NOLTA, IEICE Paper
An algorithm of identifying parameters satisfying a sufficient condition of Plum’s Newton-Kantorovich like existence theorem for nonlinear operator equations Kouta Sekine 1 a) , Akitoshi Takayasu 2 , and Shin’ichi Oishi 2 ,3 1
Graduate School of Fundamental Science and Engineering, Waseda University, 3-4-1 Okubo, Tokyo 169-8555, Japan
2
Department of Applied Mathematics, Faculty of Science and Engineering, Waseda University, 3-4-1 Okubo, Tokyo 169-8555, Japan
3
CREST, JST
a)
[email protected]
Received May 20, 2013; Revised September 8, 2013; Published January 1, 2014 Abstract: This paper presents an algorithm of identifying parameters satisfying a sufficient condition of Plum’s Newton-Kantorovich like theorem. Plum’s theorem yields a numerical existence test of solutions for nonlinear partial differential equations. The sufficient condition of Plum’s theorem is given by the nonemptiness of a region defined by one dimensional nonlinear inequalities. The aim of this paper is to develop a systematic method of constructing an inner inclusion of this region. If ρ ∈ R+ is the minimum included in this region, ρ gives the minimum of the error bounds. Moreover, if ρ ∈ R+ is the maximum included in this region, then ρ gives the maximum radius of a ball in which the exact solution is unique. In this paper, an algorithm is developed for finding ρe and ρu such that they belong to this region and become close approximations of ρ and ρ, respectively. Finally, to illustrate features of Plum’s theorem with our proposed algorithm, some numerical results compared with results by Plum’s theorem with the Newton method are presented. In addition to this, Plum’s theorem with our algorithm is also compared with Newton-Kantorovich’s theorem. One of the most important facts found in this paper is that, for some examples, ρe become smaller than error bounds obtained by Newton-Kantorovich’s theorem. Moreover, also for these examples, ρu become greater than regions indicating uniqueness of the exact solution derived by Newton-Kantorovich’s theorem. This implies that Plum’s theorem can be seen as a modification of Newton-Kantorovich’s theorem. Key Words: Computer-assisted proofs, nonlinear operator equation, Newton-Kantorovich like theorem, Finding all solution method
64 Nonlinear Theory and Its Applications, IEICE, vol. 5, no. 1, pp. 64–79
c IEICE 2014
DOI: 10.1587/nolta.5.64
1. Introduction Let X and Y be Banach spaces. In this paper, we are concerned with a problem of finding a solution u ∈ X satisfying the following nonlinear equation: F(u) = 0,
(1)
where the nonlinear operator F : X → Y is assumed to be Fr´echet differentiable. M. Plum has proved the following theorem; Theorem 1 (M. Plum [1]). Let u ˆ ∈ X be an approximate solution of (1). Let W ⊂ X be a convex closed ball centered at zero with radius ρ: W := {w ∈ X : wX ≤ ρ}. u] is nonsingular and satisfies Assuming that the Fr´echet derivative F [ˆ u]−1 Y,X ≤ C1 F [ˆ
(2)
for a certain positive constant C1 . Let C2 be a positive constant satisfying F(ˆ u)Y ≤ C2 .
(3)
Let R+ := {x ∈ R : x > 0}, where R is the set of real numbers. We assume that there exist nonlinear functions C3 : R+ → R+ and C4 : R+ → R+ satisfying 1 sup (F [ˆ u + tw] − F [ˆ u]) wdt (4) ≤ C3 (ρ)ρ. w∈W
Y
0
and
0
1
(F [ˆ u + tw1 + (1 − t)w2 ] − F [ˆ u]) dt
X,Y
≤ C4 (ρ) , ∀w1 , w2 ∈ W,
(5)
respectively. If a constant ρ satisfies C2 ≤
ρ − C3 (ρ)ρ C1
and
C1 C4 (ρ) < 1,
(6)
then there exists a solution u∗ ∈ u ˆ + W of F(u) = 0 and unique in u ˆ + W. This theorem states that if there exists a ρ satisfying (6), then there exists a solution u∗ of F(u) = 0 in u ˆ + W and u∗ is unique in u ˆ + W . Thus, in order to apply Theorem 1, to find ρ satisfying (6) is one of the most important tasks. By a series of papers M. Plum [1, 2] has shown that such ρ can be found for various interesting nonlinear partial differential equations. However, a systematic way of finding ρ seems to have not yet be published. Here, we would like to point out that if one treat problems such as numerical existence test of boundary value problems of nonlinear partial differential equations, evaluations of C1 and C2 take a lot of computational time. However, since (6) consists of one dimensional inequalities, once functional forms of C3 (ρ) and C4 (ρ) are determined, searching of a region of ρ satisfying (6) do takes little computational cost, which is almost negligible compared with those for calculating C1 and C2 . The aim of this paper is to develop such a systematic method of finding a solution region of (6). Here, we note that if one can find a minimum ρ ∈ R+ satisfying (6), ρ gives the minimum of the error bounds u∗ − u ˆ. Moreover, we also note that if one can find a maximum ρ ∈ R+ satisfying (6), then ρ gives the maximum radius of a ball u ˆ + W in which u∗ is unique. Thus, from ρ and ρ, we can also prove that there is no solution in {u ∈ X : ρ < u − u ˆX ≤ ρ}. In this paper, we will develop an algorithm of finding ρe and ρu such that they satisfy (6) and become close approximations of ρ and ρ, respectively. For this purpose, we will present an algorithm of constructing an inner inclusion of a region defined by (6) based on Moore-Jones’s algorithm of finding all solutions of one dimensional
65
nonlinear equations proposed in [3], which is based on Krawczyk’s operator [4, 5]. Here, inclusion means a subset of a region defined by (6). For example, if we solve a one dimensional nonlinear equation using the Newton method in place of Moore-Jones’s algorithm, we can get the approximate solution of the nonlinear equation. Let ρN eu be a value which is larger than this approximate solution. Then, the interval arithmetic guarantees ρN eu satisfying (6). In particular, if the initial value of the Newton method is zero, then we usually obtain the error bound ρN eu which is similar to ρe . However, by the influence of perturbation of this approximate solution, the guaranteed region can be worse than the region which is obtained by Moore-Jones’s algorithm. Moreover, when calculating ρu , the appropriate initial value of the Newton method will be needed. Because of this reason, it is hard to construct a systematic method which proves that there is no solution in {u ∈ X : ρe < u− u ˆX ≤ ρu } using the Newton method. One of the features of our method is that if a region of the solution for (6) is empty, we can prove that there is no solution. In numerical examples, we will demonstrate that ρe and ρN eu are almost the same estimates. To state a little bit more detail, let us define two nonlinear functions g1 , g2 : R+ → R by g1 (ρ) := C1 C3 (ρ)ρ − ρ + C1 C2 and g2 (ρ) := C1 C4 (ρ) − 1, respectively. We will propose an algorithm of identifying inner inclusions of regions Γe and Γu , where Γe := ρ ∈ R+ : g1 (ρ) ≤ 0 and
Γu := ρ ∈ R+ : g2 (ρ) < 0 ,
respectively. By definition, if ρ ∈ Γe ∩ Γu , ρ satisfies the sufficient condition (6) of Theorem 1. It is obvious that ρ = min{ρ : ρ ∈ Γe ∩ Γu } and ρ = max{ρ : ρ ∈ Γe ∩ Γu }. We will show in the first place that the computation cost of our method of searching ρe and ρu are negligibly small compared with those for calculating C1 and C2 for all examples we have treated. Then, for these examples, we will demonstrate that ρe becomes smaller than the error bound obtained by Newton-Kantorovich’s theorem. Moreover, we will also show that for these examples, ρu becomes greater than the region indicating uniqueness derived by Newton-Kantorovich’s theorem. This implies that Plum’s theorem (Theorem1) can be seen as a modification of Newton-Kantorovich’s theorem. The outline of this paper is as follows: In Section 2, we present an algorithm of constructing inner inclusions of Γe and Γu . In Section 3, we provide some numerical examples to illustrate our algorithm compared with the Newton method. In addition to this, Theorem 1 with our searching algorithm is also compared with Newton-Kantorovich’s theorem [6]. Before closing this section, we present a remark about Nakao’s method [7–10]. If F [ˆ u]−1 exists, the operator equation (1) can be transformed into w = T (w) u]−1 F(ˆ u) + F [ˆ u]−1 (F [ˆ u]w + F(ˆ u) − F(ˆ u + w)), = −F [ˆ
(7)
where a fixed point operator T maps X onto itself. Using this kind of a fixed point formulation and assuming T is compact, M.T. Nakao [7–10] has presented a numerical existence test for nonlinear operator equations based on Schauder’s fixed point theorem. Moreover, assuming T is further contractive, N. Yamamoto [11] has presented a method of proving the local uniqueness of a solution using Banach’s fixed point theorem [9].
2. Algorithm of generating inner inclusions of Γe and Γu For a, b ∈ R satisfying −∞ < a < b < ∞, [a, b] denotes an interval [a, b] := {x ∈ R : a ≤ x ≤ b}. Let IR be the set of intervals in R. For x ∈ IR, sup(x) ∈ R denotes y ∈ x satisfying x ≤ y for all x ∈ x.
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Similarly, inf(x) ∈ R denotes y ∈ x satisfying y ≤ x for all x ∈ x. Let F be a set of floating-point numbers obeying IEEE 754 standard [12, 13]. The floating-point predecessor and successor of a real number x ∈ R are defined by pred(x) := max{f ∈ F : f < x} and succ(x) := min{f ∈ F : f > x}, respectively. In this paper, we assume that all solutions of g1 (ρ) = 0 can be obtained by Moore-Jones’s algorithm. For this propose, we assume C3 (ρ) is continuous first and second derivatives with respect to ρ. In this case, by Sard’s lemma, with probability one we can choose a small negative ε which is a regular value of g1 [14]. If Moore-Jones’s algorithm fails to find all solutions, then instead of g1 (ρ) = 0 considering g1 (ρ) = ε, without loss of generality, we can assume that all solutions of g1 (ρ) = 0 can be obtained by Moore-Jones’s algorithm. We also consider g2 (ρ) and C4 (ρ) under the same assumptions, respectively. In this section, we propose an algorithm of obtaining such inner inclusions of Γe and Γu based on Moore-Jones’s algorithm. Figure 1(a) illustrates a behavior of the function g1 in case that there exist ρ’s satisfying g1 (ρ) = 0. Let r be a positive real number. On the interval [0, r], we first identify all solutions of g1 (ρ) = 0 by Moore-Jones’s algorithm. For example, we usually put r = ˆ u L∞ because Γe ∩ Γu relates to a maximum value of an approximate solution u ˆ. Let Ii (i = 1, 2, · · · , n) be intervals such that each Ii contains one and only one positive solution ρi of g1 (ρ) = 0. We assume that Ii ∩ Ij = φ for 1 ≤ i < j ≤ n. If g1 (ρ) = 0 have no solutions, then there does not exist any ρ > 0 satisfying (6) provided that Γe is included in [0, r]. Figure 1(b) shows an example of this case. Let g¯1 : IR → IR be an interval extension of the nonlinear function g1 . Put I0 = [0, 0] ∈ IR and In+1 = [r, r] ∈ IR. We show now how to construct an inner inclusion of Γe := {ρ ∈ [0, r] : g1 (ρ) ≤ 0}. We first calculate inf(Ii+1 ) + sup(Ii ) di := g¯1 ∈ IR, 2 for i = 0, 1, · · · , n. Put
Iie
=
[sup(Ii ), inf(Ii+1 )] if sup(di ) ≤ 0, φ if sup(di ) > 0.
˜ e of Γe is given as An inner inclusion Γ ˜e = Γ
n i=0
Iie .
˜ e. Algorithm 1 summaries this procedure of calculating Γ ˜ e ), r], we then identify all solutions of g2 (ρ) = 0 by Moore-Jones’s algorithm. On the interval [inf(Γ Let Ii (i = 1, 2, · · · , n) be intervals such that each Ii contains one and only one positive solution ρi of g2 (ρ) = 0. We assume that Ii ∩Ij = φ for 1 ≤ i < j ≤ n. If g2 (ρ) = 0 have no solutions, then there does ˜ e ), r]. Let g¯2 : IR → IR be an not exist any ρ > 0 satisfying (6) provided that Γu is included in [inf(Γ e e ˜ ˜ interval extension of the nonlinear function g2 . Put I0 = [inf(Γ ), inf(Γ )] ∈ IR and In+1 = [r, r] ∈ IR. ˜ e ), r] : g2 (ρ) < 0}. We first We show now how to construct an inner inclusion of Γu := {ρ ∈ [inf(Γ calculate inf(Ii+1 ) + sup(Ii ) di := g¯2 ∈ IR, 2 for i = 0, 1, · · · , n. Put
Iiu =
[succ(sup(Ii )), pred(inf(Ii+1 ))] if sup(di ) < 0, φ if sup(di ) ≥ 0.
˜ u of Γu is given as An inner inclusion Γ ˜u = Γ
n i=0
Iiu .
˜u. Algorithm 2 summaries this procedure of calculating Γ
67
˜e ∩ Γ ˜ u , then ρ satisfies the sufficient condition (6) of Theorem 1. Finally, we It is clear that if ρ ∈ Γ calculate ˜e ∩ Γ ˜u} ρe = min{ρ : ρ ∈ Γ and ˜e ∩ Γ ˜ u }. ρu = max{ρ : ρ ∈ Γ Then ρe and ρu become approximating of ρ and ρ, respectively. If we search tighter inclusions of all solutions of g1 (ρ) = 0 and g2 (ρ) = 0 and if Γe ∩ Γu is included in [0, r], then it is obvious ρe and ρu approach to ρ and ρ, respectively.
Fig. 1.
Example of the proposed algorithm.
˜ e = {ρ ∈ [0, r] : g1 (ρ) ≤ 0}. Algorithm 1 of obtaining Γ Find Ii ∈ IR (i = 1, 2, · · · , n) that contain ρi ∈ R+ of g1 (ρi ) = 0 in [0, r]. if ∪Ii is empty then error(‘Failure in verification’); end if I0 = 0; In+1 = r; for i = 0 : n do di = g¯1 ((inf(Ii+1 ) + sup(Ii ))/2); if sup(di ) ≤ 0 then Iie = [sup(Ii ), inf(Ii+1 )]; else Iie = φ end if end for ˜ e = ∪n I e Put Γ i=1 i e ˜ if Γ is empty then error(‘Failure in verification’); end if
68
˜ u = {ρ ∈ [inf(Γ ˜ e ), r] : g2 (ρ) < 0}. Algorithm 2 of obtaining Γ ˜ e ), r]. Find Ii ∈ IR (i = 1, 2, · · · , n) that contain ρi ∈ R+ of g2 (ρi ) = 0 in [inf(Γ if ∪Ii is empty then error(‘Failure in verification’); end if ˜ e ); I0 = inf(Γ In+1 = r; for i = 0 : n do di = g¯2 ((inf(Ii+1 ) + sup(Ii ))/2); if sup(di ) < 0 then Iiu = [succ(sup(Ii )), pred(inf(Ii+1 ))]; else Iiu = φ end if end for ˜ e = ∪n I e Put Γ i=1 i u ˜ is empty then if Γ error(‘Failure in verification’); end if
3. Illustrative examples 3.1 Functional forms of C3 (ρ) and C4 (ρ) In this paper, we are concerned with nonlinear equations which are related to the Dirichlet boundary value problem of an elliptic equation of the following form −Δu = f (u) in Ω, (8) u=0 on ∂Ω, where Ω is a bounded polygonal domain. A functional analytic formulation needed for verified computations of (8) is given by [1, 2, 7–9, 16] etc. To make this paper self-contained and to clarify notations, we briefly review such a formulation in Appendix A. From the discussion in Appendix A, a weak formula for (8) can be represented as F(u) = Au − N (u) = 0,
(9)
where F is a nonlinear mapping from H01 (Ω) → H −1 (Ω). Here, the linear operator A : H01 (Ω) → H −1 (Ω) and the nonlinear operator N : H01 (Ω) → H −1 (Ω) are defined by Au, w := (∇u, ∇w)L2 , ∀w ∈ H01 (Ω) and N (u), w := (f (u), w)L2 , ∀w ∈ H01 (Ω), respectively. We now consider to apply Theorem 1 to (9). Several methods for calculating the constant C1 and C2 are proposed in [1, 7–9, 15, 16] and so on. In this paper, for the constant C1 , we use methods of calculating in [8, 15]. We also use a method presented in [16] for the calculation of C2 . M. Plum [1, 2] has presented how to construct functions C3 (ρ) and C4 (ρ). Here, following Plum’s approach we show how to derive C3 (ρ) and C4 (ρ). First, we show how to derive C3 (ρ) in case of f (u) = αu + βu2 + γu3 , where α, β, γ ∈ R. From H¨ older’s inequality and Sobolev’s embedding theorem, C3 (ρ) can be derived as follows:
69
≤ = ≤ ≤ ≤ ≤
1 sup (F [ˆ u + tw] − F [ˆ u ]) wdt −1 w∈W 0 H 1 |((α + 2β(ˆ u + tw) + 3γ(ˆ u + tw)2 − (α + 2β u ˆ + 3γ u ˆ2 ))φ, ψ)L2 | sup sup dtwH01 φH01 ψH01 w∈W 0 φ,ψ∈H01 (Ω)\{0} 1 |((2βtw + 3γ(2ˆ utw + t2 w2 ))φ, ψ)L2 | sup sup dtwH01 φH01 ψH01 w∈W 0 φ,ψ∈H01 (Ω)\{0} 1 |(2βtwφ, ψ)L2 | + |(3γ(2ˆ ut + t2 w)wφ, ψ)L2 | sup sup dtwH01 φH01 ψH01 w∈W 0 φ,ψ∈H01 (Ω)\{0} 1
3 4 4 sup Ce,3 |β| + Ce,4 |3γ|ˆ uH01 |2t|wH01 + Ce,4 |γ||3t2 |w2H 1 dtwH01 0 w∈W 0
3 4 4 sup Ce,3 |β| + 3Ce,4 |γ|ˆ uH01 wH01 + Ce,4 |γ|w2H 1 wH01 0 w∈W
3 4 4 Ce,3 |β| + 3Ce,4 |γ|ˆ uH01 ρ + Ce,4 |γ|ρ2 ρ =: C3 (ρ)ρ.
Then, we show how to derive C4 (ρ) in case of f (u) = αu + βu2 + γu3 , where α, β, γ ∈ R. For any w1 , w2 ∈ W , from H¨ older’s inequality and Sobolev’s embedding theorem, C4 (ρ) can be derived as follows:
≤
1 0
(F [ˆ u + tw1 + (1 − t)w2 ] − F [ˆ u]) dt
1 0
sup
φ,ψ∈H01 (Ω)\{0}
H01 ,H −1
|((α + 2β(ˆ u + tw1 + (1 − t)w2 ) + 3γ(ˆ u + tw1 + (1 − t)w2 )2 − ((α + 2β u ˆ + 3γ u ˆ2 )φ, ψ)L2 | dt φH01 ψH01
≤
3 4 4 (Ce,3 |β| + 3Ce,4 |γ|ˆ uH01 )(w1 H01 + w2 H01 ) + Ce,4 |γ|(w1 2H 1 + w1 H01 w2 H01 + w2 2H 1 )
≤
3 4 4 |β| + 3Ce,4 |γ|ˆ uH01 )ρ + 3Ce,4 |γ|ρ2 =: C4 (ρ). 2(Ce,3
0
0
3.2 Numerical example for Allen-Cahn equation We now ready to present results of numerical verifications of applying Theorem 1 with Algorithms 1 and 2 for some nonlinear elliptic Dirichlet boundary value problems. We compare these results with those results obtained by Theorem 1 with the Newton method whose the initial value is zero. Here, we put 0.0001 as the perturbation of the approximate solution obtained by the Newton method. We also compare these results with those obtained by Newton-Kantorovich’s theorem [16]. All computations are carried on PC with 3.10 GHz Intel Xeon E5-2687W CPU, 128 G Byte RAM and Cent OS 6.3. We use MATLAB2012a with INTLAB version 6 [17], a toolbox for verified numerical computations. Gmsh [18] (http://geuz.org/gmsh/) is used for obtaining triangular mesh. In the first place, let us consider the following Dirichlet problem of a nonlinear elliptic partial differential equation: −Δu = λ(u + u2 − u3 ) in Ω, (10) u=0 on ∂Ω, where Ω is an unit square, (0, 1) × (0, 1) or a bounded nonconvex polygonal domain. We have calculated all approximate solutions by the finite element method with piecewise quadratic base functions on a regular triangulation. Here, we denote a mesh size by the second longest length of each side on a triangle element. Table I shows several notations which are used throughout this paper. In the following, we present verified results of (10) on convex and nonconvex domains. 3.2.1 In case of square domain We first consider the semilinear elliptic equation (10) on the square domain (0, 1) × (0, 1) with an uniform mesh. The approximate solution u ˆ for λ = 36 is presented in Fig. 2. Table II displays verification results of (10) for λ = 35.7, 36, 37.6 and 37.8. In the cases of λ = 36, 37.6 and 37.8, Newton-Kantorovich’s theorem failed. On the other hand, Theorem 1 succeeded to verify the existence of exact solutions in the cases of λ = 36 and 37.6. For λ = 37.8, both methods failed. Comparing ρe
70
Table I. C1 C2 ρe ρu ρv ρN eu ζ1 ζ2 ζv N-K C
Explanation of variables.
Norm estimation for inverse of linearized operator in (2). Residual norm in (3). By Theorem 1, the existence of exact solution is proved in u − u ˆH01 ≤ ρe . By Theorem 1, the uniqueness of exact solution is proved in u − u ˆH01 ≤ ρu . An error bound v − v ˆH01 ≤ ρv via Theorem 1. An error bound u − u ˆH01 ≤ ρN eu via Theorem 1 with the Newton method whose the initial value is zero. By Newton-Kantorovich’s theorem, existence of a solution is proved in u − u ˆH01 ≤ ζ1 . By Newton-Kantorovich’s theorem, uniqueness of a solution is proved in u − u ˆH01 ≤ ζ2 . An error bound v − v ˆH01 ≤ ζv via Newton-Kantorovich’s theorem. The verification condition of Newton Kantorovich’s theorem. (To prove the existence, the parameter describing the verified condition should be less than or equal to 1/2.)
with ρN eu , both results were almost the same estimates. Precisely saying, ρe were a little bit shaper ρN . For λ = 37.8, the Newton method converged on a minus value. On the other hand, Algorithm 1 eu and 2 proved that there is no solution satisfying (6). In Fig. 3, we show shapes of nonlinear functions ˜ e = [2.782 × 10−2 , 7.206 × 10−2 ] and g1 and g2 for λ = 36. By Algorithms 1 and 2, we obtained Γ u −2 −2 e u ˜ = [2.782 × 10 , 4.332 × 10 ] so that Γ ˜ ∩Γ ˜ = [2.782 × 10−2 , 4.332 × 10−2 ]. If ρ ∈ Γ ˜e ∩ Γ ˜u, Γ ρ satisfies the sufficient condition (6) of Theorem 1. Table III shows computational time needed for verification. As seen from this table, the computational time needed for Algorithms 1 and 2 is less than 0.5 [sec] which is negligible compared with that for calculating C1 and C2 which is more than 200 [sec]. Effect of introducing Algorithms 1 and 2 is seen from the fact that Theorem 1 with Algorithms 1 and 2 yields better estimates than those of Newton-Kantorovich’s theorem. This is reasonable because Theorem 1 uses an estimate thought C3 (ρ) while Newton-Kantorovich’s theorem uses that thought the Lipschitz continuity of F , which is an approximation of the way using C3 (ρ).
Fig. 2.
Table II. λ 35.7 36 37.6 37.8
C1 2.120 2.138 2.237 2.251
C2 9.289 × 10−3 9.406 × 10−3 1.004 × 10−2 1.012 × 10−2
Table III. ([sec]).
ρe 2.664 × 10−2 2.782 × 10−2 3.986 × 10−2 Failed
Approximate solution u ˆ for (10).
Verification results of (10) (h = 2−5 ). ρu 4.398 × 10−2 4.332 × 10−2 3.991 × 10−2 Failed
ρN eu 2.667 × 10−2 2.785 × 10−2 3.988 × 10−2 –
N-K C 0.4858 0.5063 0.6299 0.6472
ζ1 3.370 × 10−2 Failed Failed Failed
Computational time needed for verification of (10) h = 2−5 with λ 35.7 36 37.6 37.8
C1 17.77 20.27 19.81 18.17
C2 184.0 183.0 183.6 184.3
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Algorithms 1 and 2 0.5000 0.1001 0.1258 0.0503
ζ2 3.936 × 10−2 Failed Failed Failed
Fig. 3.
Nonlinear functions g1 and g2 for λ = 36.
3.2.2 In case of a bounded nonconvex polygonal domain Next, let us consider the case that Ω is a bounded nonconvex polygonal domain whose vertices are given by {(0.5, 0), (1, 0.5), (1, 1), (0.5, 0.75), (0, 1), (0, 0.5)}. Figure 4 shows this bounded nonconvex polygonal domain Ω. It is well-known that the solution does not have H 2 -regularity at a reentrant corner. It causes slow convergence. In order to improve the accuracy of approximate solutions, we used a nonuniform mesh centered at that corner as shown in Fig. 4. We consider the Dirichlet problem of the semilinear elliptic equation (10) on this nonconvex domain Ω using this nonuniform mesh. For λ = 35 and 45, approximate solutions u ˆ are shown in Fig. 5. In the cases of λ = 35, 45 and 55, verification results of (10) are shown in Table IV. Although Theorem 1 proved the existence of a solution, the verification by Newton-Kantorovich’s theorem for λ = 45 failed. For λ = 55, both methods failed. Using Algorithm 1 and 2, we proved that there is no solution satisfying (6) for λ = 55. On the other hand, the Newton method converged on a minus value. Table V shows computational time needed for verification. As seen from this table, computational time needed for Algorithms 1 and 2 is less than 0.11 [sec] which is negligible compared with that for calculating C1 and C2 which is more than 104 [sec]. As the previous example, the effect of introducing Algorithms 1 and 2 is seen from the fact that Theorem 1 with Algorithms 1 and 2 yields better estimates than those of Newton-Kantorovich’s theorem. Table IV. λ 35 45 55
C1 2.747 2.474 3.404
C2 7.909 × 10−3 1.173 × 10−2 1.436 × 10−2
Table V.
Verification result of (10) (1.646 × 10−3 ≤ h ≤ 2.644 × 10−2 ). ρe 2.510 × 10−2 4.218 × 10−2 Failed
ρu 6.319 × 10−2 5.105 × 10−2 Failed
ρN eu 2.513 × 10−2 4.222 × 10−2 –
N-K C 0.1159 0.5527 1.7939
ζ1 2.597 × 10−2 Failed Failed
ζ2 4.346 × 10−2 Failed Failed
Computational time needed for verification of (10) with ([sec]). λ 35 45 55
C1 2.468 × 103 2.483 × 103 2.482 × 103
C2 1.438 × 104 1.437 × 104 1.436 × 104
Algorithms 1 and 2 0.0994 0.1095 0.0439
3.3 Numerical example for system of elliptic partial differential equations Now, let us consider the following Dirichlet boundary value problem of a system of nonlinear elliptic partial differential equations:
72
Fig. 4.
Bounded nonconvex domain Ω.
Fig. 5. Approximate solution u ˆ of (10). ⎧ 2 3 ⎪ ⎨ −Δu = λ(u + u − u − v) in Ω, −Δv = u − 1.2v in Ω, ⎪ ⎩ u=v=0 on ∂Ω,
(11)
where Ω is an unit square, (0, 1) × (0, 1) or a bounded nonconvex polygonal domain. A numerical verification theory for (11) has been proposed by [19]. Here, following [19], we briefly sketch a verification procedure. If u is a known function, the boundary value problem: −Δv = u − 1.2v in Ω, (12) v=0 on ∂Ω, has a unique solution by Riesz’s representation theorem. Then, v can be presented as v = Bu, where B : L2 (Ω) → H01 (Ω) is a solution operator of (12). Substituting this into (11), it follows that −Δu = λ(u + u2 − u3 − Bu) in Ω, (13) u=0 on ∂Ω. Decoupling the original problem (11) into the linear Dirichlet problem (12) and the nonlinear Dirichlet problem (13), according to [19] one can verify the existence and local uniqueness of solutions for (12) and (13). For (13), we apply Theorem 1 with Algorithms 1 and 2. We have calculated all approximate solutions by the finite element method with piecewise quadratic base functions on a regular triangulation. Here, we denote a mesh size by the second longest length of each side on a triangle element. In the following, we present verification results for (11) on convex and nonconvex domains. For each case, Theorem 1 with Algorithms 1 and 2 yields better estimate than those by Newton-Kantorovich’s theorem. In addition to this, comparing ρe with ρN eu , both results were almost the same estimates. 3.3.1 In case of square domain We first consider the elliptic system (11) on the square domain (0, 1) × (0, 1). In Figs. 6 and 7, we show two approximate solutions, say Type I : (ˆ u1 , vˆ1 ) and Type II : (ˆ u2 , vˆ2 ), in the case of λ = 17.
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Verification results for (11) using Theorem 1 are given in Table VI. Also in this case, comparing ρe with ρN eu , both results were almost the same estimates. Verification results for (11) using the method proposed in [16] are also shown in Table VII. The mesh size is taken as h = 2−5 . Comparing these two tables, Theorem 1 with Algorithms 1 and 2 gives shaper error bounds and larger regions of uniqueness. Table VIII shows computational time needed for verification. As seen from this table, computational time needed for Algorithms 1 and 2 is less than 0.16 [sec] which is negligible compared with that for calculating C1 and C2 which is more than 850 [sec]. Also in this case, the effect of introducing Algorithms 1 and 2 is seen from the fact that Theorem 1 with Algorithms 1 and 2 yields better estimates than those of Newton-Kantorovich’s theorem.
Fig. 6.
Type I : u ˆ1 (left) and vˆ1 (right) of (11).
Fig. 7.
Type II : u ˆ2 (left) and vˆ2 (right) of (11).
Table VI. Type I II
C1 6.993 12.41
Verification results of (11) using Theorem 1 (h = 2−5 ).
C2 2.101 × 10−3 1.101 × 10−3
Table VII. (h = 2−5 ).
ρu 4.206 × 10−2 3.403 × 10−2
ρv 9.899 × 10−4 9.059 × 10−4
ρN eu 1.768 × 10−2 1.690 × 10−2
Verification results of (11) using the method proposed in [16]
Type I II
Table VIII. with ([sec]).
ρe 1.766 × 10−2 1.688 × 10−2
N-K C 0.2841 0.4858
ζ1 1.773 × 10−2 1.692 × 10−2
ζ2 2.938 × 10−2 2.732 × 10−2
ζv 1.059 × 10−3 9.379 × 10−4
Computational time needed for verification of (11) h = 2−5 Type I II
C1 781.2 738.8
C2 171.6 169.8
Algorithms 1 and 2 0.1588 0.1493
3.3.2 In case of a bounded nonconvex polygonal domain Next, let us consider the case that Ω is a bounded nonconvex polygonal domain whose vertices are given by {(0.5, 0), (1, 0.5), (1, 1), (0.5, 0.75), (0, 1), (0, 0.5)}.
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Figure 8 shows this bounded nonconvex polygonal domain Ω. It is also known that an exact solution does not have H 2 -regularity at a reentrant corner. It causes slow convergence. In order to improve the accuracy of approximate solutions, we used a nonuniform mesh centered at that corner as shown in Fig. 8.
Fig. 8.
Bounded nonconvex domain Ω.
We consider the nonlinear elliptic system (11) on this nonconvex domain Ω. For λ = 35, Fig. 9 shows approximate solutions u ˆ and vˆ. Table IX presents the result of verification based on Theorem 1 with Algorithms 1 and 2. We got C1 = 3.136, C2 = 6.121 × 10−3 for this nonuniform mesh triangulation. Based on Table IX, the verification parameter describing the sufficient condition of Newton-Kantorovich’s theorem was calculated as 0.2642. Comparing ρe with ρN eu , both results were almost the same estimates. For each case, Theorem 1 with Algorithms 1 and 2 yields better estimate than that by Newton-Kantorovich’s theorem. Table X shows computational time needed for verification. As seen from this table, computational time needed for Algorithms 1 and 2 is less than 0.1 [sec] which is negligible compared with that for calculating C1 and C2 which is more than 104 [sec].
Fig. 9.
Table IX. ρe 2.207 × 10−2
Verification result for (11) (7.270 × 10−4 ≤ h ≤ 2.696 × 10−2 ).
ρu 5.814 × 10−2
Table X.
Approximate solution u ˆ (left) and vˆ (right) of (11).
ρv 7.165 × 10−4
ρN eu 2.209 × 10−2
ζ1 2.276 × 10−2
ζ2 3.838 × 10−2
ζv 7.343 × 10−4
Computational time needed for verification of (11) with ([sec]). C1 1.790 × 104
C2 1.776 × 104
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Algorithms 1 and 2 0.0974
3.4 Numerical results for system of ordinary differential equations As the third example, let us consider the following system of nonlinear ordinary differential equations: ⎧ 5 2 d2 u 1 1 ⎪ 3 ⎪ u + u − = − − u − 0.2v in − 1 < x < 1, ⎪ ⎪ dx2 0.082 4 4 ⎪ ⎨ d2 v (14) − 2 =u in − 1 < x < 1, ⎪ ⎪ ⎪ dx ⎪ ⎪ ⎩ u(−1) = u(1) = v(−1) = v(1) = 0. M.T. Nakao and Y. Watanabe have studied numerical verification methods, which are called FN-Int and IN-Linz, of the above system in [10]. For (14), we also apply the same numerical verification method applied for (11). We calculated approximate solutions by the finite element method with piecewise quadratic base functions. Let N be a number of equidistant partitions for the interval [-1,1]. Approximate solutions u ˆ and vˆ are shown in Fig. 10. Verification results for (14) are provided in Table XI. In Table XII, we present verification results through Newton-Kantorovich’s theorem [16]. For five approximate solutions S1, S3, AS1, AS2 and AS3, Theorem 1 with Algorithms 1 and 2 proved the existence and local uniqueness of each solution. However, for two approximate solutions S2 and S4, both methods failed. Using Algorithm 1 and 2, we proved that there is no solution satisfying (6) for S2 and S4. On the other hand, the Newton method converged on a minus value. Comparing ρN eu with ζ1 , both results were almost the same estimates. Precisely saying, ζ1 were a little bit shaper ρN eu . For all approximate solutions, M.T. Nakao and Y. Watanabe [10] have succeeded to prove the existence of solutions which are located in neighborhood of these approximate solutions using Nakao’s IN-Linz method, which is based on Schauder’s fixed point theory. Although M.T. Nakao and Y. Watanabe [10] do not give the uniqueness results for AS1 and AS3, for these problems, the local uniqueness of each solution can be proved using Theorem 1 with Algorithms 1 and 2. Table XIII shows computational time needed for verification. As seen from this table, computational time needed for Algorithms 1 and 2 is less than 0.18 [sec] which is negligible compared with that for calculating C1 and C2 which is more than 10 [sec]. Table XI. Type S1 S2 S3 S4 AS1 AS2 AS3
N 512 1024 512 1024 1024 512 512
C1 3.282 13.90 4.162 7.521 4.857 5.202 3.643
Table XII. Type S1 S2 S3 S4 AS1 AS2 AS3
Verification result of (14) using Theorem 1.
C2 4.410 × 10−5 2.034 × 10−5 4.552 × 10−5 2.803 × 10−5 3.436 × 10−5 2.621 × 10−5 4.179 × 10−5
ρe 1.648 × 10−4 Failed 2.650 × 10−4 Failed 2.236 × 10−4 1.585 × 10−4 1.765 × 10−4
ρu 6.764 × 10−4 Failed 4.646 × 10−4 Failed 4.404 × 10−4 5.661 × 10−4 6.413 × 10−4
ρv 7.545 × 10−5 Failed 1.117 × 10−4 Failed 1.002 × 10−4 6.613 × 10−5 7.945 × 10−5
ρN eu 1.650 × 10−4 – 2.653 × 10−4 – 2.238 × 10−4 1.587 × 10−4 1.767 × 10−4
Verification result of (14) using the method proposed in [16]. N-K C 0.2139 1.3967 0.4077 0.6733 0.3787 0.2407 0.2373
ζ1 1.648 × 10−4 Failed 2.650 × 10−4 Failed 2.236 × 10−4 1.585 × 10−4 1.765 × 10−4
ζ2 2.894 × 10−4 Failed 3.789 × 10−4 Failed 3.337 × 10−4 2.726 × 10−4 3.045 × 10−4
ζv 7.545 × 10−5 Failed 1.117 × 10−4 Failed 1.002 × 10−4 6.614 × 10−5 7.945 × 10−5
Appendix A. Functional analytic formulation Let Lp (Ω), p ∈ [1, ∞) denote the functional space of p-th power Lebesgue integrable functions. For p = 2, let us define the inner product
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Fig. 10.
Approximate solutions u ˆ (blue) and vˆ (green) for (14).
(u, w)L2 :=
u(x)w(x)dx Ω
and the norm uL2 :=
(u, u)L2 .
For a fixed positive real number s, let H s (Ω) be the L2 Sobolev space of order s. The function space H01 (Ω) is defined by
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Table XIII.
Computational time needed for verification of (14) with ([sec]). Type S1 S2 S3 S4 AS1 AS2 AS3
C1 8.128 76.72 10.38 72.59 50.56 11.29 8.100
C2 3.627 18.82 3.602 18.75 18.87 3.656 3.645
Algorithms 1 and 2 0.1470 0.0809 0.1640 0.0945 0.1532 0.1727 0.1343
H01 (Ω) := {u ∈ H 1 (Ω) : u = 0 on ∂Ω} with the inner product (u, w)H01 := (∇u, ∇w)L2 and the norm uH01 := ∇uL2 . Let H −1 (Ω) be the topological dual space of H01 (Ω). We denote T u ∈ R by T, u , where T ∈ H −1 (Ω) and u ∈ H01 (Ω). The norm of T ∈ H −1 (Ω) is defined by T H −1 :=
sup
u∈H01 (Ω)\{0}
| T, u | . uH01
Let L∞ (Ω) be the space of functions that are essentially bounded on Ω with the norm uL∞ := ess sup |u(x)|. x∈Ω
Let X and Y be Banach spaces. We denote the set of bounded linear operators by L(X, Y ) with the operator norm T X,Y :=
T uY . u∈X\{0} uX sup
There exists the constant Ce,p satisfying uLp ≤ Ce,p uH01 for u ∈ H01 (Ω), p ∈ [2, ∞) from Sobolev’s embedding theorem. Embedding constants is given by the minimal eigenvalue of Laplacian, cf. Lemma 1 in [1]. Let Xh be the finite-dimensional subspace spanned by linearly independent H01 (Ω) conforming finite element base functions, where h is a mesh size (0 < h < 1). Assuming that an approximate solution u ˆ ∈ Xh is given, f [ˆ u] is essentially bounded, i.e. f [ˆ u]L∞ < 1 +∞. The weak formulation of (8) is to find u ∈ H0 (Ω) satisfying (∇u, ∇w)L2 = (f (u), w)L2 , ∀w ∈ H01 (Ω). For u ∈ H01 (Ω), we define the linear operator A : H01 (Ω) → H −1 (Ω) and the nonlinear operator N : H01 (Ω) → H −1 (Ω) by Au, w := (∇u, ∇w)L2 , ∀w ∈ H01 (Ω) and N (u), w := (f (u), w)L2 , ∀w ∈ H01 (Ω), respectively. Let F be the nonlinear operator mapping from H01 (Ω) to H −1 (Ω) described by F(u) := Au − N (u). The solution of F(u) = 0 is equivalent to the weak form of (8). The operator F is assumed to be Fr´echet differentiable. The Fr´echet derivative of F at u ˆ ∈ Xh is given by F [ˆ u] = A − N [ˆ u], where the Fr´echet derivative of nonlinear operator satisfies N [ˆ u]u, w = (f [ˆ u]u, w), ∀w ∈ H01 (Ω).
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