# Error Analysis of Galerkin's Method for Semilinear Equations

Authors: Tadashi Kawanago
Journal: Journal of Applied Mathematics
 (2012)

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## 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.


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## Body

## 1. Introduction

LetX be a real Hilbert space and Xh⊂X be a closed subspace. Here, h is a positive parameter (which will tend to zero). We denote by Ph the orthogonal projection onto Xh. We assume that
(H1)lim⁡h→0⁡∥(I-Ph)u∥=0foranyu∈X,
where I:X→X is the identity operator. We are interested in studying error analysis of Galerkin’s method for the following equation:
(1.1)f(u)∶=u-φ(u)=0.
Here, φ:U→X is a nonlinear map and U is a subset of X. We define f~h:U∩Xh→Xh by
(1.2)f~h(u)∶=Phf(u)foru∈U∩Xh.
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 equationf~h(u)=0 has a solution uh in U∩Xh for any h and there exists a monotone decreasing sequence {hk}k=1∞ with lim⁡k→∞hk=0 and u∞∈U such that uhk→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 limh→0uh=u∞ in X.(ii)
Letu*∈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 uh∈Xh for any h, which is unique for sufficiently small h and
(1.3)∥u*-uh∥≃∥(I-Ph)u*∥,
which means that ∥u*-uh∥and ∥(I-Ph)u*∥are equivalent infinitesimals as h→0.In this paper, we always assume(H1) and the following (H2) in what follows:
(H2)φ∈C1(U,X),andφ′(u)∈ℒ(X)iscompactforanyu∈U,
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 definefh:U→X by
(1.4)fh(u)∶=u-Phφ(u)foru∈U.
The map fh 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 fh(u)=0. We can treat the equation fh(u)=0 more easily than f~h(u)=0 since fh 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 Remark2.7 for further information.The paper is organized as follows. In Section2 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.
Let𝒳 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 B𝒳(u;r):={v∈𝒳;∥v-u∥<r}. The subscript will be often omitted if no possible confusion arises.(2)
For an open setV⊂𝒳, C1(V,𝒴) denotes the space of continuously differentiable functions from V to 𝒴.(3)
We denote byℒ(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)
Letϕ(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)
LetΩ be a bounded domain of Rn. We denote Lebesgue spaces by Lp(Ω)(1≤p≤∞) with the norms ∥u∥Lp(Ω)=(∫Ω|u(x)|pdx)1/p  for1≤p<∞,∥u∥L∞(Ω)=ess.sup⁡⁡{|u(x)|;x∈Ω}. We denote by H01(Ω) the completion of C0∞(Ω) (the space of C∞ functions with compact support in Ω) in the Sobolev norm: ∥u∥=∥∇u∥L2(Ω)∶=(∑k=1n‍∥∂u/∂xk∥L2(Ω)2)1/2. We denote by H-1(Ω) the Sobolev space {F∈𝒟′(Ω);∃C∈(0,∞)suchthat|F(ϕ)|≤C∥ϕ∥H01(Ω)foranyϕ∈C0∞(Ω)} with the norm ∥F∥H-1(Ω)=sup⁡⁡{|F(ϕ)|;ϕ∈C0∞(Ω)and∥ϕ∥H01(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
(2.1)T∶=φ′(u*),A∶=I-T=f′(u*),Ah∶=I-PhTPh,
for simplicity. The operator Ah 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 existh*>0and {uh}h∈(0,h*)⊂Usuch that the following (i)–(iii) hold.(i)
There existsR*>0such thatu=uhis the only solution offh(u)=0inB(u*;R*)for anyh∈(0,h*).(ii)
u=uhis an isolated solution offh(u)=0for anyh∈(0,h*).(iii)
uh→u*inXash→0with the estimate(2.2)∥u*-uh∥≤Ch∥fh(u*)∥=Ch∥(I-Ph)u*∥foranyh∈(0,h*),where{Ch}h∈(0,h*)⊂(1,2)andCh→1 as h→0.Remark 2.2.
(i) Proposition2.1(ii) is useful in our analysis below. Moreover, we immediately obtain from it that u=uhis an isolated solution of f~h(u)=0for any h∈(0,h*). This guarantees that we can always construct a Galerkin approximate solution uh by Newton’s method for small h>0.
(ii) In various contexts in applications,Xh is finite-dimensional for any h. In such contexts the assumption (H1) implies that X is separable.
(iii) We do not assumedim⁡Xh<∞. We briefly explain that it has some practical benefits. The case dim⁡Xh=∞ 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,{uh}h∈(0,h*)always denotes the sequence as described in Proposition 2.1. Since u*-uhis decomposed into the Xh-component Phu*-uhand the Xh⊥-component (I-Ph)u*, we have ∥u*-uh∥2=∥Phu*-uh∥2+∥(I-Ph)u*∥2and ∥(I-Ph)u*∥≤∥u*-uh∥. So, the last inequality and (2.2) immediately imply (2.3) below.Corollary 2.3.
We have(2.3)∥u*-uh∥≃∥fh(u*)∥=∥(I-Ph)u*∥,(2.4)∥Phu*-uh∥=o(∥(I-Ph)u*∥)ash⟶0,(2.5)∥Phu*-uh∥=o(∥u*-uh∥)ash⟶0.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 Xh-component of the error ∥Phu*-uh∥is an infinitesimal of a higher order of smallness with respect to the whole error ∥u*-uh∥as h→0.The following two results are useful for applications (see Remark2.7 below).Theorem 2.4.
We have the following:(2.6)∥u*-uh∥≃∥f(uh)∥=∥(I-Ph)φ(uh)∥≃∥A-1f(uh)∥,(2.7)∥Phu*-uh∥≃∥A-1fh(Phu*)∥≃∥PhAh-1f(Phu*)∥~∥Ph{φ(u*)-φ(Phu*)}∥.Theorem 2.5.
(i) We have(2.8)sup⁡s∈[0,1]⁡∥φ′((1-s)u*+sPhu*)(I-Ph)∥⟶0ash⟶0.
(ii) Letɛh be a positive constant for h∈(0,h*) such that
(2.9)εh≥sup⁡s∈[0,1]⁡∥φ′((1-s)u*+sPhu*)(I-Ph)∥
for any h∈(0,h*). Then, there exist constants h1∈(0,h*) and C1>0 such that
(2.10)∥Phu*-uh∥≤C1εh∥(I-Ph)u*∥foranyh∈(0,h1).In view of Theorem2.5 (i) and (ii), we can always take {εh}h∈(0,h*) in (2.10) such that εh→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 ηh∶=(2ph+qh+rh)/(1-(ph+qh+rh)), ph∶=∥A-1(I-Ph)T∥, qh∶=∥A-1PhT(I-Ph)∥and rh∶=∥A-1∥·∥φ(uh)-φ(u*)-T(uh-u*)∥/∥uh-u*∥. It follows from Proposition 2.1 (iii) and Proposition 3.1 below that ph, qh and rh converge to 0 as h→0. So, ηk→0 as h→0. Let ε^h be a positive constant for h∈(0,h*) such that ε^h≥ηh2+2ηh. Then we have
(2.11)∥Phu*-uh∥≤ε^h∥(I-Ph)u*∥foranyh∈(0,h*).
We can verify that
(2.12)ε^hislargerthan∥(I-Ph)T∥1/2forsufficientlysmallh>0.
Indeed, we immediately obtain (2.12) from
(2.13)ph≃∥(I-Ph)T∥.
We derive (2.11) and (2.13) at the end of Section 4.
(ii) When we computeɛ^h for concrete examples (e.g., examples in Section 5 below), it seems reasonable to estimate qhas qh≤C∥T(I-Ph)∥. Here, Crepresents some positive constant independent of h. Then, it is actually necessary to take ɛ^hsuch that ɛ^h≥C∥T(I-Ph)∥1/2for small h>0. On the other hand, roughly speaking, (2.9) means that we can take ɛh≈∥T(I-Ph)∥for small h>0 (See Remark 5.3 below). (We note that Proposition 3.1 below implies that ∥T(I-Ph)∥→0as h→0.)
(iii) We consider the case whereTis self-adjoint (e.g., Example 5.1 below). In this case, we have ∥(I-Ph)T∥=∥T(I-Ph)∥. So, by (2.12) ɛ^h is larger than ∥T(I-Ph)∥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*-uh∥ as output data from a numerical verification algorithm (See e.g., [5] and the references therein). By our Theorem 2.4  ∥u*-uh∥ is sufficiently close to ∥f(uh)∥ for sufficiently small h. So, Theorem 2.4 shows that we can check the accuracy of the output upper bound of ∥u*-uh∥ by finding the value of ∥f(uh)∥ when h is small. In [5] we proposed a numerical verification algorithm which also gives upper bounds of ∥Phu*-uh∥ 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 Section2.Proposition 3.1.
We assume(H1). Let K:X→X be a compact operator. Then we have the following:
(3.1)PhK⟶K,KPh⟶Kinℒ(X)ash⟶0,(3.2)PhKPh⟶Kinℒ(X)ash⟶0.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
(3.3)∥K(I-Ph)∥⟶0ash⟶0.
We proceed by contradiction. We assume that (3.3) does not hold. Then we have δ∶=limsup⁡h→0∥K(I-Ph)∥>0. Therefore, there exist {hn}n=1∞ and {un}n=1∞⊂X such that hn↘0as n→∞, ∥un∥=1for n∈Nand
(3.4)∥K(I-Phn)un∥≥δ2foranyn∈N.
Since Kis compact and (I-Phn)un converges weakly to 0, we have ∥K(I-Phn)un∥→0as n→∞. This contradicts (3.4). So, (3.3) holds. Since K*is also compact, we obtain
(3.5)∥(I-Ph)K∥=∥{(I-Ph)K}*∥=∥K*(I-Ph)∥⟶0asn⟶∞.
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, let𝒳 and 𝒴 be Banach spaces and U⊂𝒳 be an open set. We assume F∈C1(U,𝒴).Theorem 3.2.
Letu0∈U and L∈ℒ(𝒳,𝒴) be bijective. We define a map g:U→𝒳 by
(3.6)g(u):=u-L-1F(u).
Let R>0 be a constant satisfying B(u0;R)¯⊂U and b:[0,R]→[0,∞) be a non-decreasing function such that
(3.7)sup⁡⁡{∥g′(u)∥;u∈B(u0;r)¯}≤b(r)foranyr∈[0,R].
Let ε0≥0 be a constant such that
(3.8)∥L-1F(u0)∥≤ε0.
We assume that there exist constants r0and r1such that 0<r0≤r1≤R,
(3.9)ε0+∫0r0‍b(r)dr≤r0,(3.10)b(r1)<1.
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) Theorem3.2 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 anecessary condition for an isolated solution to exist. See [4, Remark  1.3] for the detail.Proof of Theorem3.2.
Though we may consider Theorem3.2 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
(3.11)g(u)-g(v)=∫01ddtg(v+t(u-v))dt=∫01g′(v+t(u-v))dt(u-v).
By (3.7) and (3.11) we have
(3.12)∥g(u)-g(v)∥≤b(r)∥u-v∥foranyr∈(0,R]andu,v∈B(u0;r)¯.
We set B0∶=B(u0;r0)¯. Let u∈B0. In view of (3.7), (3.8), and (3.11) with v∶=u0, we have
(3.13)∥g(u0)-u0∥=∥L-1F(u0)∥≤ε0,(3.14)∥g(u)-g(u0)∥≤r0∫01‍b(r0t)dt=∫0r0‍b(r)dr.
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
(3.15)F′(u):𝒳⟶𝒴isbijectiveforanyu∈B(u0;r1)¯
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.
Letu,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
(3.16)∥L-1{F(u)-F(v)}∥1+m∥L-1∥≤∥u-v∥.
Moreover, if m∥L-1∥<1 then we also obtain
(3.17)∥u-v∥≤∥L-1{F(u)-F(v)}∥1-m∥L-1∥.Proof.
The proof is similar to that of Theorem3.2. Let g:U→𝒳 be a map defined by (3.6). We have
(3.18)u-v=g(u)-g(v)+L-1{F(u)-F(v)}.
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.
Letu=u*∈U be an isolated solution of the equation F(u)=0. Let h0>0 be a positive constant, Fh∈C1(U,𝒴) and Hh∈ℒ(𝒳,𝒴)(0<h<h0). We set H∶=F′(u*). We assume that
(3.19)Fh(u*)⟶0in𝒴ash⟶0,(3.20)Fh′(u*)⟶H,Hh⟶Hinℒ(𝒳,𝒴)ash⟶0,(3.21)lim⁡r↘0⁡d(r,u)=0foranyu∈U.
Here, d(r,u):=sup⁡{∥Fh′(u)-Fh′(v)∥;0<h<h0andv∈U∩B(u;r)¯}. Then, there exist a constant h*∈(0,h0) and sequences {ch}h∈(0,h*)⊂(1,2), {uh}h∈(0,h*)⊂U such that the following (a)–(f)  hold:(a)
(3.22)ch⟶1ash⟶0,(b)
u=uh  is an isolated solution ofFh(u)=0 for anyh∈(0,h*),(c)
(3.23)∥uh-u*∥𝒳≤ch∥Hh-1Fh(u*)∥𝒳foranyh∈(0,h*),(d)
Hh is bijective with ∥Hh-Fh′(u*)∥<1/2∥Hh-1∥and∥Hh-1∥<2∥H-1∥foranyh∈(0,h*),(e)
the solution ofFh(u)=0is unique inB(u*;Rh)foranyh∈(0,h*), where
(3.24)Rh:=sup⁡{R>0;B(u*;R)¯⊂U,d(R,u*)<1∥Hh-1∥-∥Hh-Fh′(u*)∥}>0,(f)
(3.25)ch∥Hh-1Fh(u*)∥<Rhforanyh∈(0,h*).Proof.
By (3.20) and the stability property of linear operators (e.g., [3, Corollary  2.4.1]), Fh′(u*) and Hh are bijective for sufficiently small h>0 and Fh′(u*)-1→H-1, Hh-1→H-1 in ℒ(𝒴,𝒳) as h→0. Let ηh:=∥Hh-1Fh(u*)∥ and gh(u):=u-Hh-1Fh(u). We set d(r):=d(r,u*) for r>0 and define bh(r):=∥Hh-1∥{∥Hh-Fh′(u*)∥+d(r)}. Let ch:=1/{1-bh(2ηh)}, rh:=chηh and δh:=(1/2∥Hh-1∥)-∥Hh-Fh′(u*)∥. Then, we easily verify that as h→0,
(3.26)ηh⟶0,ch⟶1,rh⟶0,δh⟶12∥H-1∥(>0).
Therefore, there exist h*∈(0,h0) and ɛ∈(0,1) such that for any h∈(0,h*), Fh′(u*) is bijective with (d), 1≤ch<2, d(rh+ε)<δh and B(u*;rh+ε)¯⊂U. It follows that ∥gh′(u)∥≤bh(r) for any h∈(0,h*), r>0, and  u∈U∩B(u*;r)¯. We also have rh+ε≤Rh and
(3.27)ηh+∫0rh‍bh(r)dr≤ηh+rhbh(rh)≤rh.
Let h∈(0,h*) and R∈(rh,Rh). We apply Theorem 3.2 by setting F:=Fh, u0:=u*, L:=Hh, b:=bh, r0:=rh, r1:=R and ε0:=ηh. Then, we obtain the desired conclusions.Remark 3.6.
Theorem3.5 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 Section2. We use the notation (2.1).Proof of Proposition2.1.
We apply Theorem3.5 by putting 𝒳=𝒴:=X, F:=f, Fh:=fh, H:=A and Hh:=Ah. We show (3.19)–(3.21). By (H1) we have fh(u*)=(I-Ph)u*→0 in X as h→0. Therefore, (3.19) holds. It follows from (H2) and Proposition 3.1 that
(4.1)fh′(u*)=I-PhT⟶I-T=A,Ah⟶Ainℒ(X)ash⟶0.
So, (3.20) holds. Let r>0, u∈U and v∈U∩B(u;r)¯. Since φ′ is continuous, we have ∥fh′(u)-fh′(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, {uh}h∈(0,h*) and {Ch}h∈(0,h*) such that (a)–(f) with ch:=Ch hold. So, we immediately obtain (ii) and uh→u* in X as h→0. Since Ah-1fh(u*)=fh(u*)=(I-Ph)u*, (a) and (c) imply (2.2). So, (iii) holds. In view of (d) and (e), we have (i) with
(4.2)R*:=sup⁡{R>0;B(u*;R)¯⊂Uandd^(R)<14∥A-1∥},
where d^(R):=sup⁡{∥{φ′(v)-φ′(u*)}∥;v∈U∩B(u*;R)¯}. The proof is complete.Proof of Theorem2.4.
We setu(s,h):=(1-s)uh+su*for simplicity. Proposition 2.1 (iii) implies max⁡s∈[0,1]∥u*-u(s,h)∥=∥u*-uh∥→0 as h→0. First we show (2.6). We have f(uh)=-(I-Ph)φ(uh)=(I-Ph)f(uh), Ah-1→A-1 in ℒ(X) as h→0and
(4.3)mh:=max⁡s∈[0,1]∥Ah-f′(u(s,h))∥≤∥Ah-A∥+max⁡s∈[0,1]∥f′(u*)-f′(u(s,h))∥⟶0ash⟶0.
Since Ahf(uh)=f(uh), we have Ah-1f(uh)=f(uh). We apply Proposition 3.4 with L:=Ah, u:=uh, v:=u* and F:=f to obtain
(4.4)∥f(uh)∥1+mh∥Ah-1∥≤∥uh-u*∥≤∥f(uh)∥1-mh∥Ah-1∥forsmallh>0,
which implies ∥uh-u*∥≃∥f(uh)∥. We also have ∥uh-u*∥≃∥A-1f(uh)∥ by the above discussion with L:=Ah 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:=Ah), u:=uh, v:=Phu* and F:=fh to have
(4.5)∥Phu*-uh∥≃∥A-1fh(Phu*)∥(resp.,∥Phu*-uh∥≃∥Ah-1fh(Phu*)∥).
Since fh(Phu*)=Phf(Phu*) and Ah commutes with Ph, we have ∥Phu*-uh∥≃∥PhAh-1f(Phu*)∥. Combining (4.5) and fh(Phu*)=Ph{φ(u*)-φ(Phu*)}, we obtain ∥Phu*-uh∥~∥Ph{φ(u*)-φ(Phu*)}∥.Proof of Theorem2.5.
We setu*(s,h):=(1-s)u*+sPhu* for simplicity.(i)
It follows from (H2) and Proposition3.1 that
(4.6)∥φ′(u*)(I-Ph)∥⟶0.By (H1) and the continuity of φ′(u) at u=u* we have
(4.7)sup⁡s∈[0,1]∥φ′(u*)-φ′(u*(s,h))∥⟶0ash⟶0.
We obtain (2.8) from (4.6) and (4.7).(ii)
In the same way as (3.11) we have
(4.8)φ(u*)-φ(Phu*)=∫01‍φ′(u*(s,h))ds(I-Ph)u*.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 assumeε^h=ηh2+2ηh. First we derive (2.11). This proof is essentially the same as that of [1, Theorem 3.2]. It suffices to prove
(4.9)∥u*-uh∥≤(1+ηh)∥(I-Ph)u*∥,
which implies (2.11) in view of ∥u*-uh∥2=∥Phu*-uh∥2+∥(I-Ph)u*∥2. We have
(4.10)A(uh-u*)+(I-Ph)T(uh-u*)-PhT(I-Ph)(uh-u*)-Ph{φ(uh)-φ(u*)-T(uh-u*)}=-A(I-Ph)u*-(I-Ph)T(I-Ph)u*
It follows that
(4.11)∥uh-u*∥≤(ph+qh+rh)∥uh-u*∥+(1+ph)∥(I-Ph)u*∥,
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
(4.12)∥A-1(I-Ph)T-(I-Ph)T∥=∥K(I-Ph)T∥≤∥K(I-Ph)∥∥(I-Ph)T∥=o(1)∥(I-Ph)T∥ash⟶0.
So, (2.13) holds.


## 5. Concrete Examples

In this section we consider the following semilinear elliptic boundary value problem:(5.1)-Δu=G(x,u,∇u)inΩwithu=0on⁡∂Ω,
where Ω is a bounded convex domain in RN (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(H01(Ω),H-1(Ω)). Let L∈ℒ(H01(Ω),H-1(Ω)) be the operator defined by Lu:=-Δu. We set X:=H01(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 Xh 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)⊂R2. 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 dim⁡Xh=n-1 in the 1-dimensional case and dim⁡Xh=(n-1)2 in the 2-dimensional case. In this context the following basic estimates hold:(5.2a)∥(I-Ph)u∥L2(Ω)≤Cah∥u∥Xforanyu∈X,(5.2b)∥(I-Ph)u∥X≤Cbh∥Δu∥L2(Ω)foranyu∈X∩H2(Ω),(5.2c)∥(I-Ph)u∥L∞(Ω)≤Cch∥Δu∥L2(Ω)foranyu∈X∩H2(Ω),whereCa, Cb, and Cc 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 uh 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, uh 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:
(5.3)∥Phu*-uh∥=O(h2)ash⟶0.
For simplicity we denote u*(s,h)∶=(1-s)u*+sPhu*. We will derive (5.3) from Theorem 2.5 and the duality
(5.4)∥φ′(u*(s,h))(I-Ph)∥=∥(I-Ph)φ′(u*(s,h))*∥.We now present two examples.Example 5.1.
We consider the following Burgers equation:(5.5)-Δu+uux=ginΩ∶=(0,1)×(0,1)withu=0on⁡∂Ω.
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(-uux+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
(5.6)φ′(v)u=-L-1(vux+vxu),φ′(v)*u=L-1(vux)foranyu∈X.
By (5.2b) we have
(5.7)∥(I-Ph)φ′(u*(s,h))*u∥≤Cbh∥u*(s,h)ux∥L2(Ω)≤Cbh∥u*(s,h)∥L∞(Ω)∥u∥X.
It follows that
(5.8)∥(I-Ph)φ′(u*(s,h))*∥≤Cbh∥u*(s,h)∥L∞(Ω).
We obtain from (5.2c) that
(5.9)∥u*(s,h)∥L∞(Ω)≤∥u*∥L∞(Ω)+Cchs∥Δu*∥L2(Ω).
It follows from (5.4), (5.8), and (5.9) that
(5.10)sup⁡s∈[0,1]∥φ′(u*(s,h))(I-Ph)∥≤Cbh(∥u*∥L∞(Ω)+Cch∥Δu*∥L2(Ω)):=εh.
By (5.10), (5.2b), and Theorem 2.5 we have (5.3).Example 5.2.
We consider the Emden equation(5.11)-Δu=u2inΩ:=(0,1)×(0,1)withu=0on⁡∂Ω.
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
(5.12)∥(I-Ph)φ′(u*(s,h))u∥≤2Cbh∥u*(s,h)∥L4(Ω)∥u∥L4(Ω)≤Ch∥u*(s,h)∥∥u∥≤Ch∥u*∥∥u∥
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
(5.13)sup⁡s∈[0,1]∥φ′(u*(s,h))(I-Ph)∥≤Ch∥u*∥∶=εh.
By (5.13), (5.2b), and Theorem 2.5 we have (5.3).Remark 5.3.
This remark is related to Remark2.6.
(i) As mentioned in Remark2.6 (ii), sup⁡s∈[0,1]∥φ′((1-s)u*+sPhu*)(I-Ph)∥≈∥T(I-Ph)∥ 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-Ph)∥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-Ph)∥.
(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:
(5.14)∥Phu*-uh∥=O(h3/2)ash⟶0.
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
(5.15)∥(I-Ph)φ′(u*)∥=O(h)ash⟶0,
which is our best possible upper estimate of ∥(I-Ph)T∥. So, in view of (2.13), it is necessary to take ε^h such that ε^h≥Ch1/2 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 ∥Phu*-uh∥ 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 ∥Phu*-uh∥ 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 ∥Phu*-uh∥ derived from our Theorem 2.5. The output data as upper bounds of ∥Phu*-uh∥ 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).



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