Error Analysis of Galerkin′s Method for Semilinear Equations

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 $$ has a solution u_h in U∩X_h for any h and there exists a monotone decreasing sequence $$ with lim⁡_k→∞h_k = 0 and u_∞ ∈ U such that $$ 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 $$ has a solution u_h ∈ X_h for any h, which is unique for sufficiently small h and

  $$ (1.3)
  which means that ∥u_* − u_h∥  and ∥(I − P_h)u_*∥  are equivalent infinitesimals as h → 0.

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 Rⁿ. We denote Lebesgue spaces by L^p(Ω)(1 ≤ p ≤ ∞) with the norms $$  for$$. We denote by $$ the completion of $$ (the space of C^∞ functions with compact support in Ω) in the Sobolev norm: $$. We denote by H⁻¹(Ω) the Sobolev space $$ with the norm $$. Here, 𝒟′(Ω) stands for the set of distributions on Ω.

Proposition 2.1. There exist h_* > 0  and $$such that the following (i)–(iii) hold.

• (i)

  There exists R_* > 0  such that u = u_h  is the only solution of f_h(u) = 0  in B(u_*;   R_*)  for any h ∈ (0, h_*).

• (ii)

  u = u_h  is an isolated solution off_h(u) = 0  for anyh ∈ (0, h_*).

• (iii)

  u_h → u_*  in  X  as  h → 0  with the estimate

  $$ (2.2)
  where $$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_h  is an isolated solution of $$for 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 dim⁡X_h < ∞. We briefly explain that it has some practical benefits. The case dim⁡X_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).

Corollary 2.3. We have

Theorem 2.4. We have the following:

Theorem 2.5. (i) We have

(ii) Let ɛ_h be a positive constant for h ∈ (0, h_*) such that

for any h ∈ (0, h_*). Then, there exist constants h₁ ∈ (0, h_*) and C₁ > 0 such that

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 $$ be a positive constant for h ∈ (0, h_*) such that $$. 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 $$ for concrete examples (e.g., examples in Section 5 below), it seems reasonable to estimate q_h  as q_h ≤ C∥T(I − P_h)∥. Here, C  represents some positive constant independent of h. Then, it is actually necessary to take $$such that $$for small h > 0. On the other hand, roughly speaking, (2.9) means that we can take ɛ_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)∥ → 0  as h → 0.)

(iii) We consider the case where T  is 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) $$ is larger than $$for 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.

Proposition 3.1. We assume (H1). Let K : X → X 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 δ∶ = limsup⁡_h→0∥K(I − P_h)∥ > 0. Therefore, there exist $$ and $$ such that h_n↘0  as n → ∞, ∥u_n∥ = 1  for n ∈ N  and Since K  is compact and $$ converges weakly to 0, we have $$as n → ∞. This contradicts (3.4). So, (3.3) holds. Since K^*  is also compact, we obtain So, we have (3.1), which implies (3.2).

Theorem 3.2. Let u₀ ∈ U and L ∈ ℒ(𝒳, 𝒴) be bijective. We define a map g : U → 𝒳 by

Let R > 0 be a constant satisfying $$ and b : [0, R]→[0, ∞) be a non-decreasing function such that Let ε₀ ≥ 0 be a constant such that We assume that there exist constants r₀  and r₁  such that   0 < r₀ ≤ r₁ ≤ R, Then the equation F(u) = 0 has an isolated solution $$. Moreover, the solution of F(u) = 0 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 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

By (3.7) and (3.11) we have We set $$. Let u ∈ B₀. In view of (3.7), (3.8), and (3.11) with v∶ = u₀, we have Combining (3.9), (3.13), and (3.14), we have ∥g(u) − u₀∥ ≤ r₀, which implies g(B₀) ⊂ B₀. Therefore, in view of (3.10) and (3.12) g is a contraction on B₀. By the contraction mapping principle there exists a unique solution u = u_* on B₀ for the equation F(u) = 0. We immediately obtain from (3.10) and (3.12) that the solution of F(u) = 0 is unique on $$. Finally, it suffices to show that in order to prove that u_* is isolated. We denote by I the identity operator on 𝒳. Let $$. 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.

Proposition 3.4. Let u, v ∈ U , (1 − s)u + sv ∈ U   for 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

Moreover, if   m∥L⁻¹∥ < 1 then we also obtain

Proof. The proof is similar to that of Theorem 3.2. Let g  : U → 𝒳 be a map defined by (3.6). We have

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

Here, $$. Then, there exist a constant h_* ∈ (0, h₀) and sequences $$, $$ such that the following (a)–(f) hold:

• (a)
  $$ (3.22)

• (b)

  u = u_h  is an isolated solution of  F_h(u) = 0 for any  h ∈ (0, h_*),

• (c)
  $$ (3.23)

• (d)

  H_h is bijective with $$,

• (e)

  the solution of   F_h(u) = 0  is unique in  B(u_*; R_h)  for  any  h ∈ (0, h_*), where

  $$ (3.24)

• (f)
  $$ (3.25)

Proof. By (3.20) and the stability property of linear operators (e.g., [3, Corollary  2.4.1]), $$ and H_h are bijective for sufficiently small h > 0 and $$, $$ in ℒ(𝒴, 𝒳) as h → 0. Let $$ and $$. We set d(r): = d(r, u_*) for r > 0 and define $$. Let c_h : = 1/{1 − b_h(2η_h)}, r_h : = c_hη_h and $$. Then, we easily verify that as h → 0,

Therefore, there exist h_* ∈ (0, h₀) and ɛ ∈ (0,1) such that for any h ∈ (0, h_*), $$ is bijective with (d), 1 ≤ c_h < 2, d(r_h + ε) < δ_h and $$. It follows that $$ for any h ∈ (0, h_*),   r > 0, and $$. We also have   r_h + ε ≤ R_h   and 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.

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

So, (3.20) holds. Let r > 0, u ∈ U and $$. Since φ′ is continuous, we have $$ as r↘0, which implies (3.21). Therefore, by Theorem 3.5, there exist a small constant h_* > 0, $$ and $$ 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 $$, (a) and (c) imply (2.2). So, (iii) holds. In view of (d) and (e), we have (i) with where $$. The proof is complete.

Proof of Theorem 2.4. We set 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), $$ in ℒ(X)   as h → 0  and

Since A_hf(u_h) = f(u_h), we have $$. We apply Proposition 3.4 with L : = A_h, u : = u_h, v : = u_* and F : = f to obtain which implies ∥u_h − u_*∥≃∥f(u_h)∥. We also have ∥u_h − u_*∥≃∥A⁻¹f(u_h)∥ by the above discussion with L : = A_h replaced by L : = A. Next, we show (2.7). In the same way as above we apply Proposition 3.4 with L : = A (resp., L : = A_h), u : = u_h, v : = P_hu_* and F : = f_h to have Since f_h(P_hu_*) = P_hf(P_hu_*) and A_h commutes with P_h, we have $$. 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

  $$ (4.6)

By (H1) and the continuity of φ′(u) at u = u_* we have We obtain (2.8) from (4.6) and (4.7).

• (ii)

  In the same way as (3.11) we have

  $$ (4.8)

By this equality, (2.7) and (2.9), we have (2.10).

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 A⁻¹ = I + K  with K : = T(I − T) ⁻¹, we obtain from Proposition 3.1 that So, (2.13) holds.

Example 5.1. We consider the following Burgers equation:

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 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 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 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 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:

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 ∥(I − P_h)T∥. So, in view of (2.13), it is necessary to take $$ such that $$ 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).