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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 298640, 15 pages
http://dx.doi.org/10.1155/2012/298640
Research Article

Error Analysis of Galerkin's Method for Semilinear Equations

Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

Received 7 January 2012; Revised 16 May 2012; Accepted 8 June 2012

Academic Editor: Hui-Shen Shen

Copyright © 2012 Tadashi Kawanago. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 be a real Hilbert space and be a closed subspace. Here, is a positive parameter (which will tend to zero). We denote by the orthogonal projection onto . We assume that where is the identity operator. We are interested in studying error analysis of Galerkin’s method for the following equation: Here, is a nonlinear map and is a subset of . We define by The equation 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 . Let be a constant and . One assumes that is a completely continuous map such that . Then the following holds.(i)The equation has a solution in for any and there exists a monotone decreasing sequence with and such that in as and is a solution of (1.1). Moreover, if is the unique solution of (1.1) in , then one has in . (ii)Let be a solution of (1.1). If has a Fréchet derivative, in a neighborhood, , of and 0 is not in the spectrum of , then is the unique solution of (1.1) in and has a solution for any , which is unique for sufficiently small and which means that and are equivalent infinitesimals as .

In this paper, we always assume and the following in what follows: where is an open set. Under the conditions and , 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 under an appropriate setting of the functional spaces. See Section 5 for some concrete examples.

We define by The map is a natural extension of and is very useful in our analysis below. Obviously, is a solution of if and only if is a solution of . We can treat the equation more easily than since 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 and be Banach spaces. (1)We denote by the norm of . If is a Hilbert space, then stands for the norm induced by the inner product of . For and , we write . The subscript will be often omitted if no possible confusion arises. (2)For an open set , denotes the space of continuously differentiable functions from to .(3)We denote by the space of bounded linear operators from to and stands for . For , denotes the operator norm of . The subscript will be omitted if no possible confusion arises.(4)Let and be nonnegative functions. We write if and are infinitesimals of the same order as , that is, and as . We write if and are equivalent infinitesimals as , that is, as .(5)Let be a bounded domain of . We denote Lebesgue spaces by with the norms   for . We denote by the completion of (the space of functions with compact support in ) in the Sobolev norm: . We denote by the Sobolev space with the norm . Here, stands for the set of distributions on .

2. Main Results

In this section we describe our main results. We assume and . Let be an isolated solution of (1.1), that is, is a solution of (1.1) such that is bijective. We set for simplicity. The operator 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 and such that the following (i)–(iii) hold.(i)There exists such that is the only solution of in for any .(ii) is an isolated solution of for any . (iii) with the estimate where and as .

Remark 2.2. (i) Proposition 2.1(ii) is useful in our analysis below. Moreover, we immediately obtain from it that is an isolated solution of for any . This guarantees that we can always construct a Galerkin approximate solution by Newton’s method for small .
(ii) In various contexts in applications, is finite-dimensional for any . In such contexts the assumption implies that is separable.
(iii) We do not assume . We briefly explain that it has some practical benefits. The case 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, always denotes the sequence as described in Proposition 2.1. Since is decomposed into the -component and the -component , we have and . So, the last inequality and (2.2) immediately imply (2.3) below.

Corollary 2.3. We have

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 -component of the error is an infinitesimal of a higher order of smallness with respect to the whole error as .

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

Theorem 2.4. We have the following:

Theorem 2.5. (i) We have
(ii) Let be a positive constant for such that for any . Then, there exist constants and such that

In view of Theorem 2.5 (i) and (ii), we can always take in (2.10) such that as . 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 , , and . It follows from Proposition 2.1 (iii) and Proposition 3.1 below that , and converge to 0 as . So, as . Let be a positive constant for such that . Then we have We can verify that Indeed, we immediately obtain (2.12) from We derive (2.11) and (2.13) at the end of Section 4.
(ii) When we compute for concrete examples (e.g., examples in Section 5 below), it seems reasonable to estimate as . Here, represents some positive constant independent of . Then, it is actually necessary to take such that for small . On the other hand, roughly speaking, (2.9) means that we can take for small (See Remark 5.3 below). (We note that Proposition 3.1 below implies that as .)
(iii) We consider the case where is self-adjoint (e.g., Example 5.1 below). In this case, we have . So, by (2.12) is larger than for small .

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 as output data from a numerical verification algorithm (See e.g., [5] and the references therein). By our Theorem 2.4   is sufficiently close to for sufficiently small . So, Theorem 2.4 shows that we can check the accuracy of the output upper bound of by finding the value of when is small. In [5] we proposed a numerical verification algorithm which also gives upper bounds of 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 . Let be a compact operator. Then we have the following:

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 We proceed by contradiction. We assume that (3.3) does not hold. Then we have . Therefore, there exist and such that as , for and Since is compact and converges weakly to , we have as . This contradicts (3.4). So, (3.3) holds. Since is also compact, we obtain 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 and be Banach spaces and be an open set. We assume .

Theorem 3.2. Let and be bijective. We define a map by Let be a constant satisfying and be a non-decreasing function such that Let be a constant such that We assume that there exist constants and such that , Then the equation has an isolated solution . Moreover, the solution of is unique in .

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 is a solution of if and only if is a fixed point of . Let . We obtain By (3.7) and (3.11) we have We set . Let . In view of (3.7), (3.8), and (3.11) with , we have Combining (3.9), (3.13), and (3.14), we have , which implies . Therefore, in view of (3.10) and (3.12) is a contraction on . By the contraction mapping principle there exists a unique solution on for the equation . We immediately obtain from (3.10) and (3.12) that the solution of is unique on . Finally, it suffices to show that in order to prove that is isolated. We denote by the identity operator on . Let . Then, by (3.7) and (3.10) we have . This implies that is bijective. Since is also bijective and , (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 , for any and be bijective. We set . Then we have Moreover, if then we also obtain

Proof. The proof is similar to that of Theorem 3.2. Let be a map defined by (3.6). We have It follows from (3.11) that . Combining this inequality and (3.18), we obtain (3.16) and (3.17).

Theorem 3.5. Let be an isolated solution of the equation . Let be a positive constant, and . We set . We assume that Here, . Then, there exist a constant and sequences , such that the following (a)–(f) hold:(a) (b)   is an isolated solution of for any ,(c) (d) is bijective with ,(e) the solution of is unique in , where (f)

Proof. By (3.20) and the stability property of linear operators (e.g., [3, Corollary  2.4.1]), and are bijective for sufficiently small and , in as . Let and . We set for and define . Let , and . Then, we easily verify that as , Therefore, there exist and such that for any , is bijective with (d), , and . It follows that for any , , and . We also have and Let and . We apply Theorem 3.2 by setting , , , , , and . 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 , , , and . We show (3.19)–(3.21). By we have in as . Therefore, (3.19) holds. It follows from and Proposition 3.1 that So, (3.20) holds. Let , and . Since is continuous, we have as , which implies (3.21). Therefore, by Theorem 3.5, there exist a small constant , and such that (a)–(f) with hold. So, we immediately obtain (ii) and in as . Since , (a) and (c) imply (2.2). So, (iii) holds. In view of (d) and (e), we have (i) with where . The proof is complete.

Proof of Theorem 2.4. We set for simplicity. Proposition 2.1 (iii) implies as . First we show (2.6). We have , in as and Since , we have . We apply Proposition 3.4 with , , and to obtain which implies . We also have by the above discussion with replaced by . Next, we show (2.7). In the same way as above we apply Proposition 3.4 with (resp., ), , and to have Since and commutes with , we have . Combining (4.5) and , we obtain .

Proof of Theorem 2.5. We set for simplicity.(i) It follows from (H2) and Proposition 3.1 that By and the continuity of at we have We obtain (2.8) from (4.6) and (4.7).(ii) In the same way as (3.11) we have 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 . First we derive (2.11). This proof is essentially the same as that of [1, Theorem 3.2]. It suffices to prove which implies (2.11) in view of . We have It follows that which implies (4.9). Next we derive (2.13). Since with , we obtain from Proposition 3.1 that So, (2.13) holds.

5. Concrete Examples

In this section we consider the following semilinear elliptic boundary value problem: where is a bounded convex domain in ( ) with piecewise smooth boundary . We will rewrite (5.1) as the form (1.1) under the appropriate setting of functional spaces. We simply denote . We assume . Let be the operator defined by . We set with the norm and . Then, we have . We can rewrite (5.1) as . We choose as an approximate finite element subspace of with mesh size .

In what follows, we concentrate on the cases: and . We use finite element methods with piecewise linear and bilinear elements on the uniform (rectangular) mesh with mesh size . Then, we have in the 1-dimensional case and in the 2-dimensional case. In this context the following basic estimates hold:

where , , and are some positive constants independent of and . As in previous sections, we denote by an isolated solution of and by a finite element solution of (i.e., a solution of ) in a small neighborhood of . In view of Proposition 2.1, exists uniquely in a small neighborhood of for sufficiently small . In our examples below we show that the following error estimate holds: For simplicity we denote . We will derive (5.3) from Theorem 2.5 and the duality

We now present two examples.

Example 5.1. We consider the following Burgers equation: Here, is a given function with . As mentioned above, we rewrite (5.5) as . In the present case is a nonlinear map defined by . By the elliptic regularity property we have (see e.g., [9]). We will derive (5.3). Let . We easily verify that By (5.2b) we have It follows that We obtain from (5.2c) that It follows from (5.4), (5.8), and (5.9) that By (5.10), (5.2b), and Theorem 2.5 we have (5.3).

Example 5.2. We consider the Emden equation 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 is defined by . Let . We verify that and that is self-adjoint. By (5.2b) and Sobolev’s inequality we have for any and . Here, is a constant independent of , , and . It follows from this inequality and (5.4) that 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), holds in general. Actually, in Example 5.2 (resp., Example 5.1) our best possible upper bound of is the right-hand side of (5.13) (resp., (5.10)), which is just the same (resp., has the same order) as that of .
(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: 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 which is our best possible upper estimate of . So, in view of (2.13), it is necessary to take such that for small . Here, is a constant independent of . (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 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 as . 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 derived from our Theorem 2.5. The output data as upper bounds of in [5, Section 3] seem to have just the order of as . 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.

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