partition points increases, the distribution of these nonuniform points become extremely skewed toward the end points of the interval. This will cause a sensitivity in numerical computations, frequently requiring computations in double precision. An introduction of the singular elements in the basis and working with the uniform partition points will eliminate this problem. The coefficients in (4.18) were obtained by solving thew set of nonlinear equations of Section 3 (immediately following (3.3)) using the Newton-Raphson algorithm. Also, the Gausstype quadrature algorithm described in [?] is used to calculate all integrals. The computed errors for the spline-only solution and the singularity preserving solution are shown in the following table.

Errors

n y_{0} y_{1}
2  .032756 .004002
3  .018526 .001945
4  .012246 .001147
convergence rate ≈  1.4  1.8

References

[1] J. E. Akin, Physical bases for the design of special finite element interpolation functions, Recent Advances in Engineering Science, (G. C. Sih, ed.) Lehigh Univ. Press, (1977), 879-884.

[2] K. E. Atkinson and G. Chandler, Boundary integral equations methods for solving Laplace’s equation with nonlinear boundary conditions: The smooth boundary case, Math. Comp., 55 (1990), 455-472.

[3] Y. Cao and Y. Xu, Singularity preserving Galerkin methods for weakly singular Fredholm integral equations, Jl. Int. Eqs. and Appl., 6 (1994) 303-334.

[4] I. Graham, Singularity expansions for the solution of the second kind Fredholm integral equations with weakly singular convolution kernels, Jl. Int. Eqs., 4 (1982), 1-30.

[5] I. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp., 39 (1982), 519-533.

[6] T. J. R. Hughes, A simple scheme for developing “upwind” finite elements, Int. Jl. Num. Meth. Engng.12 (1978), 1359-1365.

[7] T. J. R. Hughes and J. E. Akin, Techniques for developing ‘special’ finite element shape functions with particular reference to singularities, Int. Jl. Num. Meth. Eng.15 (1980), 733-751.

[8] J. C. Henrich, P. S. Huyakorn, A. R. Mitchell and O. C. Zienkiewicz, An upwind finite element scheme for two-dimensional convective transport equations, Int. Jl. Num. Meth. Engng., 11 (1977), 131-143.

[9] T. J. R. Hughes, W. K. Liu and A. Brooks, Finite element analysis of incompressible viscous flows by the penalty function formulation, Jl. Comp. Phys., 30 (1979), 1-60.

[10] H. Kaneko, R. Noren and Y. Xu, Regularity of the solution of Hammerstein equations with weakly singular kernels, Int. Eqs. Op. Thy. 13 (1990), 660-670.

[11] H. Kaneko, R. Noren and Y. Xu, Numerical solutions for weakly singular Hammerstein equations and their superconvergence, J. Int. Eqs. Appl. 4 (1992), 391-407.

[12] H. Kaneko and P. Padilla, A note on the finite element method with singular basis functions, (submitted for publication).

[13] H. Kaneko and Y. Xu, Numerical solutions for weakly singular Fredholm integral equations of the second kind, Appl. Num. Math., 7 (1991), 167-177.

[14] H. Kaneko and Y. Xu, Gauss-type quadratures for weakly singular integrals and their application to Fredholm integral equations of the second kind, Math. Comp. , 62 (1994), 739-753.

[15] H. Kaneko and Y. Xu, Superconvergence of the iterated Galerkin method for Hammerstein equations, SIAM Jl. of Num. Anal.33 (1996), 1048-1064.

[16] G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, Jl. Math. Anal. Appl., 55 (1976), 32-42.

[17] J. Rice, On the degree of convergence of nonlinear spline approximation, in Approximation with Special Emphasis on Spline Functions, ed. I. J. Schoenberg, Academic Press, N.Y. (1969), 349-365.

[18] W. Rudin, Real and Complex Analysis, 3rd. ed., McGraw-Hill Book Co. New York, (1987).

[19] C. Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Int. Eqs. Op. Thy.2 (1979), 62-68.

[20] C. Schneider, Product integration for weakly singular integral equations, Math. Comp.36 (1981), 207-213.

[21] I.H. Sloan, Superconvergence, in Numerical Solution of integral equations, ed. M. A. Golberg, Plenum Press, N.Y., 35-70 (1990).

[22] G. Vainikko, Perturbed Galerkin method and general theory of approximate methods for nonlinear equations, Zh. Vychisl. Mat. Fiz. 7 (1967), 723-751. Engl. Translation, U.S.S.R. Comp. Math. and Math. Phys. 7, No. 4 (1967), 1-41.

[23] G. Vainikko and A. Pedas, The properties of solutions of weakly singular integral equations, Jl. Austral. Math. Soc. Ser. B22 (1981), 419-430.

[24] G. Vainikko and P. Ubas, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, Jl. Austral. Math. Soc. Ser. B22 (1981), 431-438.