Error Analysis of Galerkin's Method for Semilinear Equations

Abstract

We establish a general existence result for Galerkin's approximate solutions of abstract semilinear equations and conduct an error analysis. Our results may be regarded as some extension of a precedent work (Schultz 1969). The derivation of our results is, however, different from the discussion in his paper and is essentially based on the convergence theorem of Newton’s method and some techniques for deriving it. Some of our results may be applicable for investigating the quality of numerical verification methods for solutions of ordinary and partial differential equations.

1. Introduction

Let 𝑋 be a real Hilbert space and 𝑋_ℎ⊂𝑋 be a closed subspace. Here, ℎ is a positive parameter (which will tend to zero). We denote by 𝑃_ℎ the orthogonal projection onto 𝑋_ℎ. We assume that lim_ℎ→0𝐼−𝑃_ℎ𝑢=0forany𝑢∈𝑋, (H1) where 𝐼∶𝑋→𝑋 is the identity operator. We are interested in studying error analysis of Galerkin’s method for the following equation: 𝑓(𝑢)∶=𝑢−𝜑(𝑢)=0. (1.1) Here, 𝜑∶U→𝑋 is a nonlinear map and U is a subset of 𝑋. We define 𝑓_ℎ∶U∩𝑋_ℎ→𝑋_ℎ by 𝑓_ℎ(𝑢)∶=𝑃_ℎ𝑓(𝑢)for𝑢∈U∩𝑋_ℎ. (1.2) The equation 𝑓_ℎ(𝑢)=0 is the Galerkin approximate equation of (1.1). A precedent work [1] by Schultz reads as follows.

Theorem 1.1 (see [1, Theorems  3.1 and 3.2]). One assumes (H1). Let 𝑅∈(0,∞) be a constant and U={𝑢∈𝑋;‖𝑢‖≤𝑅}. One assumes that 𝜑∶U→𝑋 is a completely continuous map such that 𝜑(U)⊂U. Then the following holds.(i)The equation 𝑓_ℎ(𝑢)=0 has a solution 𝑢_ℎ in U∩𝑋_ℎ for any ℎ and there exists a monotone decreasing sequence {ℎ_𝑘}^∞_𝑘=1 with lim_𝑘→∞ℎ_𝑘=0 and 𝑢_∞∈U such that 𝑢_ℎ𝑘→𝑢_∞ in 𝑋 as 𝑘→∞ and 𝑢_∞ is a solution of (1.1). Moreover, if 𝑢_∞ is the unique solution of (1.1) in U, then one has 𝑙𝑖𝑚_ℎ→0𝑢_ℎ=𝑢_∞ in 𝑋. (ii)Let 𝑢_∗∈U be a solution of (1.1). If 𝜑 has a Fréchet derivative, in a neighborhood, 𝒩, of 𝑢_∗ and 0 is not in the spectrum of 𝑓′, then 𝑢_∗ is the unique solution of (1.1) in 𝒩 and 𝑓_ℎ(𝑢)=0 has a solution 𝑢_ℎ∈𝑋_ℎ for any ℎ, which is unique for sufficiently small ℎ and 𝑢_∗−𝑢_ℎ≃𝐼−𝑃_ℎ𝑢_∗, (1.3) which means that ‖𝑢_∗−𝑢_ℎ‖and ‖(𝐼−𝑃_ℎ)𝑢_∗‖are equivalent infinitesimals as ℎ→0.

In this paper, we always assume (H1) and the following (H2) in what follows: 𝜑∈𝐶¹(U,𝑋),and𝜑′(𝑢)∈ℒ(𝑋)iscompactforany𝑢∈U, (H2) where U⊂𝑋 is an open set. Under the conditions (H1) and (H2), we obtain results similar to Theorem 1.1 (see Proposition 2.1 and Corollary 2.3). We also establish new other results on error analysis (see Theorems 2.4 and 2.5). Our results may be regarded as some extension of Theorem 1.1. The derivation of our results is, however, different from the proof of Theorem 1.1, which is based on the Brower fixed point theorem and the equality (4.10). Our proofs are essentially based on the convergence theorem of Newton’s method (Theorem 3.2) and some techniques for deriving it. We remark that a version of the same theorem is applied in [2] to an ordinary periodic system for a purpose similar to ours.

Various ordinary and partial differential equations appearing in mathematical physics can be written in the form (1.1) with (H2) under an appropriate setting of the functional spaces. See Section 5 for some concrete examples.

We define 𝑓_ℎ∶U→𝑋 by 𝑓_ℎ(𝑢)∶=𝑢−𝑃_ℎ𝜑(𝑢)for𝑢∈U. (1.4) The map 𝑓_ℎ is a natural extension of 𝑓_ℎ and is very useful in our analysis below. Obviously, 𝑢 is a solution of 𝑓_ℎ(𝑢)=0 if and only if 𝑢 is a solution of 𝑓_ℎ(𝑢)=0. We can treat the equation 𝑓_ℎ(𝑢)=0 more easily than 𝑓_ℎ(𝑢)=0 since 𝑓_ℎ is defined globally.

One of our motivations for this study is to investigate the quality of a numerical verification method for solutions of differential equations. Some of our results in this paper may be applicable for such a purpose. See Remark 2.7 for further information.

The paper is organized as follows. In Section 2 we describe our main results. We prepare some preliminary abstract results in Section 3 and apply them to prove our main results in Section 4. In Section 5 we present some concrete examples on semilinear elliptic partial differential equations.

Notations. Let 𝒳 and 𝒴 be Banach spaces. (1)We denote by ‖⋅‖_𝒳 the norm of 𝒳. If 𝒳 is a Hilbert space, then ‖⋅‖_𝒳 stands for the norm induced by the inner product of 𝒳. For 𝑢∈𝒳 and 𝑟∈(0,∞), we write 𝐵_𝒳(𝑢;𝑟)∶={𝑣∈𝒳;‖𝑣−𝑢‖<𝑟}. The subscript will be often omitted if no possible confusion arises. (2)For an open set 𝑉⊂𝒳, 𝐶¹(𝑉,𝒴) denotes the space of continuously differentiable functions from 𝑉 to 𝒴.(3)We denote by ℒ(𝒳,𝒴) the space of bounded linear operators from 𝒳 to 𝒴 and ℒ(𝒳) stands for ℒ(𝒳,𝒳). For 𝑇∈ℒ(𝒳,𝒴), ‖𝑇‖_𝒳→𝒴 denotes the operator norm of 𝑇. The subscript will be omitted if no possible confusion arises.(4)Let 𝜙(ℎ) and 𝜓(ℎ) be nonnegative functions. We write 𝜙(ℎ)∼𝜓(ℎ) if 𝜙(ℎ) and 𝜓(ℎ) are infinitesimals of the same order as ℎ→0, that is, 𝜙(ℎ)=𝑂(1)𝜓(ℎ) and 𝜓(ℎ)=𝑂(1)𝜙(ℎ) as ℎ→0. We write 𝜙(ℎ)≃𝜓(ℎ) if 𝜙(ℎ) and 𝜓(ℎ) are equivalent infinitesimals as ℎ→0, that is, 𝜙(ℎ)={1+𝑜(1)}𝜓(ℎ) as ℎ→0.(5)Let Ω be a bounded domain of 𝐑^𝑛. We denote Lebesgue spaces by 𝐿^𝑝(Ω) (1≤𝑝≤∞) with the norms ‖𝑢‖_𝐿^𝑝_(Ω)=(∫_Ω|𝑢(𝑥)|^𝑝𝑑𝑥)^1/𝑝  for1≤𝑝<∞,‖𝑢‖_𝐿^∞_(Ω)=ess.sup{|𝑢(𝑥)|;𝑥∈Ω}. We denote by 𝐻¹₀(Ω) the completion of 𝐶^∞₀(Ω) (the space of 𝐶^∞ functions with compact support in Ω) in the Sobolev norm: ‖𝑢‖=‖∇𝑢‖_𝐿²_(Ω)∶=(∑^𝑛_𝑘=1‖𝜕𝑢/𝜕𝑥_𝑘‖²_𝐿2(Ω))^1/2. We denote by 𝐻⁻¹(Ω) the Sobolev space {𝐹∈𝒟^′(Ω);∃𝐶∈(0,∞)suchthat|𝐹(𝜙)|≤𝐶‖𝜙‖_𝐻10(Ω)forany𝜙∈𝐶^∞₀(Ω)} with the norm ‖𝐹‖_𝐻⁻¹_(Ω)=sup{|𝐹(𝜙)|;𝜙∈𝐶^∞₀(Ω)and‖𝜙‖_𝐻10(Ω)≤1}. Here, 𝒟′(Ω) stands for the set of distributions on Ω.

2. Main Results

In this section we describe our main results. We assume (H1) and (H2). Let 𝑢_∗∈U be an isolated solution of (1.1), that is, 𝑢_∗ is a solution of (1.1) such that 𝑓^′(𝑢_∗)∶𝑋→𝑋 is bijective. We set 𝑇∶=𝜑^′𝑢_∗,𝐴∶=𝐼−𝑇=𝑓^′𝑢_∗,𝐴_ℎ∶=𝐼−𝑃_ℎ𝑇𝑃_ℎ, (2.1) for simplicity. The operator 𝐴_ℎ is an almost diagonal operator introduced in [3]. First we have an existence theorem for Galerkin’s approximate solutions of (1.1).

Proposition 2.1. There exist ℎ_∗>0and {𝑢_ℎ}_{ℎ∈(0,ℎ∗)}⊂Usuch that the following (i)–(iii) hold.(i)There exists 𝑅_∗>0such that 𝑢=𝑢_ℎis the only solution of 𝑓_ℎ(𝑢)=0in 𝐵(𝑢_∗;𝑅_∗)for any ℎ∈(0,ℎ_∗).(ii)𝑢=𝑢_ℎis an isolated solution of𝑓_ℎ(𝑢)=0for anyℎ∈(0,ℎ_∗). (iii)𝑢_ℎ→𝑢_∗in𝑋asℎ→0with the estimate𝑢_∗−𝑢_ℎ≤𝐶_ℎ𝑓_ℎ𝑢_∗=𝐶_ℎ𝐼−𝑃_ℎ𝑢_∗foranyℎ∈0,ℎ_∗, (2.2)where {𝐶_ℎ}_{ℎ∈(0,ℎ∗)}⊂(1,2)and 𝐶_ℎ→1 as ℎ→0.

Remark 2.2. (i) Proposition 2.1(ii) is useful in our analysis below. Moreover, we immediately obtain from it that 𝑢=𝑢_ℎis an isolated solution of 𝑓_ℎ(𝑢)=0for any ℎ∈(0,ℎ_∗). This guarantees that we can always construct a Galerkin approximate solution 𝑢_ℎ by Newton’s method for small ℎ>0.
(ii) In various contexts in applications, 𝑋_ℎ is finite-dimensional for any ℎ. In such contexts the assumption (H1) implies that 𝑋 is separable.
(iii) We do not assume dim𝑋_ℎ<∞. We briefly explain that it has some practical benefits. The case dim𝑋_ℎ=∞ appears, for example, in the following context. We are interested in the semi-discrete approximation to a periodic system described by a partial differential equation with a periodic forcing term. We may apply a Galerkin method only in space to the original system in order to construct a simpler approximate system described by ordinary differential equations. Then, for an isolated periodic solution of the original system, our Proposition 2.1 may guarantee that in a small neighborhood of it the approximate system has a periodic solution. For example, we can actually apply Proposition 2.1 to a semi-discrete approximation to a periodic system treated in [3]. See [4, Remark  3.4] for how to rewrite the system in [3] as (1.1).

In what follows in this section, {𝑢_ℎ}_{ℎ∈(0,ℎ∗)}always denotes the sequence as described in Proposition 2.1. Since 𝑢_∗−𝑢_ℎis decomposed into the 𝑋_ℎ-component 𝑃_ℎ𝑢_∗−𝑢_ℎand the 𝑋^⟂_ℎ-component (𝐼−𝑃_ℎ)𝑢_∗, we have ‖𝑢_∗−𝑢_ℎ‖²=‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖²+‖(𝐼−𝑃_ℎ)𝑢_∗‖²and ‖(𝐼−𝑃_ℎ)𝑢_∗‖≤‖𝑢_∗−𝑢_ℎ‖. So, the last inequality and (2.2) immediately imply (2.3) below.

Corollary 2.3. We have 𝑢_∗−𝑢_ℎ≃𝑓_ℎ𝑢_∗=𝐼−𝑃_ℎ𝑢_∗,𝑃_ℎ𝑢_∗−𝑢_ℎ=𝑜𝐼−𝑃_ℎ𝑢_∗asℎ⟶0,𝑃_ℎ𝑢_∗−𝑢_ℎ=𝑜𝑢_∗−𝑢_ℎasℎ⟶0. (2.3)(2.4)(2.5)

Actually, we easily verify that (2.3), (2.4) and (2.5) are mutually equivalent. They are very general features for the Galerkin method. The estimate (2.5) means that the 𝑋_ℎ-component of the error ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖is an infinitesimal of a higher order of smallness with respect to the whole error ‖𝑢_∗−𝑢_ℎ‖as ℎ→0.

The following two results are useful for applications (see Remark 2.7 below).

Theorem 2.4. We have the following: 𝑢_∗−𝑢_ℎ≃𝑓𝑢_ℎ=𝐼−𝑃_ℎ𝜑𝑢_ℎ≃𝐴⁻¹𝑓𝑢_ℎ,𝑃_ℎ𝑢_∗−𝑢_ℎ≃𝐴⁻¹𝑓_ℎ𝑃_ℎ𝑢_∗≃𝑃_ℎ𝐴_ℎ⁻¹𝑓𝑃_ℎ𝑢_∗∼𝑃_ℎ𝜑𝑢_∗−𝜑𝑃_ℎ𝑢_∗. (2.6)(2.7)

Theorem 2.5. (i) We have sup_𝑠∈[0,1]𝜑^′(1−𝑠)𝑢_∗+𝑠𝑃_ℎ𝑢_∗𝐼−𝑃_ℎ⟶0asℎ⟶0. (2.8)
(ii) Let 𝜀_ℎ be a positive constant for ℎ∈(0,ℎ_∗) such that 𝜀_ℎ≥sup_𝑠∈[0,1]𝜑^′(1−𝑠)𝑢_∗+𝑠𝑃_ℎ𝑢_∗𝐼−𝑃_ℎ (2.9) for any ℎ∈(0,ℎ_∗). Then, there exist constants ℎ₁∈(0,ℎ_∗) and 𝐶₁>0 such that 𝑃_ℎ𝑢_∗−𝑢_ℎ≤𝐶₁𝜀_ℎ𝐼−𝑃_ℎ𝑢_∗foranyℎ∈0,ℎ₁. (2.10)

In view of Theorem 2.5 (i) and (ii), we can always take {𝜀_ℎ}_{ℎ∈(0,ℎ∗)} in (2.10) such that 𝜀_ℎ→0 as ℎ→0. The following Remarks 2.6 and 5.3 below shows that our estimate (2.10) is in general sharper than an estimate which can be derived directly from the discussion in [1].

Remark 2.6. (i) In the same way as in the proof of [1, Theorem  3.2] we can obtain an estimate related to (2.10). We set 𝜂_ℎ∶=(2𝑝_ℎ+𝑞_ℎ+𝑟_ℎ)/(1−(𝑝_ℎ+𝑞_ℎ+𝑟_ℎ)), 𝑝_ℎ∶=‖𝐴⁻¹(𝐼−𝑃_ℎ)𝑇‖, 𝑞_ℎ∶=‖𝐴⁻¹𝑃_ℎ𝑇(𝐼−𝑃_ℎ)‖and 𝑟_ℎ∶=‖𝐴⁻¹‖⋅‖𝜑(𝑢_ℎ)−𝜑(𝑢_∗)−𝑇(𝑢_ℎ−𝑢_∗)‖/‖𝑢_ℎ−𝑢_∗‖. It follows from Proposition 2.1 (iii) and Proposition 3.1 below that 𝑝_ℎ, 𝑞_ℎ and 𝑟_ℎ converge to 0 as ℎ→0. So, 𝜂_𝑘→0 as ℎ→0. Let 𝜀_ℎ̂ be a positive constant for ℎ∈(0,ℎ_∗) such that 𝜀_ℎ̂≥𝜂²_ℎ+2𝜂_ℎ. Then we have 𝑃_ℎ𝑢_∗−𝑢_ℎ≤𝜀_ℎ̂𝐼−𝑃_ℎ𝑢_∗foranyℎ∈0,ℎ_∗. (2.11) We can verify that 𝜀_ℎ̂islargerthan𝐼−𝑃_ℎ𝑇^1/2forsufficientlysmallℎ>0. (2.12) Indeed, we immediately obtain (2.12) from 𝑝_ℎ≃𝐼−𝑃_ℎ𝑇. (2.13) We derive (2.11) and (2.13) at the end of Section 4.
(ii) When we compute 𝜀_ℎ̂ for concrete examples (e.g., examples in Section 5 below), it seems reasonable to estimate 𝑞_ℎas 𝑞_ℎ≤𝐶‖𝑇(𝐼−𝑃_ℎ)‖. Here, 𝐶represents some positive constant independent of ℎ. Then, it is actually necessary to take 𝜀_ℎ̂such that 𝜀_ℎ̂≥𝐶‖𝑇(𝐼−𝑃_ℎ)‖^1/2for small ℎ>0. On the other hand, roughly speaking, (2.9) means that we can take 𝜀_ℎ≈‖𝑇(𝐼−𝑃_ℎ)‖for small ℎ>0 (See Remark 5.3 below). (We note that Proposition 3.1 below implies that ‖𝑇(𝐼−𝑃_ℎ)‖→0as ℎ→0.)
(iii) We consider the case where 𝑇is self-adjoint (e.g., Example 5.1 below). In this case, we have ‖(𝐼−𝑃_ℎ)𝑇‖=‖𝑇(𝐼−𝑃_ℎ)‖. So, by (2.12) 𝜀_ℎ̂ is larger than ‖𝑇(𝐼−𝑃_ℎ)‖^1/2for small ℎ>0.

Remark 2.7. We mention applications of our results. Some of our results may be applicable for testing the quality of a numerical verification algorithm for solutions of differential equations. In general we obtain an upper bound of ‖𝑢_∗−𝑢_ℎ‖ as output data from a numerical verification algorithm (See e.g., [5] and the references therein). By our Theorem 2.4  ‖𝑢_∗−𝑢_ℎ‖ is sufficiently close to ‖𝑓(𝑢_ℎ)‖ for sufficiently small ℎ. So, Theorem 2.4 shows that we can check the accuracy of the output upper bound of ‖𝑢_∗−𝑢_ℎ‖ by finding the value of ‖𝑓(𝑢_ℎ)‖ when ℎ is small. In [5] we proposed a numerical verification algorithm which also gives upper bounds of ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖ as output data. Our Theorem 2.5 may be applicable for testing the accuracy of such upper bounds. See Remark 5.4 for more detailed information.

3. Preliminary Abstract Results

In this section, we prepare some abstract results in order to prove our main results in Section 2.

Proposition 3.1. We assume (H1). Let 𝐾∶𝑋→𝑋 be a compact operator. Then we have the following: 𝑃_ℎ𝐾⟶𝐾,𝐾𝑃_ℎ⟶𝐾inℒ(𝑋)asℎ⟶0,𝑃_ℎ𝐾𝑃_ℎ⟶𝐾inℒ(𝑋)asℎ⟶0. (3.1)(3.2)

Proof. Though this result was proved in [6, Section 78], we give a simpler proof for the convenience of the reader. First we show that 𝐾𝐼−𝑃_ℎ⟶0asℎ⟶0. (3.3) We proceed by contradiction. We assume that (3.3) does not hold. Then we have 𝛿∶=limsup_ℎ→0‖𝐾(𝐼−𝑃_ℎ)‖>0. Therefore, there exist {ℎ_𝑛}^∞_𝑛=1 and {𝑢_𝑛}^∞_𝑛=1⊂𝑋 such that ℎ_𝑛↘0as 𝑛→∞, ‖𝑢_𝑛‖=1for 𝑛∈𝐍and 𝐾𝐼−𝑃_ℎ𝑛𝑢_𝑛≥𝛿2forany𝑛∈𝐍. (3.4) Since 𝐾is compact and (𝐼−𝑃_ℎ𝑛)𝑢_𝑛 converges weakly to 0, we have ‖𝐾(𝐼−𝑃_ℎ𝑛)𝑢_𝑛‖→0as 𝑛→∞. This contradicts (3.4). So, (3.3) holds. Since 𝐾^∗is also compact, we obtain 𝐼−𝑃_ℎ𝐾=𝐼−𝑃_ℎ𝐾^∗=𝐾^∗𝐼−𝑃_ℎ⟶0as𝑛⟶∞. (3.5) So, we have (3.1), which implies (3.2).

Next, we describe some results in a more general setting. In what follows in this section, let 𝒳 and 𝒴 be Banach spaces and U⊂𝒳 be an open set. We assume 𝐹∈𝐶¹(U,𝒴).

Theorem 3.2. Let 𝑢₀∈U and 𝐿∈ℒ(𝒳,𝒴) be bijective. We define a map 𝑔∶U→𝒳 by 𝑔(𝑢)∶=𝑢−𝐿⁻¹𝐹(𝑢). (3.6) Let 𝑅>0 be a constant satisfying 𝐵(𝑢₀;𝑅)⊂U and 𝑏∶[0,𝑅]→[0,∞) be a non-decreasing function such that sup𝑔^′(𝑢);𝑢∈𝐵𝑢₀;𝑟≤𝑏(𝑟)forany𝑟∈[0,𝑅]. (3.7) Let 𝜀₀≥0 be a constant such that 𝐿⁻¹𝐹𝑢₀≤𝜀₀. (3.8) We assume that there exist constants 𝑟₀and 𝑟₁such that 0<𝑟₀≤𝑟₁≤𝑅, 𝜀₀+^𝑟0₀𝑏(𝑟)𝑑𝑟≤𝑟₀,𝑏𝑟₁<1. (3.9)(3.10) Then the equation 𝐹(𝑢)=0 has an isolated solution 𝑢_∗∈𝐵(𝑢₀;𝑟₀). Moreover, the solution of 𝐹(𝑢)=0 is unique in 𝐵(𝑢₀;𝑟₁).

Remark 3.3. (i) Theorem 3.2 is a new version of the convergence theorem of simplified Newton’s method, which is a refinement of the classical versions such as [5, Theorem  0.1]. Actually, the former implies the latter.
(ii) The convergence theorem of simplified Newton’s method is a very strong and general principle to verify the existence of isolated solutions. The reason is, roughly speaking, that the condition of the theorem is not only a sufficient condition to guarantee an isolated solution but also virtually a necessary condition for an isolated solution to exist. See [4, Remark  1.3] for the detail.

Proof of Theorem 3.2. Though we may consider Theorem 3.2 as a corollary of [5, Theorem  1.1], we describe the proof for completeness. We easily verify that 𝑢 is a solution of 𝐹(𝑢)=0 if and only if 𝑢 is a fixed point of 𝑔(𝑢). Let 𝑢,𝑣∈𝑈. We obtain 𝑔(𝑢)−𝑔(𝑣)=¹₀𝑑𝑔((𝑑𝑡𝑣+𝑡𝑢−𝑣))𝑑𝑡=¹₀𝑔^′(𝑣+𝑡(𝑢−𝑣))𝑑𝑡(𝑢−𝑣). (3.11) By (3.7) and (3.11) we have 𝑔(𝑢)−𝑔(𝑣)≤𝑏(𝑟)‖𝑢−𝑣‖forany𝑟∈(0,𝑅]and𝑢,𝑣∈𝐵𝑢₀;𝑟. (3.12) We set 𝐵₀∶=𝐵(𝑢₀;𝑟₀). Let 𝑢∈𝐵₀. In view of (3.7), (3.8), and (3.11) with 𝑣∶=𝑢₀, we have 𝑔𝑢₀−𝑢₀=𝐿⁻¹𝐹𝑢₀≤𝜀₀,𝑔(𝑢)−𝑔𝑢₀≤𝑟₀¹₀𝑏𝑟₀𝑡𝑑𝑡=^𝑟0₀𝑏(𝑟)𝑑𝑟. (3.13)(3.14) Combining (3.9), (3.13), and (3.14), we have ‖𝑔(𝑢)−𝑢₀‖≤𝑟₀, which implies 𝑔(𝐵₀)⊂𝐵₀. Therefore, in view of (3.10) and (3.12) 𝑔 is a contraction on 𝐵₀. By the contraction mapping principle there exists a unique solution 𝑢=𝑢_∗ on 𝐵₀ for the equation 𝐹(𝑢)=0. We immediately obtain from (3.10) and (3.12) that the solution of 𝐹(𝑢)=0 is unique on 𝐵(𝑢₀;𝑟₁). Finally, it suffices to show that 𝐹^′(𝑢)∶𝒳⟶𝒴isbijectiveforany𝑢∈𝐵𝑢₀;𝑟₁ (3.15) in order to prove that 𝑢_∗ is isolated. We denote by 𝐼 the identity operator on 𝒳. Let 𝑢∈𝐵(𝑢₀;𝑟₁). Then, by (3.7) and (3.10) we have ‖𝑔^′(𝑢)‖≤𝑏(𝑟₁)<1. This implies that 𝐼−𝑔^′(𝑢)∶𝒳→𝒳 is bijective. Since 𝐿 is also bijective and 𝐹^′(𝑢)=𝐿{𝐼−𝑔^′(𝑢)}, (3.15) holds.

The next result may be considered as a refinement of [7, Theorem  3.1 (3.14)] and [8, Theorem  3.1 (3.23)].

Proposition 3.4. Let 𝑢,𝑣∈U, (1−𝑠)𝑢+𝑠𝑣∈Ufor any 𝑠∈(0,1)and 𝐿∈ℒ(𝒳,𝒴) be bijective. We set 𝑚∶=max_𝑠∈[0,1]‖𝐿−𝐹′((1−𝑠)𝑢+𝑠𝑣)‖. Then we have 𝐿⁻¹{𝐹(𝑢)−𝐹(𝑣)}1+𝑚𝐿⁻¹≤‖𝑢−𝑣‖. (3.16) Moreover, if 𝑚‖𝐿⁻¹‖<1 then we also obtain ‖‖≤𝐿𝑢−𝑣⁻¹{𝐹(𝑢)−𝐹(𝑣)}1−𝑚𝐿⁻¹. (3.17)

Proof. The proof is similar to that of Theorem 3.2. Let 𝑔∶U→𝒳 be a map defined by (3.6). We have 𝑢−𝑣=𝑔(𝑢)−𝑔(𝑣)+𝐿⁻¹{𝐹(𝑢)−𝐹(𝑣)}. (3.18) It follows from (3.11) that ‖𝑔(𝑢)−𝑔(𝑣)‖≤𝑚‖𝐿⁻¹‖‖𝑢−𝑣‖. Combining this inequality and (3.18), we obtain (3.16) and (3.17).

Theorem 3.5. Let 𝑢=𝑢_∗∈U be an isolated solution of the equation 𝐹(𝑢)=0. Let ℎ₀>0 be a positive constant, 𝐹_ℎ∈𝐶¹(U,𝒴) and 𝐻_ℎ∈ℒ(𝒳,𝒴) (0<ℎ<ℎ₀). We set 𝐻∶=𝐹^′(𝑢_∗). We assume that 𝐹_ℎ𝑢_∗⟶0in𝒴asℎ⟶0,𝐹^′_ℎ𝑢_∗⟶𝐻,𝐻_ℎ⟶𝐻inℒ(𝒳,𝒴)asℎ⟶0,lim_𝑟↘0𝑑(𝑟,𝑢)=0forany𝑢∈U. (3.19)(3.20)(3.21) Here, 𝑑(𝑟,𝑢)∶=sup{‖𝐹^′_ℎ(𝑢)−𝐹^′_ℎ(𝑣)‖;0<ℎ<ℎ₀and𝑣∈U∩𝐵(𝑢;𝑟)}. Then, there exist a constant ℎ_∗∈(0,ℎ₀) and sequences {𝑐_ℎ}_{ℎ∈(0,ℎ∗)}⊂(1,2), {𝑢_ℎ}_{ℎ∈(0,ℎ∗)}⊂U such that the following (a)–(f) hold:(a)𝑐_ℎ⟶1asℎ⟶0, (3.22)(b)𝑢=𝑢_ℎ  is an isolated solution of𝐹_ℎ(𝑢)=0 for anyℎ∈(0,ℎ_∗),(c)𝑢_ℎ−𝑢_∗𝒳≤𝑐_ℎ𝐻_ℎ⁻¹𝐹_ℎ𝑢_∗𝒳foranyℎ∈0,ℎ_∗, (3.23)(d)𝐻_ℎ is bijective with ‖𝐻_ℎ−𝐹^′_ℎ(𝑢_∗)‖<1/2‖𝐻_ℎ⁻¹‖and‖𝐻_ℎ⁻¹‖<2‖𝐻⁻¹‖foranyℎ∈(0,ℎ_∗),(e) the solution of 𝐹_ℎ(𝑢)=0is unique in𝐵(𝑢_∗;𝑅_ℎ)foranyℎ∈(0,ℎ_∗), where 𝑅_ℎ∶=sup𝑅>0;𝐵𝑢_∗;𝑅⊂U,𝑑𝑅,𝑢_∗<1𝐻_ℎ⁻¹−𝐻_ℎ−𝐹^′_ℎ𝑢_∗>0, (3.24)(f)𝑐_ℎ𝐻_ℎ⁻¹𝐹_ℎ𝑢_∗<𝑅_ℎforanyℎ∈0,ℎ_∗. (3.25)

Proof. By (3.20) and the stability property of linear operators (e.g., [3, Corollary  2.4.1]), 𝐹^′_ℎ(𝑢_∗) and 𝐻_ℎ are bijective for sufficiently small ℎ>0 and 𝐹^′_ℎ(𝑢_∗)⁻¹→𝐻⁻¹, 𝐻_ℎ⁻¹→𝐻⁻¹ in ℒ(𝒴,𝒳) as ℎ→0. Let 𝜂_ℎ∶=‖𝐻_ℎ⁻¹𝐹_ℎ(𝑢_∗)‖ and 𝑔_ℎ(𝑢)∶=𝑢−𝐻_ℎ⁻¹𝐹_ℎ(𝑢). We set 𝑑(𝑟)∶=𝑑(𝑟,𝑢_∗) for 𝑟>0 and define 𝑏_ℎ(𝑟)∶=‖𝐻_ℎ⁻¹‖{‖𝐻_ℎ−𝐹^′_ℎ(𝑢_∗)‖+𝑑(𝑟)}. Let 𝑐_ℎ∶=1/{1−𝑏_ℎ(2𝜂_ℎ)}, 𝑟_ℎ∶=𝑐_ℎ𝜂_ℎ and 𝛿_ℎ∶=(1/2‖𝐻_ℎ⁻¹‖)−‖𝐻_ℎ−𝐹^′_ℎ(𝑢_∗)‖. Then, we easily verify that as ℎ→0, 𝜂_ℎ⟶0,𝑐_ℎ⟶1,𝑟_ℎ⟶0,𝛿_ℎ⟶12𝐻⁻¹(>0). (3.26) Therefore, there exist ℎ_∗∈(0,ℎ₀) and 𝜀∈(0,1) such that for any ℎ∈(0,ℎ_∗), 𝐹^′_ℎ(𝑢_∗) is bijective with (d), 1≤𝑐_ℎ<2, 𝑑(𝑟_ℎ+𝜀)<𝛿_ℎ and 𝐵(𝑢_∗;𝑟_ℎ+𝜀)⊂U. It follows that ‖𝑔^′_ℎ(𝑢)‖≤𝑏_ℎ(𝑟) for any ℎ∈(0,ℎ_∗), 𝑟>0, and 𝑢∈U∩𝐵(𝑢_∗;𝑟). We also have 𝑟_ℎ+𝜀≤𝑅_ℎ and 𝜂_ℎ+^𝑟ℎ₀𝑏_ℎ(𝑟)𝑑𝑟≤𝜂_ℎ+𝑟_ℎ𝑏_ℎ𝑟_ℎ≤𝑟_ℎ. (3.27) Let ℎ∈(0,ℎ_∗) and 𝑅∈(𝑟_ℎ,𝑅_ℎ). We apply Theorem 3.2 by setting 𝐹∶=𝐹_ℎ, 𝑢₀∶=𝑢_∗, 𝐿∶=𝐻_ℎ, 𝑏∶=𝑏_ℎ, 𝑟₀∶=𝑟_ℎ, 𝑟₁∶=𝑅 and 𝜀₀∶=𝜂_ℎ. Then, we obtain the desired conclusions.

Remark 3.6. Theorem 3.5 is related to [7, Theorem 3.1] and [8, Theorem 3.1]. Actually, their proofs are similar to ours. Our proof is based on the convergence theorem of simplified Newton’s method, from which they may be derived similarly.

4. Proofs of Main Theorems

We prove the results in Section 2. We use the notation (2.1).

Proof of Proposition 2.1. We apply Theorem 3.5 by putting 𝒳=𝒴∶=𝑋, 𝐹∶=𝑓, 𝐹_ℎ∶=𝑓_ℎ, 𝐻∶=𝐴 and 𝐻_ℎ∶=𝐴_ℎ. We show (3.19)–(3.21). By (H1) we have 𝑓_ℎ(𝑢_∗)=(𝐼−𝑃_ℎ)𝑢_∗→0 in 𝑋 as ℎ→0. Therefore, (3.19) holds. It follows from (H2) and Proposition 3.1 that 𝑓^′_ℎ𝑢_∗=𝐼−𝑃_ℎ𝑇⟶𝐼−𝑇=𝐴,𝐴_ℎ⟶𝐴inℒ(𝑋)asℎ⟶0. (4.1) So, (3.20) holds. Let 𝑟>0, 𝑢∈U and 𝑣∈U∩𝐵(𝑢;𝑟). Since 𝜑′ is continuous, we have ‖𝑓^′_ℎ(𝑢)−𝑓^′_ℎ(𝑣)‖≤‖𝜑′(𝑢)−𝜑′(𝑣)‖→0 as 𝑟↘0, which implies (3.21). Therefore, by Theorem 3.5, there exist a small constant ℎ_∗>0, {𝑢_ℎ}_{ℎ∈(0,ℎ∗)} and {𝐶_ℎ}_{ℎ∈(0,ℎ∗)} such that (a)–(f) with 𝑐_ℎ∶=𝐶_ℎ hold. So, we immediately obtain (ii) and 𝑢_ℎ→𝑢_∗ in 𝑋 as ℎ→0. Since 𝐴_ℎ⁻¹𝑓_ℎ(𝑢_∗)=𝑓_ℎ(𝑢_∗)=(𝐼−𝑃_ℎ)𝑢_∗, (a) and (c) imply (2.2). So, (iii) holds. In view of (d) and (e), we have (i) with 𝑅_∗∶=sup𝑅>0;𝐵𝑢_∗𝑑(𝑅)<1;𝑅⊂Uand4𝐴⁻¹, (4.2) where 𝑑(𝑅)∶=sup{‖{𝜑^′(𝑣)−𝜑^′(𝑢_∗)}‖;𝑣∈U∩𝐵(𝑢_∗;𝑅)}. The proof is complete.

Proof of Theorem 2.4. We set 𝑢(𝑠,ℎ)∶=(1−𝑠)𝑢_ℎ+𝑠𝑢_∗for simplicity. Proposition 2.1 (iii) implies max_𝑠∈[0,1]‖𝑢_∗−𝑢(𝑠,ℎ)‖=‖𝑢_∗−𝑢_ℎ‖→0 as ℎ→0. First we show (2.6). We have 𝑓(𝑢_ℎ)=−(𝐼−𝑃_ℎ)𝜑(𝑢_ℎ)=(𝐼−𝑃_ℎ)𝑓(𝑢_ℎ), 𝐴_ℎ⁻¹→𝐴⁻¹ in ℒ(𝑋) as ℎ→0and 𝑚_ℎ∶=max_𝑠∈[0,1]𝐴_ℎ(𝑢(≤𝐴−𝑓′𝑠,ℎ))_ℎ−𝐴+max_[]𝑠∈0,1𝑓^′𝑢_∗−𝑓^′(𝑢(𝑠,ℎ))⟶0asℎ⟶0. (4.3) Since 𝐴_ℎ𝑓(𝑢_ℎ)=𝑓(𝑢_ℎ), we have 𝐴_ℎ⁻¹𝑓(𝑢_ℎ)=𝑓(𝑢_ℎ). We apply Proposition 3.4 with 𝐿∶=𝐴_ℎ, 𝑢∶=𝑢_ℎ, 𝑣∶=𝑢_∗ and 𝐹∶=𝑓 to obtain 𝑓𝑢_ℎ1+𝑚_ℎ𝐴_ℎ⁻¹≤𝑢_ℎ−𝑢_∗≤𝑓𝑢_ℎ1−𝑚_ℎ𝐴_ℎ⁻¹forsmallℎ>0, (4.4) which implies ‖𝑢_ℎ−𝑢_∗‖≃‖𝑓(𝑢_ℎ)‖. We also have ‖𝑢_ℎ−𝑢_∗‖≃‖𝐴⁻¹𝑓(𝑢_ℎ)‖ by the above discussion with 𝐿∶=𝐴_ℎ replaced by 𝐿∶=𝐴. Next, we show (2.7). In the same way as above we apply Proposition 3.4 with 𝐿∶=𝐴 (resp., 𝐿∶=𝐴_ℎ), 𝑢∶=𝑢_ℎ, 𝑣∶=𝑃_ℎ𝑢_∗ and 𝐹∶=𝑓_ℎ to have 𝑃_ℎ𝑢_∗−𝑢_ℎ≃𝐴⁻¹𝑓_ℎ𝑃_ℎ𝑢_∗resp.,𝑃_ℎ𝑢_∗−𝑢_ℎ≃𝐴_ℎ⁻¹𝑓_ℎ𝑃_ℎ𝑢_∗. (4.5) Since 𝑓_ℎ(𝑃_ℎ𝑢_∗)=𝑃_ℎ𝑓(𝑃_ℎ𝑢_∗) and 𝐴_ℎ commutes with 𝑃_ℎ, we have ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖≃‖𝑃_ℎ𝐴_ℎ⁻¹𝑓(𝑃_ℎ𝑢_∗)‖. Combining (4.5) and 𝑓_ℎ(𝑃_ℎ𝑢_∗)=𝑃_ℎ{𝜑(𝑢_∗)−𝜑(𝑃_ℎ𝑢_∗)}, we obtain ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖∼‖𝑃_ℎ{𝜑(𝑢_∗)−𝜑(𝑃_ℎ𝑢_∗)}‖.

Proof of Theorem 2.5. We set 𝑢_∗(𝑠,ℎ)∶=(1−𝑠)𝑢_∗+𝑠𝑃_ℎ𝑢_∗ for simplicity.(i) It follows from (H2) and Proposition 3.1 that 𝜑′𝑢_∗𝐼−𝑃_ℎ⟶0. (4.6)By (H1) and the continuity of 𝜑′(𝑢) at 𝑢=𝑢_∗ we have sup_𝑠∈[0,1]𝜑′𝑢_∗−𝜑′𝑢_∗(𝑠,ℎ)⟶0asℎ⟶0. (4.7) We obtain (2.8) from (4.6) and (4.7).(ii) In the same way as (3.11) we have 𝜑𝑢_∗−𝜑𝑃_ℎ𝑢_∗=¹₀𝜑^′𝑢_∗(𝑠,ℎ)𝑑𝑠𝐼−𝑃_ℎ𝑢_∗. (4.8)By this equality, (2.7) and (2.9), we have (2.10).

Finally we derive (2.11) and (2.12).

Proof of (2.11) and (2.13). Without loss of generality we assume 𝜀_ℎ̂=𝜂²_ℎ+2𝜂_ℎ. First we derive (2.11). This proof is essentially the same as that of [1, Theorem 3.2]. It suffices to prove 𝑢_∗−𝑢_ℎ≤1+𝜂_ℎ𝐼−𝑃_ℎ𝑢_∗, (4.9) which implies (2.11) in view of ‖𝑢_∗−𝑢_ℎ‖²=‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖²+‖(𝐼−𝑃_ℎ)𝑢_∗‖². We have 𝐴𝑢_ℎ−𝑢_∗+𝐼−𝑃_ℎ𝑇𝑢_ℎ−𝑢_∗−𝑃_ℎ𝑇𝐼−𝑃_ℎ𝑢_ℎ−𝑢_∗−𝑃_ℎ𝜑𝑢_ℎ−𝜑𝑢_∗−𝑇𝑢_ℎ−𝑢_∗=−𝐴𝐼−𝑃_ℎ𝑢_∗−𝐼−𝑃_ℎ𝑇𝐼−𝑃_ℎ𝑢_∗ (4.10) It follows that 𝑢_ℎ−𝑢_∗≤𝑝_ℎ+𝑞_ℎ+𝑟_ℎ𝑢_ℎ−𝑢_∗+1+𝑝_ℎ𝐼−𝑃_ℎ𝑢_∗, (4.11) which implies (4.9). Next we derive (2.13). Since 𝐴⁻¹=𝐼+𝐾with 𝐾∶=𝑇(𝐼−𝑇)⁻¹, we obtain from Proposition 3.1 that 𝐴⁻¹𝐼−𝑃_ℎ𝑇−𝐼−𝑃_ℎ𝑇=𝐾𝐼−𝑃_ℎ𝑇≤𝐾𝐼−𝑃_ℎ𝐼−𝑃_ℎ𝑇=𝑜(1)𝐼−𝑃_ℎ𝑇asℎ⟶0. (4.12) So, (2.13) holds.

5. Concrete Examples

In this section we consider the following semilinear elliptic boundary value problem: −Δ𝑢=𝐺(𝑥,𝑢,∇𝑢)inΩwith𝑢=0on𝜕Ω, (5.1) where Ω is a bounded convex domain in 𝐑^𝑁 (𝑁≤3) with piecewise smooth boundary 𝜕Ω. We will rewrite (5.1) as the form (1.1) under the appropriate setting of functional spaces. We simply denote 𝐺(𝑢)∶=𝐺(⋅,𝑢,∇𝑢). We assume 𝐺∈𝐶¹(𝐻¹₀(Ω),𝐻⁻¹(Ω)). Let 𝐿∈ℒ(𝐻¹₀(Ω),𝐻⁻¹(Ω)) be the operator defined by 𝐿𝑢∶=−Δ𝑢. We set 𝑋∶=𝐻¹₀(Ω) with the norm ‖𝑢‖_𝑋∶=‖∇𝑢‖_𝐿²_(Ω) and 𝜑(𝑢)∶=𝐿⁻¹𝐺(𝑢). Then, we have 𝜑,𝑓∈𝐶¹(𝑋,𝑋). We can rewrite (5.1) as 𝑓(𝑢)=0. We choose 𝑋_ℎ as an approximate finite element subspace of 𝑋 with mesh size ℎ.

In what follows, we concentrate on the cases: Ω=(0,1)⊂𝐑 and Ω=(0,1)×(0,1)⊂𝐑^𝟐. We use finite element methods with piecewise linear and bilinear elements on the uniform (rectangular) mesh with mesh size ℎ=1/𝑛 (𝑛∈𝐍). Then, we have dim𝑋_ℎ=𝑛−1 in the 1-dimensional case and dim𝑋_ℎ=(𝑛−1)² in the 2-dimensional case. In this context the following basic estimates hold:𝐼−𝑃_ℎ𝑢_𝐿2(Ω)≤𝐶_aℎ‖𝑢‖_𝑋forany𝑢∈𝑋,𝐼−𝑃_ℎ𝑢_𝑋≤𝐶_bℎ‖Δ𝑢‖_𝐿²_(Ω)forany𝑢∈𝑋∩𝐻²(Ω),𝐼−𝑃_ℎ𝑢_𝐿∞(Ω)≤𝐶_cℎ‖Δ𝑢‖_𝐿²_(Ω)forany𝑢∈𝑋∩𝐻²(Ω), (5.2a)(5.2b)(5.2c)

where 𝐶_a, 𝐶_b, and 𝐶_c are some positive constants independent of ℎ and 𝑢. As in previous sections, we denote by 𝑢_∗ an isolated solution of 𝑓(𝑢)=0 and by 𝑢_ℎ a finite element solution of 𝑓(𝑢)=0 (i.e., a solution of 𝑓_ℎ(𝑢)=0) in a small neighborhood of 𝑢_∗. In view of Proposition 2.1, 𝑢_ℎ exists uniquely in a small neighborhood of 𝑢_∗ for sufficiently small ℎ>0. In our examples below we show that the following error estimate holds: 𝑃_ℎ𝑢_∗−𝑢_ℎ=𝑂ℎ²asℎ⟶0. (5.3) For simplicity we denote 𝑢_∗(𝑠,ℎ)∶=(1−𝑠)𝑢_∗+𝑠𝑃_ℎ𝑢_∗. We will derive (5.3) from Theorem 2.5 and the duality 𝜑^′𝑢_∗(𝑠,ℎ)𝐼−𝑃_ℎ=𝐼−𝑃_ℎ𝜑^′𝑢_∗(𝑠,ℎ)^∗. (5.4)

We now present two examples.

Example 5.1. We consider the following Burgers equation: −Δ𝑢+𝑢𝑢_𝑥=𝑔inΩ∶=(0,1)×(0,1)with𝑢=0on𝜕Ω. (5.5) Here, 𝑔(𝑥,𝑦)is a given function with 𝑔∈𝐿²(Ω). As mentioned above, we rewrite (5.5) as 𝑓(𝑢)∶=𝑢−𝜑(𝑢)=0. In the present case 𝜑∶𝑋→𝑋 is a nonlinear map defined by 𝜑(𝑢)∶=𝐿⁻¹(−𝑢𝑢_𝑥+𝑔). By the elliptic regularity property we have 𝑢_∗∈𝐻²(Ω) (see e.g., [9]). We will derive (5.3). Let 𝑢,𝑣∈𝑋. We easily verify that 𝜑^′(𝑣)𝑢=−𝐿⁻¹𝑣𝑢_𝑥+𝑣_𝑥𝑢,𝜑^′(𝑣)^∗𝑢=𝐿⁻¹𝑣𝑢_𝑥forany𝑢∈𝑋. (5.6) By (5.2b) we have 𝐼−𝑃_ℎ𝜑^′𝑢_∗(𝑠,ℎ)^∗𝑢≤𝐶_bℎ𝑢_∗(𝑠,ℎ)𝑢_{𝑥𝐿2(Ω)}≤𝐶_bℎ𝑢_∗(𝑠,ℎ)_𝐿∞(Ω)‖𝑢‖_𝑋. (5.7) It follows that 𝐼−𝑃_ℎ𝜑′𝑢_∗(𝑠,ℎ)^∗≤𝐶_bℎ𝑢_∗(𝑠,ℎ)_𝐿∞(Ω). (5.8) We obtain from (5.2c) that 𝑢_∗(𝑠,ℎ)_𝐿∞(Ω)≤𝑢_{∗𝐿∞(Ω)}+𝐶_cℎ𝑠Δ𝑢_∗𝐿2(Ω). (5.9) It follows from (5.4), (5.8), and (5.9) that sup_𝑠∈[0,1]𝜑^′𝑢_∗(𝑠,ℎ)𝐼−𝑃_ℎ≤𝐶_bℎ𝑢_{∗𝐿∞(Ω)}+𝐶_cℎΔ𝑢_∗𝐿2(Ω)∶=𝜀_ℎ. (5.10) By (5.10), (5.2b), and Theorem 2.5 we have (5.3).

Example 5.2. We consider the Emden equation −Δ𝑢=𝑢²inΩ∶=(0,1)×(0,1)with𝑢=0on𝜕Ω. (5.11) We omit the one-dimensional case since it is easier. We can treat the present case in a similar way to Example 5.1. We rewrite (5.11) as (1.1). In the present case 𝜑∶𝑋→𝑋 is defined by 𝜑(𝑢)∶=𝐿⁻¹(𝑢²). Let 𝑢,𝑣∈𝑋. We verify that 𝜑′(𝑣)𝑢=2𝐿⁻¹(𝑣𝑢) and that 𝜑^′(𝑣) is self-adjoint. By (5.2b) and Sobolev’s inequality we have 𝐼−𝑃_ℎ𝜑^′𝑢_∗(𝑠,ℎ)𝑢≤2𝐶_bℎ𝑢_∗(𝑠,ℎ)_𝐿4(Ω)‖𝑢‖_𝐿⁴_(Ω)≤𝐶ℎ𝑢_∗(𝑠,ℎ)‖𝑢‖≤𝐶ℎ𝑢_∗‖𝑢‖ (5.12) for any 𝑠∈[0,1] and 𝑢∈𝑋. Here, 𝐶>0 is a constant independent of 𝑠, ℎ, and 𝑢. It follows from this inequality and (5.4) that sup_𝑠∈[0,1]𝜑^′𝑢_∗(𝑠,ℎ)𝐼−𝑃_ℎ≤𝐶ℎ𝑢_∗∶=𝜀_ℎ. (5.13) By (5.13), (5.2b), and Theorem 2.5 we have (5.3).

Remark 5.3. This remark is related to Remark 2.6.
(i) As mentioned in Remark 2.6 (ii), sup_𝑠∈[0,1]‖𝜑^′((1−𝑠)𝑢_∗+𝑠𝑃_ℎ𝑢_∗)(𝐼−𝑃_ℎ)‖≈‖𝑇(𝐼−𝑃_ℎ)‖ holds in general. Actually, in Example 5.2 (resp., Example 5.1) our best possible upper bound of sup_𝑠∈[0,1]‖𝜑′(𝑢_∗(𝑠,ℎ))(𝐼−𝑃_ℎ)‖is the right-hand side of (5.13) (resp., (5.10)), which is just the same (resp., has the same order) as that of ‖𝑇(𝐼−𝑃_ℎ)‖.
(ii) We pointed out that our estimate (2.10) is in general sharper than (2.11), which is directly derived from the discussion in [1]. In order to show it concretely, we apply (2.11) to the equations in Examples 5.1 and 5.2. In both cases our best possible error estimate is the following: 𝑃_ℎ𝑢_∗−𝑢_ℎ=𝑂ℎ^3/2asℎ⟶0. (5.14) Compare (5.14) with (5.3), which is based on (2.10). Though we omit the detailed derivation of (5.14), we show here that we cannot obtain a better estimate than (5.14) if we use (2.11) as a basic estimate. By the same discussion in Examples 5.1 and 5.2 we have 𝐼−𝑃_ℎ𝜑^′𝑢_∗=𝑂(ℎ)asℎ⟶0, (5.15) which is our best possible upper estimate of ‖(𝐼−𝑃_ℎ)𝑇‖. So, in view of (2.13), it is necessary to take 𝜀_ℎ̂ such that 𝜀_ℎ̂≥𝐶ℎ^1/2 for small ℎ. Here, 𝐶>0 is a constant independent of ℎ. (Compare this estimate with (5.10) and (5.13).) Therefore, we cannot improve (5.14) if we use (2.11) and (5.2b) as basic estimates.

Remark 5.4. Various numerical verification algorithms for solutions of differential equations were proposed up to now (see e.g., [10]). Some of them give upper bounds of ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖ as output data (see [5]). Theorem 2.5 may be applicable for checking the accuracy of such output upper bounds since we can apply it to given problems in order to compute the concrete order of ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖ as ℎ→0. For example, we treated problems (5.5) and (5.11) as concrete numerical examples in [5], where we proposed a numerical verification algorithm based on a convergence theorem of Newton’s method. In these problems (5.3) is the theoretical estimate of ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖ derived from our Theorem 2.5. The output data as upper bounds of ‖𝑃_ℎ𝑢_∗−𝑢_ℎ‖ in [5, Section 3] seem to have just the order of ℎ² as ℎ→0. So, the accuracy of such output upper bounds in [5, Section 3] is satisfactory as long as we judge it by the theoretical estimate (5.3).

Acknowledgments

The author would like to express his sincere gratitude to Professor Takuya Tsuchiya and Professor Atsushi Yagi for their valuable comments and encouragement. He is grateful to the referee for constructive comments.