A Galerkin Methods for a singularly perturbed bilocal problem

Costica Mustata
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Abstract


A bilocal singularly perturbed problem is solved using Galerkin’s method in a space in which the test functions are weighted primitives of wavelets. This method provides a “good” numerical solution of this problem.

Authors

Adrian Muresan
“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Costica Mustata

Costică Mustăţa


“Tiberiu Popoviciu” Institute of Numerical Analysis, Romanian Academy, Romania

Keywords

Paper coordinates

A.C. Mureşan, C. Mustăţa, A Galerkin Methods for a singularly perturbed bilocal problem, Bull. Şt. Univ. Baia Mare, Seria B, Fascicola Matematică-informatică, 15 (1999) nos. 1-2, 89-102, https://www.jstor.org/stable/44001741

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Buletinul ştiinţific al Universitatii Baia Mare,

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Sinus Association

DOI

https://www.jstor.org/stable/44001741

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12221201

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[1] I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. pure Appl. Math. 41 (1998), pp. 909-996.
[2] R. Głowiński, W.N. Lawton, M. Ravachol, E. Tenenbaum, Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. In: R. Głowiński, A. Lichnewsky, ads., Computing Methods in Applied Sciences and Engineering, SIAM, Philadelphia (1990), pp. 55-120.
[3] P.W. Hemker, A numerical study of stiff two-point boundary problems, Amsterdam, 1997.
[4] J.-C. Xu, W.-C. Shann, Galerkin – wavelet methods for two point boundary value problems. Numer. Math., 63 (1992), pp. 123-142.
[5] H. Yserentant, On the multi – level splitting of finite element spaces. Numer. Math. 49 (1986), pp. 379-412.

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1999-Mustata-BAM-A-Galerkin-methods-for-a-singularly-perturbed-bilocal-problem

A Galerkin Method for a Singularly Perturbed Bilocal Problem*

A.C.Muresan and C.Mustata

Abstract

A bilocal singularly perturbed problem is solved using Galerkin's method in a space in which the test functions are weighted primitives of wavelets. This method provides a "good" numerical solution of this problem.

In the study of convection - diffusion problems, the following boundary value singularly perturbed problem appears:
( P ) { − ε u ′ ′ ( x ) + a ( x ) u ′ ( x ) = f ( x ) , for x ∈ ( 0 , 1 ) u ( 0 ) = u ( 1 ) = 0 ( P ) − ε u ′ ′ ( x ) + a ( x ) u ′ ( x ) = f ( x ) ,  for  x ∈ ( 0 , 1 ) u ( 0 ) = u ( 1 ) = 0 (P){[-epsiu^('')(x)+a(x)u^(')(x)=f(x)","" for "x in(0","1)],[u(0)=u(1)=0]:}(P)\left\{\begin{array}{c} -\varepsilon u^{\prime \prime}(x)+a(x) u^{\prime}(x)=f(x), \text { for } x \in(0,1) \\ u(0)=u(1)=0 \end{array}\right.(P){−εu′′(x)+a(x)u′(x)=f(x), for x∈(0,1)u(0)=u(1)=0
where 0 < ε ≪ 1 , a ( x ) > α > 0 , x ∈ [ 0 , 1 ] 0 < ε ≪ 1 , a ( x ) > α > 0 , x ∈ [ 0 , 1 ] 0 < epsi≪1,a(x) > alpha > 0,x in[0,1]0<\varepsilon \ll 1, a(x)>\alpha>0, x \in[0,1]0<ε≪1,a(x)>α>0,x∈[0,1] and functions a a aaa and f f fff are sufficiently smooth.
The exact solution of problem ( P ) ( P ) (P)(P)(P) has a boundary layer in x = 1 x = 1 x=1x=1x=1. Because of its presence, certain numerical methods (finite element method, centered finite difference method) lead to numerical solutions with oscillations in the area of the boundary layer, abnormal from the physical point of view.
The piecewise polinomial test functions are replaced by wavelets, within the finite element method, in the work of Glowinski, Lawton, Ravachol and Tenenbaum [2]. Many examples provided show the great potential which wavelets have in the numerical solving of differential equations. Unfortunately, some disavantages may occur : the weak regularity of wavelets does not allow the use of small order wavelets; orthogonality of wavelets does not play a significant role.
Disadvantages in the use of wavelets can be partially eliminated if primitives of wavelets as test functions ([4]) are used.
In the present paper, Galerkin's method is not applied, for problem ( P ) ( P ) (P)(P)(P) (in the space H 0 1 [ 0 , 1 ] H 0 1 [ 0 , 1 ] H_(0)^(1)[0,1]H_{0}^{1}[0,1]H01[0,1] ). First, the space H 0 1 [ 0 , 1 ] H 0 1 [ 0 , 1 ] H_(0)^(1)[0,1]H_{0}^{1}[0,1]H01[0,1] turns "conveniently" into the space G H 0 1 [ 0 , 1 ] G H 0 1 [ 0 , 1 ] GH_(0)^(1)[0,1]G H_{0}^{1}[0,1]GH01[0,1], which is the image of H 0 1 [ 0 , 1 ] H 0 1 [ 0 , 1 ] H_(0)^(1)[0,1]H_{0}^{1}[0,1]H01[0,1] by G u := u o g , u ∈ H 0 1 [ 0 , 1 ] G u := u o g , u ∈ H 0 1 [ 0 , 1 ] Gu:=uog,u inH_(0)^(1)[0,1]G u:=u o g, u \in H_{0}^{1}[0,1]Gu:=uog,u∈H01[0,1] and g : [ 0 , 1 ] − > [ 0 , 1 ] g : [ 0 , 1 ] − > [ 0 , 1 ] g:[0,1]- > [0,1]g:[0,1]->[0,1]g:[0,1]−>[0,1] with g ( 0 ) = 0 , g ( 1 ) = 1 g ( 0 ) = 0 , g ( 1 ) = 1 g(0)=0,g(1)=1g(0)=0, g(1)=1g(0)=0,g(1)=1 and ∃ M > 0 ∃ M > 0 EE M > 0\exists M>0∃M>0, such that 0 ≤ g ′ ( y ) ≤ M , y ∈ [ 0 , 1 ] 0 ≤ g ′ ( y ) ≤ M , y ∈ [ 0 , 1 ] 0 <= g^(')(y) <= M,y in[0,1]0 \leq g^{\prime}(y) \leq M, y \in[0,1]0≤g′(y)≤M,y∈[0,1]. Problem ( P P PPP ) is transcribed in G H 0 1 [ 0 , 1 ] G H 0 1 [ 0 , 1 ] GH_(0)^(1)[0,1]G H_{0}^{1}[0,1]GH01[0,1] and Galerkin's method is applied in order to solve the new problem. Weighted primitives of Haar's system are used as test functions (weighted primitives of Daubechies wavelets of the first order).
Accordingly, the problem in G H 0 1 [ 0 , 1 ] G H 0 1 [ 0 , 1 ] GH_(0)^(1)[0,1]G H_{0}^{1}[0,1]GH01[0,1] will have a Galerkin solution with attenuated oscillations in the area of the boundary layer. Getting back to problem ( P P PPP ), it becomes out that a very good solution from the numerical point of view, is obtained. The numerical example fairly confirms it.
We consider the standard spaces. Let
L 2 [ 0 , 1 ] := { v : [ 0 , 1 ] − > R / v is measurable, and ‖ v ‖ L 2 ∣ 0 , 1 ] < ∞ } , L 2 [ 0 , 1 ] := v : [ 0 , 1 ] − > R / v  is measurable, and  ‖ v ‖ L 2 ∣ 0 , 1 < ∞ , L^(2)[0,1]:={v:[0,1]- > R//v" is measurable, and "||v||_({:L^(2)∣0,1]) < oo},L^{2}[0,1]:=\left\{v:[0,1]->R / v \text { is measurable, and }\|v\|_{\left.L^{2} \mid 0,1\right]}<\infty\right\},L2[0,1]:={v:[0,1]−>R/v is measurable, and ‖v‖L2∣0,1]<∞},
norm on L 2 [ 0 , 1 ] L 2 [ 0 , 1 ] L^(2)[0,1]L^{2}[0,1]L2[0,1] being:
‖ v ‖ L 2 [ 0 , 1 ] := ( ∫ 0 1 | v ( x ) | 2 d x ) 1 2 ‖ v ‖ L 2 [ 0 , 1 ] := ∫ 0 1   | v ( x ) | 2 d x 1 2 ||v||_(L^(2)[0,1]):=(int_(0)^(1)|v(x)|^(2)dx)^((1)/(2))\|v\|_{L^{2}[0,1]}:=\left(\int_{0}^{1}|v(x)|^{2} d x\right)^{\frac{1}{2}}‖v‖L2[0,1]:=(∫01|v(x)|2dx)12
Let
H 1 [ 0 , 1 ] := { v ∈ L 2 [ 0 , 1 ] / v ( k ) ∈ L 2 [ 0 , 1 ] for k = 0 , 1 } H 1 [ 0 , 1 ] := v ∈ L 2 [ 0 , 1 ] / v ( k ) ∈ L 2 [ 0 , 1 ]  for  k = 0 , 1 H^(1)[0,1]:={v inL^(2)[0,1]//v^((k))inL^(2)[0,1]" for "k=0,1}H^{1}[0,1]:=\left\{v \in L^{2}[0,1] / v^{(k)} \in L^{2}[0,1] \text { for } k=0,1\right\}H1[0,1]:={v∈L2[0,1]/v(k)∈L2[0,1] for k=0,1}
with norm
‖ v ‖ 1 := ( ∫ 0 1 | v ( x ) | 2 d x + ∫ 0 1 | v l ( x ) | 2 d x ) 1 2 ‖ v ‖ 1 := ∫ 0 1   | v ( x ) | 2 d x + ∫ 0 1   v l ( x ) 2 d x 1 2 ||v||_(1):=(int_(0)^(1)|v(x)|^(2)dx+int_(0)^(1)|v^(l)(x)|^(2)dx)^((1)/(2))\|v\|_{1}:=\left(\int_{0}^{1}|v(x)|^{2} d x+\int_{0}^{1}\left|v^{l}(x)\right|^{2} d x\right)^{\frac{1}{2}}‖v‖1:=(∫01|v(x)|2dx+∫01|vl(x)|2dx)12
and seminorm
| v | 1 := ( ∫ 0 1 | v ′ ( x ) | 2 d x ) 1 2 | v | 1 := ∫ 0 1   v ′ ( x ) 2 d x 1 2 |v|_(1):=(int_(0)^(1)|v^(')(x)|^(2)dx)^((1)/(2))|v|_{1}:=\left(\int_{0}^{1}\left|v^{\prime}(x)\right|^{2} d x\right)^{\frac{1}{2}}|v|1:=(∫01|v′(x)|2dx)12
and subspace
H 0 1 [ 0 , 1 ] := { v ∈ H 1 [ 0 , 1 ] / v ( 0 ) = v ( 1 ) = 0 } . H 0 1 [ 0 , 1 ] := v ∈ H 1 [ 0 , 1 ] / v ( 0 ) = v ( 1 ) = 0 . H_(0)^(1)[0,1]:={v inH^(1)[0,1]//v(0)=v(1)=0}.H_{0}^{1}[0,1]:=\left\{v \in H^{1}[0,1] / v(0)=v(1)=0\right\} .H01[0,1]:={v∈H1[0,1]/v(0)=v(1)=0}.
Seminorm | ⋅ | 1 | ⋅ | 1 |*|_(1)|\cdot|_{1}|⋅|1 is norm (equivalent to ‖ ⋅ ‖ 1 ‖ ⋅ ‖ 1 ||*||_(1)\|\cdot\|_{1}‖⋅‖1 ) on the space H 0 1 [ 0 , 1 ] H 0 1 [ 0 , 1 ] H_(0)^(1)[0,1]H_{0}^{1}[0,1]H01[0,1].
Let g : [ 0 , 1 ] − > [ 0 , 1 ] g : [ 0 , 1 ] − > [ 0 , 1 ] g:[0,1]- > [0,1]g:[0,1]->[0,1]g:[0,1]−>[0,1] a differe. atiable function on [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1] so that:
g ( 0 ) = 0 , g ( 1 ) = 1 ∃ M > 0 a.i. 0 ≤ g ′ ( y ) ≤ M , for ∀ y ∈ [ 0 , 1 ] . g ( 0 ) = 0 , g ( 1 ) = 1 ∃ M > 0  a.i.  0 ≤ g ′ ( y ) ≤ M ,  for  ∀ y ∈ [ 0 , 1 ] . {:[g(0)=0","g(1)=1],[EE M > 0" a.i. "0 <= g^(')(y) <= M","" for "AA y in[0","1].]:}\begin{aligned} & g(0)=0, g(1)=1 \\ & \exists M>0 \text { a.i. } 0 \leq g^{\prime}(y) \leq M, \text { for } \forall y \in[0,1] . \end{aligned}g(0)=0,g(1)=1∃M>0 a.i. 0≤g′(y)≤M, for ∀y∈[0,1].
We note J ( y ) = g ′ ( y ) J ( y ) = g ′ ( y ) J(y)=g^(')(y)J(y)=g^{\prime}(y)J(y)=g′(y) and we define G H 0 1 [ 0 , 1 ] G H 0 1 [ 0 , 1 ] GH_(0)^(1)[0,1]G H_{0}^{1}[0,1]GH01[0,1] to be the image of space H 0 1 [ 0 , 1 ] H 0 1 [ 0 , 1 ] H_(0)^(1)[0,1]H_{0}^{1}[0,1]H01[0,1] by transformation G u := u ∘ g G u := u ∘ g Gu:=u@gG u:=u \circ gGu:=u∘g.

Lemma 1

‖ v ‖ G H 0 1 ≡ { ∫ 0 1 1 J ( y ) | v ′ ( y ) | 2 d y } 1 2 ‖ v ‖ G H 0 1 ≡ ∫ 0 1   1 J ( y ) v ′ ( y ) 2 d y 1 2 ||v||_(GH_(0)^(1))-={int_(0)^(1)(1)/(J(y))|v^(')(y)|^(2)dy}^((1)/(2))\|v\|_{G H_{0}^{1}} \equiv\left\{\int_{0}^{1} \frac{1}{J(y)}\left|v^{\prime}(y)\right|^{2} d y\right\}^{\frac{1}{2}}‖v‖GH01≡{∫011J(y)|v′(y)|2dy}12
is norm on the space G H 0 1 G H 0 1 GH_(0)^(1)G H_{\mathbf{0}}^{1}GH01 and the following inequality takes place :
‖ v ‖ L ∞ [ 0 , 1 ] ≡ sup { | v ( y ) | : 0 ≤ y ≤ 1 } ≤ ‖ v ‖ G H 0 1 ‖ v ‖ L ∞ [ 0 , 1 ] ≡ sup { | v ( y ) | : 0 ≤ y ≤ 1 } ≤ ‖ v ‖ G H 0 1 ||v||_(L^(oo)[0,1])-=s u p{|v(y)|:0 <= y <= 1} <= ||v||_(GH_(0)^(1))\|v\|_{L^{\infty}[0,1]} \equiv \sup \{|v(y)|: 0 \leq y \leq 1\} \leq\|v\|_{G H_{0}^{1}}‖v‖L∞[0,1]≡sup{|v(y)|:0≤y≤1}≤‖v‖GH01 for ∀ v ∈ C H 0 1 ∀ v ∈ C H 0 1 AA v in CH_(0)^(1)\forall v \in C H_{0}^{1}∀v∈CH01.

Proof.

Because for ∀ v ∈ G H 0 1 ( 0 , 1 ) , ∃ u ∈ H 0 1 ( 0 , 1 ) ∀ v ∈ G H 0 1 ( 0 , 1 ) , ∃ u ∈ H 0 1 ( 0 , 1 ) AA v in GH_(0)^(1)(0,1),EE u inH_(0)^(1)(0,1)\forall v \in G H_{0}^{1}(0,1), \exists u \in H_{0}^{1}(0,1)∀v∈GH01(0,1),∃u∈H01(0,1) so that v ( y ) = u ( g ( y ) ) v ( y ) = u ( g ( y ) ) v(y)=u(g(y))v(y)=u(g(y))v(y)=u(g(y)) we have
‖ v ‖ G H 0 1 ( 0 , 1 ) = ( ∫ 0 1 1 J ( y ) | v ′ ( y ) | 2 d y ) 1 2 = ( ∫ 0 1 J ( y ) | u ′ ( g ( y ) ) | 2 d y ) 1 2 = | u | 1 ‖ v ‖ G H 0 1 ( 0 , 1 ) = ∫ 0 1   1 J ( y ) v ′ ( y ) 2 d y 1 2 = ∫ 0 1   J ( y ) u ′ ( g ( y ) ) 2 d y 1 2 = | u | 1 ||v||_(GH_(0)^(1)(0,1))=(int_(0)^(1)(1)/(J(y))|v^(')(y)|^(2)dy)^((1)/(2))=(int_(0)^(1)J(y)|u^(')(g(y))|^(2)dy)^((1)/(2))=|u|_(1)\|v\|_{G H_{0}^{1}(0,1)}=\left(\int_{0}^{1} \frac{1}{J(y)}\left|v^{\prime}(y)\right|^{2} d y\right)^{\frac{1}{2}}=\left(\int_{0}^{1} J(y)\left|u^{\prime}(g(y))\right|^{2} d y\right)^{\frac{1}{2}}=|u|_{1}‖v‖GH01(0,1)=(∫011J(y)|v′(y)|2dy)12=(∫01J(y)|u′(g(y))|2dy)12=|u|1
so ‖ . ‖ G H 0 1 ( 0 , 1 ) ‖ . ‖ G H 0 1 ( 0 , 1 ) ||.||_(GH_(0)^(1)(0,1))\|.\|_{G H_{0}^{1}(0,1)}‖.‖GH01(0,1) is norm on space G H 0 1 G H 0 1 GH_(0)^(1)G H_{0}^{1}GH01.
Let v ∈ G H 0 1 ( 0 , 1 ) ⟹ v v ∈ G H 0 1 ( 0 , 1 ) ⟹ v v in GH_(0)^(1)(0,1)Longrightarrow vv \in G H_{0}^{1}(0,1) \Longrightarrow vv∈GH01(0,1)⟹v is absolutely continuous, and v ( 0 ) = 0 v ( 0 ) = 0 v(0)=0v(0)=0v(0)=0, thus.
v ( y ) = ∫ 0 y v ′ ( t ) d t = ∫ 0 y J ( t ) ( 1 J ( t ) v ′ ( t ) ) d t , for ∀ y ∈ [ 0 , 1 ] v ( y ) = ∫ 0 y   v ′ ( t ) d t = ∫ 0 y   J ( t ) 1 J ( t ) v ′ ( t ) d t ,  for  ∀ y ∈ [ 0 , 1 ] v(y)=int_(0)^(y)v^(')(t)dt=int_(0)^(y)sqrt(J(t))((1)/(sqrt(J(t)))v^(')(t))dt," for "AA y in[0,1]v(y)=\int_{0}^{y} v^{\prime}(t) d t=\int_{0}^{y} \sqrt{J(t)}\left(\frac{1}{\sqrt{J(t)}} v^{\prime}(t)\right) d t, \text { for } \forall y \in[0,1]v(y)=∫0yv′(t)dt=∫0yJ(t)(1J(t)v′(t))dt, for ∀y∈[0,1]
By use of Cauchy-Schwartz inequality, we obtain
| v ( y ) | ≤ ( ∫ 0 1 J ( t ) d t ) 1 2 ( ∫ 0 1 1 J ( t ) | v ′ ( t ) | 2 d t ) 1 2 = ‖ v ‖ c ; H 0 1 . q.e.d. | v ( y ) | ≤ ∫ 0 1   J ( t ) d t 1 2 ∫ 0 1   1 J ( t ) v ′ ( t ) 2 d t 1 2 = ‖ v ‖ c ; H 0 1 . q.e.d.  |v(y)| <= (int_(0)^(1)J(t)dt)^((1)/(2))(int_(0)^(1)(1)/(J(t))|v^(')(t)|^(2)dt)^((1)/(2))=||v||_(c;H_(0)^(1))". q.e.d. "|v(y)| \leq\left(\int_{0}^{1} J(t) d t\right)^{\frac{1}{2}}\left(\int_{0}^{1} \frac{1}{J(t)}\left|v^{\prime}(t)\right|^{2} d t\right)^{\frac{1}{2}}=\|v\|_{c ; H_{0}^{1}} \text {. q.e.d. }|v(y)|≤(∫01J(t)dt)12(∫011J(t)|v′(t)|2dt)12=‖v‖c;H01. q.e.d. 
For the sake of simplicity, we consider a ( x ) ≡ 1 a ( x ) ≡ 1 a(x)-=1a(x) \equiv 1a(x)≡1 in ( P P PPP ) and consequently we have the following singularly perturbed problem :
( P 1 ) { − ε u ′ ′ ( x ) + u ′ ( x ) = f ( x ) , for x ∈ ( 0 , 1 ) u ( 0 ) = u ( 1 ) = 0 ( P 1 ) − ε u ′ ′ ( x ) + u ′ ( x ) = f ( x ) ,  for  x ∈ ( 0 , 1 ) u ( 0 ) = u ( 1 ) = 0 (P1){[-epsiu^('')(x)+u^(')(x)=f(x)","" for "x in(0","1)],[u(0)=u(1)=0]:}(P 1)\left\{\begin{array}{c} -\varepsilon u^{\prime \prime}(x)+u^{\prime}(x)=f(x), \text { for } x \in(0,1) \\ u(0)=u(1)=0 \end{array}\right.(P1){−εu′′(x)+u′(x)=f(x), for x∈(0,1)u(0)=u(1)=0
Making the change of variable x = g ( y ) x = g ( y ) x=g(y)x=g(y)x=g(y) we obtain:
( P 2 ) { − ε ( 1 J ( y ) v ′ ( y ) ) ′ + v ′ ( y ) = J ( y ) F ( y ) , for x ∈ ( 0 , 1 ) v ( 0 ) = v ( 1 ) = 0 ( P 2 ) − ε 1 J ( y ) v ′ ( y ) ′ + v ′ ( y ) = J ( y ) F ( y ) ,  for  x ∈ ( 0 , 1 ) v ( 0 ) = v ( 1 ) = 0 (P2){[-epsi((1)/(J(y))v^(')(y))^(')+v^(')(y)=J(y)F(y)","" for "x in(0","1)],[v(0)=v(1)=0]:}(P 2)\left\{\begin{array}{c} -\varepsilon\left(\frac{1}{J(y)} v^{\prime}(y)\right)^{\prime}+v^{\prime}(y)=J(y) F(y), \text { for } x \in(0,1) \\ v(0)=v(1)=0 \end{array}\right.(P2){−ε(1J(y)v′(y))′+v′(y)=J(y)F(y), for x∈(0,1)v(0)=v(1)=0
where F ( y ) = f ( g ( y ) ) F ( y ) = f ( g ( y ) ) F(y)=f(g(y))F(y)=f(g(y))F(y)=f(g(y)), and v ( y ) = u ( g ( y ) ) v ( y ) = u ( g ( y ) ) v(y)=u(g(y))v(y)=u(g(y))v(y)=u(g(y)).
Problem (P1) has one solution,
(1) { u ∈ H 0 1 ( 0 , 1 ) , such that ε ∫ 0 1 u ′ ( x ) w ′ ( x ) d x + ∫ 0 1 u ′ ( x ) w ( x ) d x = ∫ 0 1 f ( x ) w ( x ) d x , ∀ w ∈ H 0 1 ( 0 , 1 ) . u ∈ H 0 1 ( 0 , 1 ) ,  such that  ε ∫ 0 1   u ′ ( x ) w ′ ( x ) d x + ∫ 0 1   u ′ ( x ) w ( x ) d x = ∫ 0 1   f ( x ) w ( x ) d x , ∀ w ∈ H 0 1 ( 0 , 1 ) . {[u inH_(0)^(1)(0","1)","" such that "],[epsiint_(0)^(1)u^(')(x)w^(')(x)dx+int_(0)^(1)u^(')(x)w(x)dx=int_(0)^(1)f(x)w(x)dx","AA w inH_(0)^(1)(0","1).]:}\left\{\begin{array}{c}u \in H_{0}^{1}(0,1), \text { such that } \\ \varepsilon \int_{0}^{1} u^{\prime}(x) w^{\prime}(x) d x+\int_{0}^{1} u^{\prime}(x) w(x) d x=\int_{0}^{1} f(x) w(x) d x, \forall w \in H_{0}^{1}(0,1) .\end{array}\right.{u∈H01(0,1), such that ε∫01u′(x)w′(x)dx+∫01u′(x)w(x)dx=∫01f(x)w(x)dx,∀w∈H01(0,1).
and problem ( P 2 P 2 P2P 2P2 ) will have one solution,
(2) { v ∈ G H 0 1 ( 0 , 1 ) , such that ε ∫ 0 1 1 J ( y ) v ′ ( y ) w ′ ( y ) d y + ∫ 0 1 v ′ ( y ) w ( y ) d y = ∫ 0 1 F ( y ) J ( y ) w ( y ) , ∀ w ∈ G H 0 1 ( 0 , 1 ) . v ∈ G H 0 1 ( 0 , 1 ) ,  such that  ε ∫ 0 1   1 J ( y ) v ′ ( y ) w ′ ( y ) d y + ∫ 0 1   v ′ ( y ) w ( y ) d y = ∫ 0 1   F ( y ) J ( y ) w ( y ) , ∀ w ∈ G H 0 1 ( 0 , 1 ) . {[v in GH_(0)^(1)(0","1)","" such that "],[epsiint_(0)^(1)(1)/(J(y))v^(')(y)w^(')(y)dy+int_(0)^(1)v^(')(y)w(y)dy=int_(0)^(1)F(y)J(y)w(y)","AA w in GH_(0)^(1)(0","1).]:}\left\{\begin{array}{c}v \in G H_{0}^{1}(0,1), \text { such that } \\ \varepsilon \int_{0}^{1} \frac{1}{J(y)} v^{\prime}(y) w^{\prime}(y) d y+\int_{0}^{1} v^{\prime}(y) w(y) d y=\int_{0}^{1} F(y) J(y) w(y), \forall w \in G H_{0}^{1}(0,1) .\end{array}\right.{v∈GH01(0,1), such that ε∫011J(y)v′(y)w′(y)dy+∫01v′(y)w(y)dy=∫01F(y)J(y)w(y),∀w∈GH01(0,1). (see [3]).
We will describe further the approximation scheme of the solution of pröblem (2).
Let L > 0 L > 0 L > 0L>0L>0 fixed natural number, and Π L : 0 = y 0 < y 1 < … < y N = 1 Π L : 0 = y 0 < y 1 < … < y N = 1 Pi^(L):0=y_(0) < y_(1) < dots < y_(N)=1\Pi^{L}: 0=y_{0}<y_{1}<\ldots<y_{N}=1ΠL:0=y0<y1<…<yN=1 a uniform division of interval [ 0 , 1 ] [ 0 , 1 ] [0,1][0,1][0,1], where N = 2 L N = 2 L N=2^(L)N=2^{L}N=2L and y j = j 2 L , 0 ≤ j ≤ 2 L y j = j 2 L , 0 ≤ j ≤ 2 L y_(j)=(j)/(2^(L)),0 <= j <= 2^(L)y_{j}=\frac{j}{2^{L}}, 0 \leq j \leq 2^{L}yj=j2L,0≤j≤2L. We define set V L V L V^(L)V^{L}VL as being the subspace of G H 0 1 G H 0 1 GH_(0)^(1)G H_{0}^{1}GH01, which contains those functions w w www satisfying,
(3) ( 1 J ( y ) w ′ ( y ) ) ′ = 0 1 J ( y ) w ′ ( y ) ′ = 0 ((1)/(J(y))w^(')(y))^(')=0\left(\frac{1}{J(y)} w^{\prime}(y)\right)^{\prime}=0(1J(y)w′(y))′=0 where y j < y < y j + 1 y j < y < y j + 1 y_(j) < y < y_(j+1)y_{j}<y<y_{j+1}yj<y<yj+1 for each 0 ≤ j ≤ 2 L − 1 0 ≤ j ≤ 2 L − 1 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-10≤j≤2L−1.
In V L V L V^(L)V^{L}VL we consider the system of functions:
(4) Φ L , k ( y ) = ∫ 0 y J ( s ) χ L , k − 1 ( s ) d s ∫ 0 1 J ( s ) χ L , k − 1 ( s ) d s − ∫ 0 y J ( s ) χ L , k ( s ) d s ∫ 0 1 J ( s ) χ L , k ( s ) d s  (4)  Φ L , k ( y ) = ∫ 0 y   J ( s ) χ L , k − 1 ( s ) d s ∫ 0 1   J ( s ) χ L , k − 1 ( s ) d s − ∫ 0 y   J ( s ) χ L , k ( s ) d s ∫ 0 1   J ( s ) χ L , k ( s ) d s " (4) "Phi_(L,k)(y)=(int_(0)^(y)J(s)chi_(L,k-1)(s)ds)/(int_(0)^(1)J(s)chi_(L,k-1)(s)ds)-(int_(0)^(y)J(s)chi_(L,k)(s)ds)/(int_(0)^(1)J(s)chi_(L,k)(s)ds)\text { (4) } \Phi_{L, k}(y)=\frac{\int_{0}^{y} J(s) \chi_{L, k-1}(s) d s}{\int_{0}^{1} J(s) \chi_{L, k-1}(s) d s}-\frac{\int_{0}^{y} J(s) \chi_{L, k}(s) d s}{\int_{0}^{1} J(s) \chi_{L, k}(s) d s} (4) ΦL,k(y)=∫0yJ(s)χL,k−1(s)ds∫01J(s)χL,k−1(s)ds−∫0yJ(s)χL,k(s)ds∫01J(s)χL,k(s)ds
for 1 ≤ k ≤ 2 L − 1 1 ≤ k ≤ 2 L − 1 1 <= k <= 2^(L)-11 \leq k \leq 2^{L}-11≤k≤2L−1, where χ L , k χ L , k chi_(L,k)\chi_{L, k}χL,k is characteristic function of interval [ k 2 L , k + 1 2 L ] k 2 L , k + 1 2 L [(k)/(2^(L)),(k+1)/(2^(L))]\left[\frac{k}{2^{L}}, \frac{k+1}{2^{L}}\right][k2L,k+12L]. Obviously Φ L , k ∈ V L Φ L , k ∈ V L Phi_(L,k)inV^(L)\Phi_{L, k} \in V^{L}ΦL,k∈VL for 1 ≤ k ≤ 2 L − 1 1 ≤ k ≤ 2 L − 1 1 <= k <= 2^(L)-11 \leq k \leq 2^{L}-11≤k≤2L−1 and Φ L , k ( 0 ) = Φ L , k ( 1 ) = 0 Φ L , k ( 0 ) = Φ L , k ( 1 ) = 0 Phi_(L,k)(0)=Phi_(L,k)(1)=0\Phi_{L, k}(0)=\Phi_{L, k}(1)=0ΦL,k(0)=ΦL,k(1)=0.
Because in addition,
Φ L , k ( y j ) = δ k , j , 1 ≤ k ≤ 2 L − 1 , 0 ≤ j ≤ 2 L Φ L , k y j = δ k , j , 1 ≤ k ≤ 2 L − 1 , 0 ≤ j ≤ 2 L Phi_(L,k)(y_(j))=delta_(k,j),1 <= k <= 2^(L)-1,0 <= j <= 2^(L)\Phi_{L, k}\left(y_{j}\right)=\delta_{k, j}, 1 \leq k \leq 2^{L}-1,0 \leq j \leq 2^{L}ΦL,k(yj)=δk,j,1≤k≤2L−1,0≤j≤2L
very function g ∈ V L g ∈ V L g inV^(L)g \in V^{L}g∈VL can be represented using base { Φ L , k } k = 1 2 L − 1 Φ L , k k = 1 2 L − 1 {Phi_(L,k)}_(k=1)^(2^(L)-1)\left\{\Phi_{L, k}\right\}_{k=1}^{2^{L}-1}{ΦL,k}k=12L−1 thus
g ( y ) = ∑ k = 1 2 L − 1 g ( y k ) Φ L , k ( y ) . g ( y ) = ∑ k = 1 2 L − 1   g y k Φ L , k ( y ) . g(y)=sum_(k=1)^(2^(L)-1)g(y_(k))Phi_(L,k)(y).g(y)=\sum_{k=1}^{2^{L}-1} g\left(y_{k}\right) \Phi_{L, k}(y) .g(y)=∑k=12L−1g(yk)ΦL,k(y).
onsequently { Φ L , k } , 1 ≤ k ≤ 2 L − 1 Φ L , k , 1 ≤ k ≤ 2 L − 1 {Phi_(L,k)},1 <= k <= 2^(L)-1\left\{\Phi_{L, k}\right\}, 1 \leq k \leq 2^{L}-1{ΦL,k},1≤k≤2L−1 form a base in V L V L V^(L)V^{L}VL.
For v ∈ G H 0 1 ( 0 , 1 ) v ∈ G H 0 1 ( 0 , 1 ) v in GH_(0)^(1)(0,1)v \in G H_{0}^{1}(0,1)v∈GH01(0,1), we define the interpolant w L ∗ w L ∗ w_(L)^(**)w_{L}^{*}wL∗ from V L V L V^(L)V^{L}VL as being the nique element from V L V L V^(L)V^{L}VL which satisfies
w L ∗ ( y j ) = v ( y j ) , 0 ≤ j ≤ 2 L . w L ∗ y j = v y j , 0 ≤ j ≤ 2 L . w_(L)^(**)(y_(j))=v(y_(j)),0 <= j <= 2^(L).w_{L}^{*}\left(y_{j}\right)=v\left(y_{j}\right), 0 \leq j \leq 2^{L} .wL∗(yj)=v(yj),0≤j≤2L.
emma 2 Let v v vvv be the solution of problem ( P 2 P 2 P2P 2P2 ) and be w L ∗ w L ∗ w_(L)^(**)w_{L}^{*}wL∗ the unique V L − V L − V^(L)-V^{L}-VL− nterpolant . Then,
‖ w L ∗ − v ‖ L ∞ [ y j , y j + 1 ] ≤ h g ′ ( ξ j ) | ∫ x j x j + 1 u ′ ′ ( x ) d x | , 0 ≤ j ≤ 2 L − 1 w L ∗ − v L ∞ y j , y j + 1 ≤ h g ′ ξ j ∫ x j x j + 1   u ′ ′ ( x ) d x , 0 ≤ j ≤ 2 L − 1 ||w_(L)^(**)-v||_(L^(oo)[y_(j),y_(j+1)]) <= hg^(')(xi_(j))|int_(x_(j))^(x_(j+1))u^('')(x)dx|,0 <= j <= 2^(L)-1\left\|w_{L}^{*}-v\right\|_{L^{\infty}\left[y_{j}, y_{j+1}\right]} \leq h g^{\prime}\left(\xi_{j}\right)\left|\int_{x_{j}}^{x_{j+1}} u^{\prime \prime}(x) d x\right|, 0 \leq j \leq 2^{L}-1‖wL∗−v‖L∞[yj,yj+1]≤hg′(ξj)|∫xjxj+1u′′(x)dx|,0≤j≤2L−1
here h = 1 2 L , ξ j ∈ ( y j , y j + 1 ) h = 1 2 L , ξ j ∈ y j , y j + 1 h=(1)/(2^(L)),xi_(j)in(y_(j),y_(j+1))h=\frac{1}{2^{L}}, \xi_{j} \in\left(y_{j}, y_{j+1}\right)h=12L,ξj∈(yj,yj+1) and x j = g ( y j ) x j = g y j x_(j)=g(y_(j))x_{j}=g\left(y_{j}\right)xj=g(yj) for 0 ≤ j ≤ 2 L − 1 0 ≤ j ≤ 2 L − 1 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-10≤j≤2L−1.
Proof.
Consider a certain interval [ y j , y j + 1 ] , 0 ≤ j ≤ 2 L − 1 y j , y j + 1 , 0 ≤ j ≤ 2 L − 1 [y_(j),y_(j+1)],0 <= j <= 2^(L)-1\left[y_{j}, y_{j+1}\right], 0 \leq j \leq 2^{L}-1[yj,yj+1],0≤j≤2L−1. Because w L ∗ − v w L ∗ − v w_(L)^(**)-vw_{L}^{*}-vwL∗−v ancels at the end of this interval, we obtain, integrating by parts :
y j y j + 1 1 J ( t ) { w L ∗ ′ ( t ) − v ′ ( t ) } 2 d t = ∫ y j y j + 1 1 J ( t ) [ w L ∗ ′ ( t ) − v ′ ( t ) ] [ w L ∗ ′ ( t ) − v ′ ( t ) ] d t = y j y j + 1 1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) 2 d t = ∫ y j y j + 1   1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) w L ∗ ′ ( t ) − v ′ ( t ) d t = _(y_(j))^(y_(j+1))(1)/(J(t)){w_(L)^(**')(t)-v^(')(t)}^(2)dt=int_(y_(j))^(y_(j+1))(1)/(J(t))[w_(L)^(**')(t)-v^(')(t)][w_(L)^(**')(t)-v^(')(t)]dt={ }_{y_{j}}^{y_{j+1}} \frac{1}{J(t)}\left\{w_{L}^{* \prime}(t)-v^{\prime}(t)\right\}^{2} d t=\int_{y_{j}}^{y_{j+1}} \frac{1}{J(t)}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right]\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right] d t=yjyj+11J(t){wL∗′(t)−v′(t)}2dt=∫yjyj+11J(t)[wL∗′(t)−v′(t)][wL∗′(t)−v′(t)]dt=
1 J ( t ) [ w L ∗ ′ ( t ) − v ′ ( t ) ] [ w L ∗ ( t ) − v ( t ) ] | y j y j + 1 − ∫ y j y j + 1 { 1 J ( t ) [ w L ∗ ′ ( t ) − v ′ ( t ) ] } ′ [ w L ∗ ( t ) − v ( t ) ] d t = 1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) w L ∗ ( t ) − v ( t ) y j y j + 1 − ∫ y j y j + 1   1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) ′ w L ∗ ( t ) − v ( t ) d t = (1)/(J(t))[w_(L)^(**')(t)-v^(')(t)][w_(L)^(**)(t)-v(t)]|_(y_(j))^(y_(j+1))-int_(y_(j))^(y_(j+1)){(1)/(J(t))[w_(L)^(**')(t)-v^(')(t)]}^(')[w_(L)^(**)(t)-v(t)]dt=\left.\frac{1}{J(t)}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right]\left[w_{L}^{*}(t)-v(t)\right]\right|_{y_{j}} ^{y_{j+1}}-\int_{y_{j}}^{y_{j+1}}\left\{\frac{1}{J(t)}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right]\right\}^{\prime}\left[w_{L}^{*}(t)-v(t)\right] d t=1J(t)[wL∗′(t)−v′(t)][wL∗(t)−v(t)]|yjyj+1−∫yjyj+1{1J(t)[wL∗′(t)−v′(t)]}′[wL∗(t)−v(t)]dt=
ecause w L ∗ ( y j ) − v ( y j ) = 0 , w L ∗ ( y j + 1 ) − v ( y j + 1 ) = 0 w L ∗ y j − v y j = 0 , w L ∗ y j + 1 − v y j + 1 = 0 w_(L)^(**)(y_(j))-v(y_(j))=0,w_(L)^(**)(y_(j+1))-v(y_(j+1))=0w_{L}^{*}\left(y_{j}\right)-v\left(y_{j}\right)=0, w_{L}^{*}\left(y_{j+1}\right)-v\left(y_{j+1}\right)=0wL∗(yj)−v(yj)=0,wL∗(yj+1)−v(yj+1)=0 and v , w L ∗ ∈ G H 0 1 ( 0 , 1 ) ⟹ u , u L ∗ ∈ H 0 1 ( 0 , 1 ) v , w L ∗ ∈ G H 0 1 ( 0 , 1 ) ⟹ u , u L ∗ ∈ H 0 1 ( 0 , 1 ) v,w_(L)^(**)in GH_(0)^(1)(0,1)Longrightarrow u,u_(L)^(**)inH_(0)^(1)(0,1)v, w_{L}^{*} \in G H_{0}^{1}(0,1) \Longrightarrow u, u_{L}^{*} \in H_{0}^{1}(0,1)v,wL∗∈GH01(0,1)⟹u,uL∗∈H01(0,1) so that v ( y ) = u ( g ( y ) ) , w L ∗ ( y ) = u L ∗ ( g ( y ) ) ⟹ v ′ ( y ) = ( y ) u ′ ( g ( y ) ) , w L ∗ ′ ( y ) = J ( y ) u L ∗ ′ ( g ( y ) ) ⟹ 1 J ( t ) [ w L ∗ ′ ( y ) − v ′ ( y ) ] = J ( y ) [ u L ∗ ( g ( y ) ) − u ′ ( g ( y ) ) ] v ( y ) = u ( g ( y ) ) , w L ∗ ( y ) = u L ∗ ( g ( y ) ) ⟹ v ′ ( y ) = ( y ) u ′ ( g ( y ) ) , w L ∗ ′ ( y ) = J ( y ) u L ∗ ′ ( g ( y ) ) ⟹ 1 J ( t ) w L ∗ ′ ( y ) − v ′ ( y ) = J ( y ) u L ∗ ( g ( y ) ) − u ′ ( g ( y ) ) v(y)=u(g(y)),w_(L)^(**)(y)=u_(L)^(**)(g(y))Longrightarrowv^(')(y)=(y)u^(')(g(y)),w_(L)^(**')(y)=J(y)u_(L)^(**')(g(y))Longrightarrow(1)/(J(t))[w_(L)^(**')(y)-v^(')(y)]=J(y)[u_(L)^(**)(g(y))-u^(')(g(y))]v(y)=u(g(y)), w_{L}^{*}(y)=u_{L}^{*}(g(y)) \Longrightarrow v^{\prime}(y)= (y) u^{\prime}(g(y)), w_{L}^{* \prime}(y)=J(y) u_{L}^{* \prime}(g(y)) \Longrightarrow \frac{1}{J(t)}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right]=J(y)\left[u_{L}^{*}(g(y))-u^{\prime}(g(y))\right]v(y)=u(g(y)),wL∗(y)=uL∗(g(y))⟹v′(y)=(y)u′(g(y)),wL∗′(y)=J(y)uL∗′(g(y))⟹1J(t)[wL∗′(y)−v′(y)]=J(y)[uL∗(g(y))−u′(g(y))]
= − ∫ y j y j + 1 ( 1 J ( t ) w L ∗ ′ ( t ) ) [ w L ∗ ( t ) − v ( t ) ] d t + ∫ y j y j + 1 [ 1 J ( t ) v ′ ( t ) ] ′ [ w L ∗ ( t ) − v ( t ) ] d t = = − ∫ y j y j + 1   1 J ( t ) w L ∗ ′ ( t ) w L ∗ ( t ) − v ( t ) d t + ∫ y j y j + 1   1 J ( t ) v ′ ( t ) ′ w L ∗ ( t ) − v ( t ) d t = =-int_(y_(j))^(y_(j+1))((1)/(J(t))w_(L)^(**')(t))[w_(L)^(**)(t)-v(t)]dt+int_(y_(j))^(y_(j+1))[(1)/(J(t))v^(')(t)]^(')[w_(L)^(**)(t)-v(t)]dt==-\int_{y_{j}}^{y_{j+1}}\left(\frac{1}{J(t)} w_{L}^{* \prime}(t)\right)\left[w_{L}^{*}(t)-v(t)\right] d t+\int_{y_{j}}^{y_{j+1}}\left[\frac{1}{J(t)} v^{\prime}(t)\right]^{\prime}\left[w_{L}^{*}(t)-v(t)\right] d t==−∫yjyj+1(1J(t)wL∗′(t))[wL∗(t)−v(t)]dt+∫yjyj+1[1J(t)v′(t)]′[wL∗(t)−v(t)]dt=
(5) ∫ y j y j + 1 v ′ ( t ) − J ( t ) ε [ w L ∗ ( t ) − v ( t ) ] d t ≤ | ∫ y j y j + 1 v ′ ( t ) − J ( t ) ε d t | ‖ w L ∗ − v ‖ L ∞ | y j , y j + 1 | (5) ∫ y j y j + 1   v ′ ( t ) − J ( t ) ε w L ∗ ( t ) − v ( t ) d t ≤ ∫ y j y j + 1   v ′ ( t ) − J ( t ) ε d t w L ∗ − v L ∞ y j , y j + 1 {:(5)int_(y_(j))^(y_(j)+1)(v^(')(t)-J(t))/(epsi)[w_(L)^(**)(t)-v(t)]dt <= |int_(y_(j))^(y_(j+1))(v^(')(t)-J(t))/(epsi)dt|||w_(L)^(**)-v||_(L^(oo)|y_(j),y_(j+1)|):}\begin{equation*} \int_{y_{j}}^{y_{j}+1} \frac{v^{\prime}(t)-J(t)}{\varepsilon}\left[w_{L}^{*}(t)-v(t)\right] d t \leq\left|\int_{y_{j}}^{y_{j+1}} \frac{v^{\prime}(t)-J(t)}{\varepsilon} d t\right|\left\|w_{L}^{*}-v\right\|_{L^{\infty}\left|y_{j}, y_{j+1}\right|} \tag{5} \end{equation*}(5)∫yjyj+1v′(t)−J(t)ε[wL∗(t)−v(t)]dt≤|∫yjyj+1v′(t)−J(t)εdt|‖wL∗−v‖L∞|yj,yj+1|
For ∀ y ∈ [ y j , y j + 1 ] , 0 ≤ j ≤ 2 L − 1 ∀ y ∈ y j , y j + 1 , 0 ≤ j ≤ 2 L − 1 AA y in[y_(j),y_(j+1)],0 <= j <= 2^(L)-1\forall y \in\left[y_{j}, y_{j+1}\right], 0 \leq j \leq 2^{L}-1∀y∈[yj,yj+1],0≤j≤2L−1, we have :
w L ∗ ( y ) − ι ′ ( y ) = ∫ y j y j + 1 [ w L ∗ ′ ( t ) − v ′ ( t ) ] d t = ∫ y j y j + 1 J ( t ) 1 J ( t ) [ w L ∗ ′ ( t ) − v ′ ( t ) ] d t ≤ ≤ ( ∫ y j y j + 1 J ( t ) d t ) 1 2 { ∫ y j y j + 1 1 J ( t ) [ w L ∗ ′ ( t ) − v ′ ( t ) ] 2 } 1 2 ⟹ ‖ w L ∗ − v ‖ L ∞ [ y j , y j + 1 ] ≤ { ∫ y j y j + 1 J ( t ) d t } 1 2 { ∫ y j y j + 1 1 J ( t ) [ w L ∗ ′ ( t ) − v ′ ( t ) ] 2 d t } 1 2 w L ∗ ( y ) − ι ′ ( y ) = ∫ y j y j + 1   w L ∗ ′ ( t ) − v ′ ( t ) d t = ∫ y j y j + 1   J ( t ) 1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) d t ≤ ≤ ∫ y j y j + 1   J ( t ) d t 1 2 ∫ y j y j + 1   1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) 2 1 2 ⟹ w L ∗ − v L ∞ y j , y j + 1 ≤ ∫ y j y j + 1   J ( t ) d t 1 2 ∫ y j y j + 1   1 J ( t ) w L ∗ ′ ( t ) − v ′ ( t ) 2 d t 1 2 {:[w_(L)^(**)(y)-iota^(')(y)=int_(y_(j))^(y_(j+1))[w_(L)^(**')(t)-v^(')(t)]dt=int_(y_(j))^(y_(j+1))sqrt(J(t))(1)/(sqrt(J(t)))[w_(L)^(**')(t)-v^(')(t)]dt <= ],[ <= (int_(y_(j))^(y_(j+1))J(t)dt)^((1)/(2)){int_(y_(j))^(y_(j+1))(1)/(J(t))[w_(L)^(**')(t)-v^(')(t)]^(2)}^((1)/(2))],[Longrightarrow||w_(L)^(**)-v||_(L^(oo)[y_(j),y_(j+1)]) <= {int_(y_(j))^(y_(j+1))J(t)dt}^((1)/(2)){int_(y_(j))^(y_(j+1))(1)/(J(t))[w_(L)^(**')(t)-v^(')(t)]^(2)dt}^((1)/(2))]:}\begin{aligned} w_{L}^{*}(y)-\iota^{\prime}(y) & =\int_{y_{j}}^{y_{j+1}}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right] d t=\int_{y_{j}}^{y_{j+1}} \sqrt{J(t)} \frac{1}{\sqrt{J(t)}}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right] d t \leq \\ & \leq\left(\int_{y_{j}}^{y_{j+1}} J(t) d t\right)^{\frac{1}{2}}\left\{\int_{y_{j}}^{y_{j+1}} \frac{1}{J(t)}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right]^{2}\right\}^{\frac{1}{2}} \\ \Longrightarrow & \left\|w_{L}^{*}-v\right\|_{L^{\infty}\left[y_{j}, y_{j+1}\right]} \leq\left\{\int_{y_{j}}^{y_{j+1}} J(t) d t\right\}^{\frac{1}{2}}\left\{\int_{y_{j}}^{y_{j+1}} \frac{1}{J(t)}\left[w_{L}^{* \prime}(t)-v^{\prime}(t)\right]^{2} d t\right\}^{\frac{1}{2}} \end{aligned}wL∗(y)−ι′(y)=∫yjyj+1[wL∗′(t)−v′(t)]dt=∫yjyj+1J(t)1J(t)[wL∗′(t)−v′(t)]dt≤≤(∫yjyj+1J(t)dt)12{∫yjyj+11J(t)[wL∗′(t)−v′(t)]2}12⟹‖wL∗−v‖L∞[yj,yj+1]≤{∫yjyj+1J(t)dt}12{∫yjyj+11J(t)[wL∗′(t)−v′(t)]2dt}12
Using this and inequality (5), we obtain:
‖ w L ∗ − | ‖ L ∞ | y j , y j + 1 | ≤ ( ∫ y j y j + 1 J ( t ) d t ) | ∫ y j y j + 1 v ′ ( t ) − J ( t ) ε d t | = h g ′ ( ξ j ) | ∫ y j y j + 1 u ′ ′ ( x ) d x w L ∗ − ‖ L ∞ y j , y j + 1 ≤ ∫ y j y j + 1   J ( t ) d t ∫ y j y j + 1   v ′ ( t ) − J ( t ) ε d t = h g ′ ξ j ∫ y j y j + 1   u ′ ′ ( x ) d x ||w_(L)^(**)-|||_(L^(oo)|y_(j),y_(j+1)|) <= (int_(y_(j))^(y_(j+1))J(t)dt)|int_(y_(j))^(y_(j+1))(v^(')(t)-J(t))/(epsi)dt|=hg^(')(xi_(j))|int_(y_(j))^(y_(j+1))u^('')(x)dx:}\left\|w_{L}^{*}-\left|\|_{L^{\infty}\left|y_{j}, y_{j+1}\right|} \leq\left(\int_{y_{j}}^{y_{j+1}} J(t) d t\right)\right| \int_{y_{j}}^{y_{j+1}} \frac{v^{\prime}(t)-J(t)}{\varepsilon} d t\left|=h g^{\prime}\left(\xi_{j}\right)\right| \int_{y_{j}}^{y_{j+1}} u^{\prime \prime}(x) d x\right.‖wL∗−|‖L∞|yj,yj+1|≤(∫yjyj+1J(t)dt)|∫yjyj+1v′(t)−J(t)εdt|=hg′(ξj)|∫yjyj+1u′′(x)dx
where h = 1 2 L , ξ j ∈ ( y j , y j + 1 ) h = 1 2 L , ξ j ∈ y j , y j + 1 h=(1)/(2^(L)),xi_(j)in(y_(j),y_(j+1))h=\frac{1}{2^{L}}, \xi_{j} \in\left(y_{j}, y_{j+1}\right)h=12L,ξj∈(yj,yj+1) and x j = g ( y j ) x j = g y j x_(j)=g(y_(j))x_{j}=g\left(y_{j}\right)xj=g(yj) for 0 ≤ j ≤ 2 L − 1 0 ≤ j ≤ 2 L − 1 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-10≤j≤2L−1.
Let w L ( g ( y ) ) = w L ∗ ( y ) w L ( g ( y ) ) = w L ∗ ( y ) w_(L)(g(y))=w_(L)^(**)(y)w_{L}(g(y))=w_{L}^{*}(y)wL(g(y))=wL∗(y) for ∀ y ∈ [ y j , y j + 1 ] ∀ y ∈ y j , y j + 1 AA y in[y_(j),y_(j+1)]\forall y \in\left[y_{j}, y_{j+1}\right]∀y∈[yj,yj+1]; then, we have
u ( x ) − w L ( x ) = u ( g ( y ) ) − w L ( g ( y ) ) = v ( y ) − w L ∗ ( y ) ⟹ | u ( x ) − w L ( x ) | = | v ( y ) − w L ∗ ( y ) | ≤ ‖ v − w L ∗ ‖ L ∞ [ y j , y j + 1 ] ⟹ max x ∈ { | x j , x j + 1 | | u ( x ) − w L ( x ) | ≤ ‖ v − w L ∗ ‖ L ∞ [ y j , y j + 1 ] ≤ h g ′ ( ξ j ) | ∫ y j y j + 1 u ′ ′ ( x ) d x | u ( x ) − w L ( x ) = u ( g ( y ) ) − w L ( g ( y ) ) = v ( y ) − w L ∗ ( y ) ⟹ u ( x ) − w L ( x ) = v ( y ) − w L ∗ ( y ) ≤ v − w L ∗ L ∞ y j , y j + 1 ⟹ max x ∈ x j , x j + 1   u ( x ) − w L ( x ) ≤ v − w L ∗ L ∞ y j , y j + 1 ≤ h g ′ ξ j ∫ y j y j + 1   u ′ ′ ( x ) d x {:[u(x)-w_(L)(x)=u(g(y))-w_(L)(g(y))=v(y)-w_(L)^(**)(y)],[Longrightarrow|u(x)-w_(L)(x)|=|v(y)-w_(L)^(**)(y)| <= ||v-w_(L)^(**)||_(L^(oo)[y_(j),y_(j+1)])],[Longrightarrowmax_(x in{|x_(j),x_(j+1)|:})|u(x)-w_(L)(x)| <= ||v-w_(L)^(**)||_(L^(oo)[y_(j),y_(j+1)]) <= hg^(')(xi_(j))|int_(y_(j))^(y_(j+1))u^('')(x)dx|]:}\begin{gathered} u(x)-w_{L}(x)=u(g(y))-w_{L}(g(y))=v(y)-w_{L}^{*}(y) \\ \Longrightarrow\left|u(x)-w_{L}(x)\right|=\left|v(y)-w_{L}^{*}(y)\right| \leq\left\|v-w_{L}^{*}\right\|_{L^{\infty}\left[y_{j}, y_{j+1}\right]} \\ \Longrightarrow \max _{x \in\left\{\left|x_{j}, x_{j+1}\right|\right.}\left|u(x)-w_{L}(x)\right| \leq\left\|v-w_{L}^{*}\right\|_{L^{\infty}\left[y_{j}, y_{j+1}\right]} \leq h g^{\prime}\left(\xi_{j}\right)\left|\int_{y_{j}}^{y_{j+1}} u^{\prime \prime}(x) d x\right| \end{gathered}u(x)−wL(x)=u(g(y))−wL(g(y))=v(y)−wL∗(y)⟹|u(x)−wL(x)|=|v(y)−wL∗(y)|≤‖v−wL∗‖L∞[yj,yj+1]⟹maxx∈{|xj,xj+1||u(x)−wL(x)|≤‖v−wL∗‖L∞[yj,yj+1]≤hg′(ξj)|∫yjyj+1u′′(x)dx|
and, therefore,
‖ u − w L ‖ L ∞ [ x j , x j + 1 ] ≤ h g ′ ( ξ j ) | ∫ y j y j + 1 u ′ ′ ( x ) d x | .q.e.d. u − w L L ∞ x j , x j + 1 ≤ h g ′ ξ j ∫ y j y j + 1   u ′ ′ ( x ) d x .q.e.d.  ||u-w_(L)||_(L^(oo)[x_(j),x_(j+1)]) <= hg^(')(xi_(j))|int_(y_(j))^(y_(j+1))u^('')(x)dx|".q.e.d. "\left\|u-w_{L}\right\|_{L^{\infty}\left[x_{j}, x_{j+1}\right]} \leq h g^{\prime}\left(\xi_{j}\right)\left|\int_{y_{j}}^{y_{j+1}} u^{\prime \prime}(x) d x\right| \text {.q.e.d. }‖u−wL‖L∞[xj,xj+1]≤hg′(ξj)|∫yjyj+1u′′(x)dx|.q.e.d. 
With a view to obtain an evaluation of approximation error by Galerkin method, we will compare Galerkin approximation to the approximation by V L V L V^(L)V^{L}VL - interpolant of solution of problem (P2).
Theorem 3 Let v v vvv be the solution of problem (P2), let w L # w L # w_(L)^(#)w_{L}^{\#}wL# be the Galerkin approximation of it, and w L ∗ w L ∗ w_(L)^(**)w_{L}^{*}wL∗ its interpolant in space V L V L V^(L)V^{L}VL. Then, the following inequalities
‖ w L # − v ‖ L ∞ [ 0 , 1 ] ≤ 2 ‖ w L ∗ − v ‖ L ∞ [ 0 , 1 ] w L # − v L ∞ [ 0 , 1 ] ≤ 2 w L ∗ − v L ∞ [ 0 , 1 ] ||w_(L)^(#)-v||_(L^(oo)[0,1]) <= 2||w_(L)^(**)-v||_(L^(oo)[0,1])\left\|w_{L}^{\#}-v\right\|_{L^{\infty}[0,1]} \leq 2\left\|w_{L}^{*}-v\right\|_{L^{\infty}[0,1]}‖wL#−v‖L∞[0,1]≤2‖wL∗−v‖L∞[0,1]
holds.
Proof. We will use the fact that on every interval ( y j , y j + 1 ) , 0 ≤ j ≤ 2 L − 1 y j , y j + 1 , 0 ≤ j ≤ 2 L − 1 (y_(j),y_(j+1)),0 <= j <= 2^(L)-1\left(y_{j}, y_{j+1}\right), 0 \leq j \leq 2^{L}-1(yj,yj+1),0≤j≤2L−1, the functions from V L V L V^(L)V^{L}VL satisfy the differential equation
( 1 J ( y ) w L ∗ ′ ( y ) ) ′ = 0 1 J ( y ) w L ∗ ′ ( y ) ′ = 0 ((1)/(J(y))w_(L)^(**')(y))^(')=0\left(\frac{1}{J(y)} w_{L}^{* \prime}(y)\right)^{\prime}=0(1J(y)wL∗′(y))′=0
Let M k = ∫ 0 1 { 1 J ( y ) w L ∗ ′ ( y ) Φ L , k ′ ( y ) + w L ∗ ′ ( y ) Φ L , k ( y ) − F ( y ) J ( y ) Φ L , k ( y ) } d y , 1 ≤ k ≤ 2 L M k = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) Φ L , k ′ ( y ) + w L ∗ ′ ( y ) Φ L , k ( y ) − F ( y ) J ( y ) Φ L , k ( y ) d y , 1 ≤ k ≤ 2 L M_(k)=int_(0)^(1){(1)/(J(y))w_(L)^(**')(y)Phi_(L,k)^(')(y)+w_(L)^(**')(y)Phi_(L,k)(y)-F(y)J(y)Phi_(L,k)(y)}dy,1 <= k <= 2^(L)M_{k}=\int_{\mathbf{0}}^{1}\left\{\frac{1}{J(y)} w_{L}^{* \prime}(y) \Phi_{L, k}^{\prime}(y)+w_{L}^{* \prime}(y) \Phi_{L, k}(y)-F(y) J(y) \Phi_{L, k}(y)\right\} d y, 1 \leq k \leq 2^{L}Mk=∫01{1J(y)wL∗′(y)ΦL,k′(y)+wL∗′(y)ΦL,k(y)−F(y)J(y)ΦL,k(y)}dy,1≤k≤2L. -1 . Because
∫ 0 1 { 1 J ( y ) v ′ ( y ) Φ L , k ′ ( y ) + v ′ ( y ) Φ L , k ( y ) − F ( y ) J ( y ) Φ L , k ( y ˙ ) } d y = 0 , 1 ≤ k ≤ 2 L − ∫ 0 1   1 J ( y ) v ′ ( y ) Φ L , k ′ ( y ) + v ′ ( y ) Φ L , k ( y ) − F ( y ) J ( y ) Φ L , k ( y ˙ ) d y = 0 , 1 ≤ k ≤ 2 L − int_(0)^(1){(1)/(J(y))v^(')(y)Phi_(L,k)^(')(y)+v^(')(y)Phi_(L,k)(y)-F(y)J(y)Phi_(L,k)((y^(˙)))}dy=0,1 <= k <= 2^(L)-\int_{0}^{1}\left\{\frac{1}{J(y)} v^{\prime}(y) \Phi_{L, k}^{\prime}(y)+v^{\prime}(y) \Phi_{L, k}(y)-F(y) J(y) \Phi_{L, k}(\dot{y})\right\} d y=0,1 \leq k \leq 2^{L}-∫01{1J(y)v′(y)ΦL,k′(y)+v′(y)ΦL,k(y)−F(y)J(y)ΦL,k(y˙)}dy=0,1≤k≤2L−
it results that
(6) M k = ∫ 0 1 { 1 J ( y ) [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ′ ( y ) + [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ( y ) } d y M k = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ′ ( y ) + w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ( y ) d y quadM_(k)=int_(0)^(1){(1)/(J(y))[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)^(')(y)+[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)(y)}dy\quad M_{k}=\int_{0}^{1}\left\{\frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}^{\prime}(y)+\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}(y)\right\} d yMk=∫01{1J(y)[wL∗′(y)−v′(y)]ΦL,k′(y)+[wL∗′(y)−v′(y)]ΦL,k(y)}dy.
Similarly, we have
∫ 0 1 { 1 J ( y ) w L # ′ ( y ) Φ L , k ′ ( y ) + w L # ′ ( y ) Φ L , k ( y ) − F ( y ) J ( y ) Φ L , k ( y ) } d y = 0 ∫ 0 1   1 J ( y ) w L # ′ ( y ) Φ L , k ′ ( y ) + w L # ′ ( y ) Φ L , k ( y ) − F ( y ) J ( y ) Φ L , k ( y ) d y = 0 int_(0)^(1){(1)/(J(y))w_(L)^(#')(y)Phi_(L,k)^(')(y)+w_(L)^(#')(y)Phi_(L,k)(y)-F(y)J(y)Phi_(L,k)(y)}dy=0\int_{0}^{1}\left\{\frac{1}{J(y)} w_{L}^{\# \prime}(y) \Phi_{L, k}^{\prime}(y)+w_{L}^{\# \prime}(y) \Phi_{L, k}(y)-F(y) J(y) \Phi_{L, k}(y)\right\} d y=0∫01{1J(y)wL#′(y)ΦL,k′(y)+wL#′(y)ΦL,k(y)−F(y)J(y)ΦL,k(y)}dy=0
thus
(7) M k = ∫ 0 1 { 1 J ( y ) [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] Φ L , k ′ ( y ) + [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] Φ L , k ( y ) } d y M k = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − w L # ′ ( y ) Φ L , k ′ ( y ) + w L ∗ ′ ( y ) − w L # ′ ( y ) Φ L , k ( y ) d y M_(k)=int_(0)^(1){(1)/(J(y))[w_(L)^(**')(y)-w_(L)^(#')(y)]Phi_(L,k)^(')(y)+[w_(L)^(**')(y)-w_(L)^(#')(y)]Phi_(L,k)(y)}dyM_{k}=\int_{0}^{1}\left\{\frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right] \Phi_{L, k}^{\prime}(y)+\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right] \Phi_{L, k}(y)\right\} d yMk=∫01{1J(y)[wL∗′(y)−wL#′(y)]ΦL,k′(y)+[wL∗′(y)−wL#′(y)]ΦL,k(y)}dy.
We also have
∫ 0 1 1 J ( y ) [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ′ ( y ) d y = = ∫ 0 1 1 J ( y ) [ w L ∗ ′ ( y ) − v ′ ( y ) ] [ J ( y ) χ L , k ( y ) ∫ 0 1 J ( s ) χ L , k ( s ) d s − J ( y ) χ L , k + 1 ( y ) ∫ 0 1 J ( s ) χ L , k + 1 ( s ) d s ] d y = = ∫ k − 1 2 L k 2 L [ w L ∗ ′ ( y ) − v ′ ( y ) ] d y ∫ k − 1 2 L k 2 L J ( s ) d s − ∫ k 2 L k + 1 2 L [ w L ∗ ′ ( y ) − v ′ ( y ) ] d y ∫ k 2 L k + 1 2 L J ( s ) d s = = [ w L ∗ ( y ) − v ( y ) ] | k − 1 2 L k 2 L − [ w L ∗ ( y ) − v ( y ) ] | k 2 L k + 1 2 L ∫ k − 1 2 L k 2 L J ( s ) d s = 0 ∫ k 2 L k + 1 2 L J ( s ) d s ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ′ ( y ) d y = = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − v ′ ( y ) J ( y ) χ L , k ( y ) ∫ 0 1   J ( s ) χ L , k ( s ) d s − J ( y ) χ L , k + 1 ( y ) ∫ 0 1   J ( s ) χ L , k + 1 ( s ) d s d y = = ∫ k − 1 2 L k 2 L   w L ∗ ′ ( y ) − v ′ ( y ) d y ∫ k − 1 2 L k 2 L   J ( s ) d s − ∫ k 2 L k + 1 2 L   w L ∗ ′ ( y ) − v ′ ( y ) d y ∫ k 2 L k + 1 2 L   J ( s ) d s = = w L ∗ ( y ) − v ( y ) k − 1 2 L k 2 L − w L ∗ ( y ) − v ( y ) k 2 L k + 1 2 L ∫ k − 1 2 L k 2 L   J ( s ) d s = 0 ∫ k 2 L k + 1 2 L   J ( s ) d s {:[int_(0)^(1)(1)/(J(y))[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)^(')(y)dy=],[=int_(0)^(1)(1)/(J(y))[w_(L)^(**')(y)-v^(')(y)][(J(y)chi_(L,k)(y))/(int_(0)^(1)J(s)chi_(L,k)(s)ds)-(J(y)chi_(L,k+1)(y))/(int_(0)^(1)J(s)chi_(L,k+1)(s)ds)]dy=],[=(int_((k-1)/(2L))^((k)/(2L))[w_(L)^(**')(y)-v^(')(y)]dy)/(int_((k-1)/(2L))^((k)/(2L))J(s)ds)-(int_((k)/(2L))^((k+1)/(2L))[w_(L)^(**')(y)-v^(')(y)]dy)/(int_((k)/(2L))^((k+1)/(2L))J(s)ds)=],[=([w_(L)^(**)(y)-v(y)]|_((k-1)/(2L))^((k)/(2L))-([w_(L)^(**)(y)-v(y)]|_((k)/(2L))^((k+1)/(2L)))/(int_((k-1)/(2L))^((k)/(2L))J(s)ds)=0)/(int_((k)/(2L))^((k+1)/(2L))J(s)ds)]:}\begin{gathered} \int_{0}^{1} \frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}^{\prime}(y) d y= \\ =\int_{0}^{1} \frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right]\left[\frac{J(y) \chi_{L, k}(y)}{\int_{0}^{1} J(s) \chi_{L, k}(s) d s}-\frac{J(y) \chi_{L, k+1}(y)}{\int_{0}^{1} J(s) \chi_{L, k+1}(s) d s}\right] d y= \\ =\frac{\int_{\frac{k-1}{2 L}}^{\frac{k}{2 L}}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] d y}{\int_{\frac{k-1}{2 L}}^{\frac{k}{2 L}} J(s) d s}-\frac{\int_{\frac{k}{2 L}}^{\frac{k+1}{2 L}}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] d y}{\int_{\frac{k}{2 L}}^{\frac{k+1}{2 L}} J(s) d s}= \\ =\frac{\left.\left[w_{L}^{*}(y)-v(y)\right]\right|_{\frac{k-1}{2 L}} ^{\frac{k}{2 L}}-\frac{\left.\left[w_{L}^{*}(y)-v(y)\right]\right|_{\frac{k}{2 L}} ^{\frac{k+1}{2 L}}}{\int_{\frac{k-1}{2 L}}^{\frac{k}{2 L}} J(s) d s}=0}{\int_{\frac{k}{2 L}}^{\frac{k+1}{2 L}} J(s) d s} \end{gathered}∫011J(y)[wL∗′(y)−v′(y)]ΦL,k′(y)dy==∫011J(y)[wL∗′(y)−v′(y)][J(y)χL,k(y)∫01J(s)χL,k(s)ds−J(y)χL,k+1(y)∫01J(s)χL,k+1(s)ds]dy==∫k−12Lk2L[wL∗′(y)−v′(y)]dy∫k−12Lk2LJ(s)ds−∫k2Lk+12L[wL∗′(y)−v′(y)]dy∫k2Lk+12LJ(s)ds==[wL∗(y)−v(y)]|k−12Lk2L−[wL∗(y)−v(y)]|k2Lk+12L∫k−12Lk2LJ(s)ds=0∫k2Lk+12LJ(s)ds
and using this in (6), we obtain
(8) M k = ∫ 0 1 [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ( y ) d y  (8)  M k = ∫ 0 1   w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ( y ) d y " (8) "M_(k)=int_(0)^(1)[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)(y)dy\text { (8) } M_{k}=\int_{0}^{1}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}(y) d y (8) Mk=∫01[wL∗′(y)−v′(y)]ΦL,k(y)dy
Considering w L ∗ = ∑ k = 1 2 L − 1 u k ∗ Φ L , k w L ∗ = ∑ k = 1 2 L − 1   u k ∗ Φ L , k w_(L)^(**)=sum_(k=1)^(2^(L)-1)u_(k)^(**)Phi_(L,k)w_{L}^{*}=\sum_{k=1}^{2^{L}-1} u_{k}^{*} \Phi_{L, k}wL∗=∑k=12L−1uk∗ΦL,k, and w L # = ∑ k = 1 2 L − 1 u k # Φ L , k w L # = ∑ k = 1 2 L − 1   u k # Φ L , k w_(L)^(#)=sum_(k=1)^(2^(L)-1)u_(k)^(#)Phi_(L,k)w_{L}^{\#}=\sum_{k=1}^{2^{L}-1} u_{k}^{\#} \Phi_{L, k}wL#=∑k=12L−1uk#ΦL,k, from (8) we obtain :
∑ k = 1 2 L − 1 ( u k ∗ − u k # ) M k = ∑ k = 1 2 L − 1 ( u k ∗ − u k # ) ∫ 0 1 [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ( y ) d y = = ∫ 0 1 ∑ k = 1 2 L − 1 ( u k ∗ − u k # ) [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ( y ) d y = ∫ 0 1 [ w L ∗ ( y ) − w L # ( y ) ] [ w L ∗ ′ ( y ) − v ′ ( y ) ] d y ∑ k = 1 2 L − 1   u k ∗ − u k # M k = ∑ k = 1 2 L − 1   u k ∗ − u k # ∫ 0 1   w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ( y ) d y = = ∫ 0 1   ∑ k = 1 2 L − 1   u k ∗ − u k # w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ( y ) d y = ∫ 0 1   w L ∗ ( y ) − w L # ( y ) w L ∗ ′ ( y ) − v ′ ( y ) d y {:[sum_(k=1)^(2^(L)-1)(u_(k)^(**)-u_(k)^(#))M_(k)=sum_(k=1)^(2^(L)-1)(u_(k)^(**)-u_(k)^(#))int_(0)^(1)[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)(y)dy=],[=int_(0)^(1)sum_(k=1)^(2^(L)-1)(u_(k)^(**)-u_(k)^(#))[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)(y)dy=int_(0)^(1)[w_(L)^(**)(y)-w_(L)^(#)(y)][w_(L)^(**')(y)-v^(')(y)]dy]:}\begin{gathered} \sum_{k=1}^{2^{L}-1}\left(u_{k}^{*}-u_{k}^{\#}\right) M_{k}=\sum_{k=1}^{2^{L}-1}\left(u_{k}^{*}-u_{k}^{\#}\right) \int_{0}^{1}\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}(y) d y= \\ =\int_{0}^{1} \sum_{k=1}^{2^{L}-1}\left(u_{k}^{*}-u_{k}^{\#}\right)\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}(y) d y=\int_{0}^{1}\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] d y \end{gathered}∑k=12L−1(uk∗−uk#)Mk=∑k=12L−1(uk∗−uk#)∫01[wL∗′(y)−v′(y)]ΦL,k(y)dy==∫01∑k=12L−1(uk∗−uk#)[wL∗′(y)−v′(y)]ΦL,k(y)dy=∫01[wL∗(y)−wL#(y)][wL∗′(y)−v′(y)]dy
ıd similarly, using (7)
(9) ∑ k = 1 2 L − 1 ( u k ∗ − u k # ) M k =  (9)  ∑ k = 1 2 L − 1   u k ∗ − u k # M k = " (9) "sum_(k=1)^(2^(L)-1)(u_(k)^(**)-u_(k)^(#))M_(k)=\text { (9) } \sum_{k=1}^{2^{L}-1}\left(u_{k}^{*}-u_{k}^{\#}\right) M_{k}= (9) ∑k=12L−1(uk∗−uk#)Mk=
e find
∑ k = 1 2 L − 1 ( u k ∗ − u k # ) ∫ 0 1 { 1 J ( y ) [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] Φ L , k ′ ( y ) + [ w L ∗ ′ ( y ) − v ′ ( y ) ] Φ L , k ( y ) } d y = = ∫ 0 1 { 1 J ( y ) [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] 2 + [ w L ∗ ′ ( y ) − v ′ ( y ) ] [ w L ∗ ( y ) − w L # ( y ) ] } d y ∑ k = 1 2 L − 1   u k ∗ − u k # ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − w L # ′ ( y ) Φ L , k ′ ( y ) + w L ∗ ′ ( y ) − v ′ ( y ) Φ L , k ( y ) d y = = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − w L # ′ ( y ) 2 + w L ∗ ′ ( y ) − v ′ ( y ) w L ∗ ( y ) − w L # ( y ) d y {:[sum_(k=1)^(2^(L)-1)(u_(k)^(**)-u_(k)^(#))int_(0)^(1){(1)/(J(y))[w_(L)^(**')(y)-w_(L)^(#')(y)]Phi_(L,k)^(')(y)+[w_(L)^(**')(y)-v^(')(y)]Phi_(L,k)(y)}dy=],[=int_(0)^(1){(1)/(J(y))[w_(L)^(**')(y)-w_(L)^(#')(y)]^(2)+[w_(L)^(**')(y)-v^(')(y)][w_(L)^(**)(y)-w_(L)^(#)(y)]}dy]:}\begin{aligned} & \sum_{k=1}^{2^{L}-1}\left(u_{k}^{*}-u_{k}^{\#}\right) \int_{0}^{1}\left\{\frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right] \Phi_{L, k}^{\prime}(y)+\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] \Phi_{L, k}(y)\right\} d y= \\ & =\int_{0}^{1}\left\{\frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right]^{2}+\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right]\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\right\} d y \end{aligned}∑k=12L−1(uk∗−uk#)∫01{1J(y)[wL∗′(y)−wL#′(y)]ΦL,k′(y)+[wL∗′(y)−v′(y)]ΦL,k(y)}dy==∫01{1J(y)[wL∗′(y)−wL#′(y)]2+[wL∗′(y)−v′(y)][wL∗(y)−wL#(y)]}dy
From (8) and (9) it results :
∫ 0 1 [ w L ∗ ( y ) − w L # ( y ) ] [ w L ∗ ′ ( y ) − v ′ ( y ) ] d y = = ∫ 0 1 1 J ( y ) [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] 2 d y + ∫ 0 1 [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] [ w L ∗ ( y ) − w L # ( y ) ] d y but ∫ 0 1 [ w L ∗ ( y ) − w L # ( y ) ] [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] d y = 1 2 w L ∗ ( y ) − w L # ( y ) | 0 1 = 0 ⟹ ∫ 0 1 [ w L ∗ ( y ) − w L # ( y ) ] [ w L ∗ ′ ( y ) − v ′ ( y ) ] d y = ∫ 0 1 1 J ( y ) [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] 2 d y ⟹∣ w L ∗ − w L # ‖ G H 0 1 2 = ∫ 0 1 [ w L ∗ ( y ) − w L # ( y ) ] [ w L ∗ ′ ( y ) − v ′ ( y ) ] d y = [ w L ∗ ( y ) − w L # ( y ) ] [ w L ∗ ( y ) − v ( y ) ] | 0 1 − ∫ 0 1 [ w L ∗ ′ ( y ) − w L # ′ ( y ) ] [ w L ∗ ( y ) − v ( y ) ] d y = ∫ 0 1 [ w L # ′ ( y ) − w L ∗ ′ ( y ) ] [ w L ∗ ( y ) − v ( y ) ] d y ≤ ( ∫ 0 1 [ w L # ′ ( y ) − w L ∗ ′ ( y ) ] 2 ) 1 2 ( ∫ 0 1 [ u L ∗ ( y ) − v ( y ) ] 2 d y ) ∫ 0 1   w L ∗ ( y ) − w L # ( y ) w L ∗ ′ ( y ) − v ′ ( y ) d y = = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − w L # ′ ( y ) 2 d y + ∫ 0 1   w L ∗ ′ ( y ) − w L # ′ ( y ) w L ∗ ( y ) − w L # ( y ) d y  but  ∫ 0 1   w L ∗ ( y ) − w L # ( y ) w L ∗ ′ ( y ) − w L # ′ ( y ) d y = 1 2 w L ∗ ( y ) − w L # ( y ) 0 1 = 0 ⟹ ∫ 0 1   w L ∗ ( y ) − w L # ( y ) w L ∗ ′ ( y ) − v ′ ( y ) d y = ∫ 0 1   1 J ( y ) w L ∗ ′ ( y ) − w L # ′ ( y ) 2 d y ⟹∣ w L ∗ − w L # ‖ G H 0 1 2 = ∫ 0 1   w L ∗ ( y ) − w L # ( y ) w L ∗ ′ ( y ) − v ′ ( y ) d y = w L ∗ ( y ) − w L # ( y ) w L ∗ ( y ) − v ( y ) 0 1 − ∫ 0 1   w L ∗ ′ ( y ) − w L # ′ ( y ) w L ∗ ( y ) − v ( y ) d y = ∫ 0 1   w L # ′ ( y ) − w L ∗ ′ ( y ) w L ∗ ( y ) − v ( y ) d y ≤ ∫ 0 1   w L # ′ ( y ) − w L ∗ ′ ( y ) 2 1 2 ∫ 0 1   u L ∗ ( y ) − v ( y ) 2 d y {:[qquadint_(0)^(1)[w_(L)^(**)(y)-w_(L)^(#)(y)][w_(L)^(**')(y)-v^(')(y)]dy=],[=int_(0)^(1)(1)/(J(y))[w_(L)^(**')(y)-w_(L)^(#')(y)]^(2)dy+int_(0)^(1)[w_(L)^(**')(y)-w_(L)^(#')(y)][w_(L)^(**)(y)-w_(L)^(#)(y)]dy],[" but "],[qquadint_(0)^(1)[w_(L)^(**)(y)-w_(L)^(#)(y)][w_(L)^(**')(y)-w_(L)^(#')(y)]dy=(1)/(2)w_(L)^(**)(y)-w_(L)^(#)(y)|_(0)^(1)=0],[ Longrightarrowint_(0)^(1)[w_(L)^(**)(y)-w_(L)^(#)(y)][w_(L)^(**')(y)-v^(')(y)]dy=int_(0)^(1)(1)/(J(y))[w_(L)^(**')(y)-w_(L)^(#')(y)]^(2)dy],[quad Longrightarrow ∣w_(L)^(**)-w_(L)^(#)||_(GH_(0)^(1))^(2)=int_(0)^(1)[w_(L)^(**)(y)-w_(L)^(#)(y)][w_(L)^(**')(y)-v^(')(y)]dy=],[[w_(L)^(**)(y)-w_(L)^(#)(y)][w_(L)^(**)(y)-v(y)]|_(0)^(1)-int_(0)^(1)[w_(L)^(**')(y)-w_(L)^(#')(y)][w_(L)^(**)(y)-v(y)]dy=],[int_(0)^(1)[w_(L)^(#')(y)-w_(L)^(**')(y)][w_(L)^(**)(y)-v(y)]dy <= (int_(0)^(1)[w_(L)^(#')(y)-w_(L)^(**')(y)]^(2))^((1)/(2))(int_(0)^(1)[u_(L)^(**)(y)-v(y)]^(2)dy)]:}\begin{aligned} & \qquad \int_{0}^{1}\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] d y= \\ & =\int_{0}^{1} \frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right]^{2} d y+\int_{0}^{1}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right]\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right] d y \\ & \text { but } \\ & \qquad \int_{0}^{1}\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right] d y=\frac{1}{2} w_{L}^{*}(y)-\left.w_{L}^{\#}(y)\right|_{0} ^{1}=0 \\ & \Longrightarrow \int_{0}^{1}\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] d y=\int_{0}^{1} \frac{1}{J(y)}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right]^{2} d y \\ & \quad \Longrightarrow \mid w_{L}^{*}-w_{L}^{\#} \|_{G H_{0}^{1}}^{2}=\int_{0}^{1}\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\left[w_{L}^{* \prime}(y)-v^{\prime}(y)\right] d y= \\ & {\left.\left[w_{L}^{*}(y)-w_{L}^{\#}(y)\right]\left[w_{L}^{*}(y)-v(y)\right]\right|_{0} ^{1}-\int_{0}^{1}\left[w_{L}^{* \prime}(y)-w_{L}^{\# \prime}(y)\right]\left[w_{L}^{*}(y)-v(y)\right] d y=} \\ & \int_{0}^{1}\left[w_{L}^{\# \prime}(y)-w_{L}^{* \prime}(y)\right]\left[w_{L}^{*}(y)-v(y)\right] d y \leq\left(\int_{0}^{1}\left[w_{L}^{\# \prime}(y)-w_{L}^{* \prime}(y)\right]^{2}\right)^{\frac{1}{2}}\left(\int_{0}^{1}\left[u_{L}^{*}(y)-v(y)\right]^{2} d y\right) \end{aligned}∫01[wL∗(y)−wL#(y)][wL∗′(y)−v′(y)]dy==∫011J(y)[wL∗′(y)−wL#′(y)]2dy+∫01[wL∗′(y)−wL#′(y)][wL∗(y)−wL#(y)]dy but ∫01[wL∗(y)−wL#(y)][wL∗′(y)−wL#′(y)]dy=12wL∗(y)−wL#(y)|01=0⟹∫01[wL∗(y)−wL#(y)][wL∗′(y)−v′(y)]dy=∫011J(y)[wL∗′(y)−wL#′(y)]2dy⟹∣wL∗−wL#‖GH012=∫01[wL∗(y)−wL#(y)][wL∗′(y)−v′(y)]dy=[wL∗(y)−wL#(y)][wL∗(y)−v(y)]|01−∫01[wL∗′(y)−wL#′(y)][wL∗(y)−v(y)]dy=∫01[wL#′(y)−wL∗′(y)][wL∗(y)−v(y)]dy≤(∫01[wL#′(y)−wL∗′(y)]2)12(∫01[uL∗(y)−v(y)]2dy)
= ( ∫ 0 1 J ( y ) 1 J ( y ) [ w L # ′ ( y ) − w L ∗ ′ ( y ) ] 2 ) 1 2 ‖ w L ∗ − v L ‖ L 2 [ 0 , 1 ] ≤ ≤ ( ∫ 0 1 J ( y ) d y ) 1 2 ( ∫ 0 1 1 J ( y ) [ w L # ′ ( y ) − w L ∗ ′ ( y ) ] 2 ) 1 2 ‖ w L ∗ − v L ‖ L 2 [ 0 , 1 ] = = ‖ w L # − w L ∗ ‖ G H 0 1 ‖ w L ∗ − v L ‖ L 2 [ 0 , 1 ] ⟹ ‖ w L # − w L ∗ ‖ G H 0 1 ≤ ‖ w L ∗ − v L ‖ L 2 [ 0 , 1 ] = ∫ 0 1   J ( y ) 1 J ( y ) w L # ′ ( y ) − w L ∗ ′ ( y ) 2 1 2 w L ∗ − v L L 2 [ 0 , 1 ] ≤ ≤ ∫ 0 1   J ( y ) d y 1 2 ∫ 0 1   1 J ( y ) w L # ′ ( y ) − w L ∗ ′ ( y ) 2 1 2 w L ∗ − v L L 2 [ 0 , 1 ] = = w L # − w L ∗ G H 0 1 w L ∗ − v L L 2 [ 0 , 1 ] ⟹ w L # − w L ∗ G H 0 1 ≤ w L ∗ − v L L 2 [ 0 , 1 ] {:[=(int_(0)^(1)J(y)(1)/(J(y))[w_(L)^(#')(y)-w_(L)^(**')(y)]^(2))^((1)/(2))||w_(L)^(**)-v_(L)||_(L^(2)[0,1]) <= ],[ <= (int_(0)^(1)J(y)dy)^((1)/(2))(int_(0)^(1)(1)/(J(y))[w_(L)^(#')(y)-w_(L)^(**')(y)]^(2))^((1)/(2))||w_(L)^(**)-v_(L)||_(L^(2)[0,1])=],[=||w_(L)^(#)-w_(L)^(**)||_(GH_(0)^(1))||w_(L)^(**)-v_(L)||_(L^(2)[0,1])],[Longrightarrow||w_(L)^(#)-w_(L)^(**)||_(GH_(0)^(1)) <= ||w_(L)^(**)-v_(L)||_(L^(2)[0,1])]:}\begin{gathered} =\left(\int_{0}^{1} J(y) \frac{1}{J(y)}\left[w_{L}^{\# \prime}(y)-w_{L}^{* \prime}(y)\right]^{2}\right)^{\frac{1}{2}}\left\|w_{L}^{*}-v_{L}\right\|_{L^{2}[0,1]} \leq \\ \leq\left(\int_{0}^{1} J(y) d y\right)^{\frac{1}{2}}\left(\int_{0}^{1} \frac{1}{J(y)}\left[w_{L}^{\# \prime}(y)-w_{L}^{* \prime}(y)\right]^{2}\right)^{\frac{1}{2}}\left\|w_{L}^{*}-v_{L}\right\|_{L^{2}[0,1]}= \\ =\left\|w_{L}^{\#}-w_{L}^{*}\right\|_{G H_{0}^{1}}\left\|w_{L}^{*}-v_{L}\right\|_{L^{2}[0,1]} \\ \Longrightarrow\left\|w_{L}^{\#}-w_{L}^{*}\right\|_{G H_{0}^{1}} \leq\left\|w_{L}^{*}-v_{L}\right\|_{L^{2}[0,1]} \end{gathered}=(∫01J(y)1J(y)[wL#′(y)−wL∗′(y)]2)12‖wL∗−vL‖L2[0,1]≤≤(∫01J(y)dy)12(∫011J(y)[wL#′(y)−wL∗′(y)]2)12‖wL∗−vL‖L2[0,1]==‖wL#−wL∗‖GH01‖wL∗−vL‖L2[0,1]⟹‖wL#−wL∗‖GH01≤‖wL∗−vL‖L2[0,1]
We have
‖ w L ∗ − v L ‖ L 2 [ 0 , 1 ] ≤ ‖ w L ∗ − v L ‖ L ∞ [ 0 , 1 ] ⇒ ‖ w L # − v ‖ L ∞ [ 0 , 1 ] ≤ ‖ w L # − w L ∗ ‖ L ∞ [ 0 , 1 ] + ‖ w L ∗ − v ‖ L ∞ [ 0 , 1 ] ≤ ‖ w L # − w L ∗ ‖ G H 0 1 + ‖ w L ∗ − v ‖ L α ≤ ‖ w L ∗ − v L ‖ L 2 [ 0 , 1 ] + ‖ w L ∗ − v ‖ L ∞ [ 0 , 1 ] ≤ 2 ‖ w L ∗ − v ‖ L ∞ [ 0 , 1 ] . q.e.d. w L ∗ − v L L 2 [ 0 , 1 ] ≤ w L ∗ − v L L ∞ [ 0 , 1 ] ⇒ w L # − v L ∞ [ 0 , 1 ] ≤ w L # − w L ∗ L ∞ [ 0 , 1 ] + w L ∗ − v L ∞ [ 0 , 1 ] ≤ w L # − w L ∗ G H 0 1 + w L ∗ − v L α ≤ w L ∗ − v L L 2 [ 0 , 1 ] + w L ∗ − v L ∞ [ 0 , 1 ] ≤ 2 w L ∗ − v L ∞ [ 0 , 1 ] .  q.e.d.  {:[||w_(L)^(**)-v_(L)||_(L^(2)[0,1]) <= ||w_(L)^(**)-v_(L)||_(L^(oo)[0,1])],[=>||w_(L)^(#)-v||_(L^(oo)[0,1]) <= ||w_(L)^(#)-w_(L)^(**)||_(L^(oo)[0,1])+||w_(L)^(**)-v||_(L^(oo)[0,1]) <= ||w_(L)^(#)-w_(L)^(**)||_(GH_(0)^(1))+||w_(L)^(**)-v||_(L^(alpha))],[ <= ||w_(L)^(**)-v_(L)||_(L^(2)[0,1])+||w_(L)^(**)-v||_(L^(oo)[0,1]) <= 2||w_(L)^(**)-v||_(L^(oo)[0,1])." q.e.d. "]:}\begin{gathered} \left\|w_{L}^{*}-v_{L}\right\|_{L^{2}[0,1]} \leq\left\|w_{L}^{*}-v_{L}\right\|_{L^{\infty}[0,1]} \\ \Rightarrow\left\|w_{L}^{\#}-v\right\|_{L^{\infty}[0,1]} \leq\left\|w_{L}^{\#}-w_{L}^{*}\right\|_{L^{\infty}[0,1]}+\left\|w_{L}^{*}-v\right\|_{L^{\infty}[0,1]} \leq\left\|w_{L}^{\#}-w_{L}^{*}\right\|_{G H_{0}^{1}}+\left\|w_{L}^{*}-v\right\|_{L^{\alpha}} \\ \leq\left\|w_{L}^{*}-v_{L}\right\|_{L^{2}[0,1]}+\left\|w_{L}^{*}-v\right\|_{L^{\infty}[0,1]} \leq 2\left\|w_{L}^{*}-v\right\|_{L^{\infty}[0,1]} . \text { q.e.d. } \end{gathered}‖wL∗−vL‖L2[0,1]≤‖wL∗−vL‖L∞[0,1]⇒‖wL#−v‖L∞[0,1]≤‖wL#−wL∗‖L∞[0,1]+‖wL∗−v‖L∞[0,1]≤‖wL#−wL∗‖GH01+‖wL∗−v‖Lα≤‖wL∗−vL‖L2[0,1]+‖wL∗−v‖L∞[0,1]≤2‖wL∗−v‖L∞[0,1]. q.e.d. 
We similarly define space V L − 1 V L − 1 V_(L-1)V_{L-1}VL−1, and within this space { Φ L − 1 , k } k = 1 2 L − 1 − 1 Φ L − 1 , k k = 1 2 L − 1 − 1 {Phi_(L-1,k)}_(k=1)^(2^(L-1)-1)\left\{\Phi_{L-1, k}\right\}_{k=1}^{2^{L-1}-1}{ΦL−1,k}k=12L−1−1 is วase, where
Φ L − 1 , k ( y ) = ∫ 0 y J ( s ) χ L − 1 , k − 1 ( s ) d s ∫ 0 1 J ( s ) χ L − 1 , k − 1 ( s ) d s − ∫ 0 y J ( s ) χ L − 1 , k ( s ) d s ∫ 0 1 J ( s ) χ L − 1 , k ( s ) d s Φ L − 1 , k ( y ) = ∫ 0 y   J ( s ) χ L − 1 , k − 1 ( s ) d s ∫ 0 1   J ( s ) χ L − 1 , k − 1 ( s ) d s − ∫ 0 y   J ( s ) χ L − 1 , k ( s ) d s ∫ 0 1   J ( s ) χ L − 1 , k ( s ) d s Phi_(L-1,k)(y)=(int_(0)^(y)J(s)chi_(L-1,k-1)(s)ds)/(int_(0)^(1)J(s)chi_(L-1,k-1)(s)ds)-(int_(0)^(y)J(s)chi_(L-1,k)(s)ds)/(int_(0)^(1)J(s)chi_(L-1,k)(s)ds)\Phi_{L-1, k}(y)=\frac{\int_{0}^{y} J(s) \chi_{L-1, k-1}(s) d s}{\int_{0}^{1} J(s) \chi_{L-1, k-1}(s) d s}-\frac{\int_{0}^{y} J(s) \chi_{L-1, k}(s) d s}{\int_{0}^{1} J(s) \chi_{L-1, k}(s) d s}ΦL−1,k(y)=∫0yJ(s)χL−1,k−1(s)ds∫01J(s)χL−1,k−1(s)ds−∫0yJ(s)χL−1,k(s)ds∫01J(s)χL−1,k(s)ds
We have :
Φ L − 1 , k ( y ) = H 0 L − 1 , k Φ L , 2 k − 1 ( y ) + H 1 L − 1 , k Φ L , 2 k ( y ) + H 2 L − 1 , k Φ L , 2 k + 1 ( y ) Φ L − 1 , k ( y ) = H 0 L − 1 , k Φ L , 2 k − 1 ( y ) + H 1 L − 1 , k Φ L , 2 k ( y ) + H 2 L − 1 , k Φ L , 2 k + 1 ( y ) Phi_(L-1,k)(y)=H_(0)^(L-1,k)Phi_(L,2k-1)(y)+H_(1)^(L-1,k)Phi_(L,2k)(y)+H_(2)^(L-1,k)Phi_(L,2k+1)(y)\Phi_{L-1, k}(y)=H_{0}^{L-1, k} \Phi_{L, 2 k-1}(y)+H_{1}^{L-1, k} \Phi_{L, 2 k}(y)+H_{2}^{L-1, k} \Phi_{L, 2 k+1}(y)ΦL−1,k(y)=H0L−1,kΦL,2k−1(y)+H1L−1,kΦL,2k(y)+H2L−1,kΦL,2k+1(y)
where
Obviously V L − 1 ⊂ V L V L − 1 ⊂ V L V^(L-1)subV^(L)V^{L-1} \subset V^{L}VL−1⊂VL.
Let Ψ L − 1 , k ( y ) = Φ L , 2 k − 1 ( y ) , k = 1 , … , 2 L − 1 Ψ L − 1 , k ( y ) = Φ L , 2 k − 1 ( y ) , k = 1 , … , 2 L − 1 Psi_(L-1,k)(y)=Phi_(L,2k-1)(y),k=1,dots,2^(L-1)\Psi_{L-1, k}(y)=\Phi_{L, 2 k-1}(y), k=1, \ldots, 2^{L-1}ΨL−1,k(y)=ΦL,2k−1(y),k=1,…,2L−1 and
W L − 1 = span { Ψ L − 1 , k / k = 1 , … , 2 L − 1 } . W L − 1 = span Ψ L − 1 , k / k = 1 , … , 2 L − 1 . W^(L-1)=span{Psi_(L-1,k)//k=1,dots,2^(L-1)}.W^{L-1}=\operatorname{span}\left\{\Psi_{L-1, k} / k=1, \ldots, 2^{L-1}\right\} .WL−1=span{ΨL−1,k/k=1,…,2L−1}.
Functions Ψ L − 1 , k ( y ) Ψ L − 1 , k ( y ) Psi_(L-1,k)(y)\Psi_{L-1, k}(y)ΨL−1,k(y) have as support the interval [ 2 k − 2 2 L , 2 k 2 L ] 2 k − 2 2 L , 2 k 2 L [(2k-2)/(2^(L)),(2k)/(2^(L))]\left[\frac{2 k-2}{2^{L}}, \frac{2 k}{2^{L}}\right][2k−22L,2k2L] and obviously supp Ψ L − 1 , k ⋂ supp Ψ L − 1 , l = Φ supp Ψ L − 1 , k ⋂ supp Ψ L − 1 , l = Φ suppPsi_(L-1,k)nnn suppPsi_(L-1,l)=Phi\operatorname{supp} \Psi_{L-1, k} \bigcap \operatorname{supp} \Psi_{L-1, l}=\PhisuppΨL−1,k⋂suppΨL−1,l=Φ if k ≠ l k ≠ l k!=lk \neq lk≠l. System { Ψ L − 1 , k } k = 1 2 L − 1 Ψ L − 1 , k k = 1 2 L − 1 {Psi_(L-1,k)}_(k=1)^(2L-1)\left\{\Psi_{L-1, k}\right\}_{k=1}^{2 L-1}{ΨL−1,k}k=12L−1 forms a base of space W L − 1 W L − 1 W^(L-1)W^{L-1}WL−1 and we also have W L − 1 ⊂ V L − 1 W L − 1 ⊂ V L − 1 W^(L-1)subV^(L-1)W^{L-1} \subset V^{L-1}WL−1⊂VL−1.
We will show that V L − 1 ⊕ W L − 1 = V L V L − 1 ⊕ W L − 1 = V L V^(L-1)o+W^(L-1)=V^(L)V^{L-1} \oplus W^{L-1}=V^{L}VL−1⊕WL−1=VL. The fact that V L − 1 + W L − 1 = I − 1 V L − 1 + W L − 1 = I − 1 V^(L-1)+W^(L-1)=I^(-1)V^{L-1}+W^{L-1}=I^{-1}VL−1+WL−1=I−1. is obvious; we will further prove that
V L − 1 ⊥ W L − 1 . V L − 1 ⊥ W L − 1 . V^(L-1)_|_W^(L-1).V^{L-1} \perp W^{L-1} .VL−1⊥WL−1.
We have
∫ 0 1 1 J ( y ) Ψ L − 1 , k ′ ( y ) Φ L − 1 , k ′ ( y ) d y = ∫ k − 1 2 L − 1 k − 1 2 2 L − 1 J ( y ) ∫ k − 1 2 L − 1 k − 1 2 − 1 J ( s ) d s ∫ k 2 L − 1 k 2 L − 1 J ( s ) d s − ∫ k − 1 2 L − 1 k 2 L − 1 J ( y ) ∫ k − 2 2 L − 1 k 2 L − 1 J ( s ) d s ∫ k − 1 2 L − 1 k 2 L − 1 J ( s ) d s d y = = 1 ∫ k − 1 2 L − 1 k 2 L − 1 J ( s ) d s − 1 ∫ k − 1 2 L − 1 k 2 L − 1 J ( s ) d s = 0 ∫ 0 1   1 J ( y ) Ψ L − 1 , k ′ ( y ) Φ L − 1 , k ′ ( y ) d y = ∫ k − 1 2 L − 1 k − 1 2 2 L − 1   J ( y ) ∫ k − 1 2 L − 1 k − 1 2 − 1   J ( s ) d s ∫ k 2 L − 1 k 2 L − 1   J ( s ) d s − ∫ k − 1 2 L − 1 k 2 L − 1   J ( y ) ∫ k − 2 2 L − 1 k 2 L − 1   J ( s ) d s ∫ k − 1 2 L − 1 k 2 L − 1   J ( s ) d s d y = = 1 ∫ k − 1 2 L − 1 k 2 L − 1   J ( s ) d s − 1 ∫ k − 1 2 L − 1 k 2 L − 1   J ( s ) d s = 0 {:[int_(0)^(1)(1)/(J(y))Psi_(L-1,k)^(')(y)Phi_(L-1,k)^(')(y)dy=int_((k-1)/(2L-1))^((k-(1)/(2))/(2L-1))(J(y))/(int_((k-1)/(2L-1))^((k-1)/(2)-1))J(s)dsint_((k)/(2L-1))^((k)/(2L-1))J(s)ds],[-int_((k-1)/(2L-1))^((k)/(2L-1))(J(y))/(int_((k-2)/(2L-1))^((k)/(2L-1))J(s)dsint_((k-1)/(2L-1))^((k)/(2L-1))J(s)ds)dy=],[=(1)/(int_((k-1)/(2L-1))^((k)/(2L-1))J(s)ds)-(1)/(int_((k-1)/(2L-1))^((k)/(2L-1))J(s)ds)=0]:}\begin{gathered} \int_{0}^{1} \frac{1}{J(y)} \Psi_{L-1, k}^{\prime}(y) \Phi_{L-1, k}^{\prime}(y) d y=\int_{\frac{k-1}{2 L-1}}^{\frac{k-\frac{1}{2}}{2 L-1}} \frac{J(y)}{\int_{\frac{k-1}{2 L-1}}^{\frac{k-1}{2}-1}} J(s) d s \int_{\frac{k}{2 L-1}}^{\frac{k}{2 L-1}} J(s) d s \\ -\int_{\frac{k-1}{2 L-1}}^{\frac{k}{2 L-1}} \frac{J(y)}{\int_{\frac{k-2}{2 L-1}}^{\frac{k}{2 L-1}} J(s) d s \int_{\frac{k-1}{2 L-1}}^{\frac{k}{2 L-1}} J(s) d s} d y= \\ =\frac{1}{\int_{\frac{k-1}{2 L-1}}^{\frac{k}{2 L-1}} J(s) d s}-\frac{1}{\int_{\frac{k-1}{2 L-1}}^{\frac{k}{2 L-1}} J(s) d s}=0 \end{gathered}∫011J(y)ΨL−1,k′(y)ΦL−1,k′(y)dy=∫k−12L−1k−122L−1J(y)∫k−12L−1k−12−1J(s)ds∫k2L−1k2L−1J(s)ds−∫k−12L−1k2L−1J(y)∫k−22L−1k2L−1J(s)ds∫k−12L−1k2L−1J(s)dsdy==1∫k−12L−1k2L−1J(s)ds−1∫k−12L−1k2L−1J(s)ds=0
Similarly,
∫ 0 1 1 J ( y ) Ψ L − 1 , k + 1 ′ ( y ) Φ L − 1 , k ′ ( y ) d y = 0 ∫ 0 1   1 J ( y ) Ψ L − 1 , k + 1 ′ ( y ) Φ L − 1 , k ′ ( y ) d y = 0 int_(0)^(1)(1)/(J(y))Psi_(L-1,k+1)^(')(y)Phi_(L-1,k)^(')(y)dy=0\int_{0}^{1} \frac{1}{J(y)} \Psi_{L-1, k+1}^{\prime}(y) \Phi_{L-1, k}^{\prime}(y) d y=0∫011J(y)ΨL−1,k+1′(y)ΦL−1,k′(y)dy=0
Consequently, subspaces V L − 1 V L − 1 V^(L-1)V^{L-1}VL−1 and W L − 1 W L − 1 W^(L-1)W^{L-1}WL−1 are orthogonal related to the scalar product,
( f , g ) G H 0 1 := ∫ 0 1 1 J ( y ) f ′ ( y ) g ′ ( y ) d y ( f , g ) G H 0 1 := ∫ 0 1   1 J ( y ) f ′ ( y ) g ′ ( y ) d y (f,g)_(GH_(0)^(1)):=int_(0)^(1)(1)/(J(y))f^(')(y)g^(')(y)dy(f, g)_{G H_{0}^{1}}:=\int_{0}^{1} \frac{1}{J(y)} f^{\prime}(y) g^{\prime}(y) d y(f,g)GH01:=∫011J(y)f′(y)g′(y)dy
which means that the sum is direct..
The procedure can continue, leading finally to decomposition of space V L V L V^(L)V^{L}VL, so that :
V L = V 1 ⨁ W 1 ⨁ W 2 ⨁ … ⨁ W L − 1 V L = V 1 ⨁ W 1 ⨁ W 2 ⨁ … ⨁ W L − 1 V^(L)=V^(1)bigoplusW^(1)bigoplusW^(2)bigoplus dots bigoplusW^(L-1)V^{L}=V^{1} \bigoplus W^{1} \bigoplus W^{2} \bigoplus \ldots \bigoplus W^{L-1}VL=V1⨁W1⨁W2⨁…⨁WL−1
This decomposition allows the consideration of another base in space V L V L V^(L)V^{L}VL, namely :
{ Ψ j k } k = 1 , 2 j − 1 j = 0 , L − 1 Ψ j k k = 1 , 2 j − 1 j = 0 , L − 1 {Psi_(jk)}_({:[k=1","2^(j)-1],[j=0","L-1]:})\left\{\Psi_{j k}\right\}_{\substack{k=1,2^{j}-1 \\ j=0, L-1}}{Ψjk}k=1,2j−1j=0,L−1
where Ψ j k ( y ) = Φ j + 1 , 2 k − 1 ( y ) Ψ j k ( y ) = Φ j + 1 , 2 k − 1 ( y ) Psi_(jk)(y)=Phi_(j+1,2k-1)(y)\Psi_{j k}(y)=\Phi_{j+1,2 k-1}(y)Ψjk(y)=Φj+1,2k−1(y).
We also have
Ψ j k ( y ) = { ∫ k − 1 2 j y J ( s ) ψ j k ( s ) d s k − 1 2 2 j J ( s ) ψ j k ( s ) d s , if y ∈ [ k − 1 2 j , k − 1 2 2 j ] ∫ k − 1 2 j ∫ y k 2 j J ( s ) ψ j k ( s ) d s , if y ∈ [ k − 1 2 2 j , k 2 j ] ∫ k − 1 2 2 j J ( s ) ψ j k ( s ) d s , in rest Ψ j k ( y ) = ∫ k − 1 2 j y   J ( s ) ψ j k ( s ) d s k − 1 2 2 j J ( s ) ψ j k ( s ) d s ,  if  y ∈ k − 1 2 j , k − 1 2 2 j ∫ k − 1 2 j   ∫ y k 2 j   J ( s ) ψ j k ( s ) d s ,  if  y ∈ k − 1 2 2 j , k 2 j ∫ k − 1 2 2 j J ( s ) ψ j k ( s ) d s   ,  in rest  Psi_(jk)(y)={[(int_((k-1)/(2^(j)))^(y)J(s)psi_(jk)(s)ds)/((k-(1)/(2))/(2^(j))J(s)psi_(jk)(s)ds)","" if "y in[(k-1)/(2^(j)),(k-(1)/(2))/(2^(j))]],[(int_((k-1)/(2^(j))))/(int_(y)^((k)/(2^(j)))J(s)psi_(jk)(s)ds)","" if "y in[(k-(1)/(2))/(2^(j)),(k)/(2^(j))]],[int_((k-(1)/(2))/(2^(j))J(s)psi_(jk)(s)ds)","" in rest "]:}\Psi_{j k}(y)=\left\{\begin{array}{l} \frac{\int_{\frac{k-1}{2^{j}}}^{y} J(s) \psi_{j k}(s) d s}{\frac{k-\frac{1}{2}}{2^{j}} J(s) \psi_{j k}(s) d s}, \text { if } y \in\left[\frac{k-1}{2^{j}}, \frac{k-\frac{1}{2}}{2^{j}}\right] \\ \frac{\int_{\frac{k-1}{2^{j}}}}{\int_{y}^{\frac{k}{2^{j}}} J(s) \psi_{j k}(s) d s}, \text { if } y \in\left[\frac{k-\frac{1}{2}}{2^{j}}, \frac{k}{2^{j}}\right] \\ \int_{\frac{k-\frac{1}{2}}{2^{j}} J(s) \psi_{j k}(s) d s}, \text { in rest } \end{array}\right.Ψjk(y)={∫k−12jyJ(s)ψjk(s)dsk−122jJ(s)ψjk(s)ds, if y∈[k−12j,k−122j]∫k−12j∫yk2jJ(s)ψjk(s)ds, if y∈[k−122j,k2j]∫k−122jJ(s)ψjk(s)ds, in rest 
where ψ j k ( y ) = 2 j ψ ( 2 j y − k ) ψ j k ( y ) = 2 j ψ 2 j y − k psi_(jk)(y)=sqrt(2^(j))psi(2^(j)y-k)\psi_{j k}(y)=\sqrt{2^{j}} \psi\left(2^{j} y-k\right)ψjk(y)=2jψ(2jy−k) and ψ ψ psi\psiψ is Haar's function
ψ ( x ) = { 1 if 0 ≤ x < 1 2 − 1 if 1 2 ≤ x < 1 0 otherwise ψ ( x ) = 1  if  0 ≤ x < 1 2 − 1  if  1 2 ≤ x < 1 0  otherwise  psi(x)={[1" if "0 <= x < (1)/(2)],[-1" if "(1)/(2) <= x < 1],[0" otherwise "]:}\psi(x)=\left\{\begin{array}{c} 1 \text { if } 0 \leq x<\frac{1}{2} \\ -1 \text { if } \frac{1}{2} \leq x<1 \\ 0 \text { otherwise } \end{array}\right.ψ(x)={1 if 0≤x<12−1 if 12≤x<10 otherwise 

Numerical example.

We consider the following problem :
− ε u ′ ′ ( x ) + u ′ ( x ) = 1 , for x ∈ ( 0 , 1 ) u ( 0 ) = u ( 1 ) = 0 . − ε u ′ ′ ( x ) + u ′ ( x ) = 1 ,  for  x ∈ ( 0 , 1 ) u ( 0 ) = u ( 1 ) = 0 . {:[-epsiu^('')(x)+u^(')(x)=1","" for "x in(0","1)],[u(0)=u(1)=0.]:}\begin{aligned} -\varepsilon u^{\prime \prime}(x)+u^{\prime}(x) & =1, \text { for } x \in(0,1) \\ u(0) & =u(1)=0 . \end{aligned}−εu′′(x)+u′(x)=1, for x∈(0,1)u(0)=u(1)=0.
which have the solution u ( x ) = exp ⁡ ( x ε ) − exp ⁡ ( 1 ε ) 1 − exp ⁡ ( 1 ε ) + x − 1 u ( x ) = exp ⁡ x ε − exp ⁡ 1 ε 1 − exp ⁡ 1 ε + x − 1 u(x)=(exp((x)/( epsi))-exp((1)/(epsi)))/(1-exp((1)/(epsi)))+x-1u(x)=\frac{\exp \left(\frac{x}{\varepsilon}\right)-\exp \left(\frac{1}{\varepsilon}\right)}{1-\exp \left(\frac{1}{\varepsilon}\right)}+x-1u(x)=exp⁡(xε)−exp⁡(1ε)1−exp⁡(1ε)+x−1.
We do the change of variable x = g ( y ) x = g ( y ) x=g(y)x=g(y)x=g(y) where g ( y ) = 1 − ( 1 − y ) p + 1 g ( y ) = 1 − ( 1 − y ) p + 1 g(y)=1-(1-y)^(p+1)g(y)=1-(1-y)^{p+1}g(y)=1−(1−y)p+1.
In the Figures 1., 2., 3. the exact solution and the approximate solution are presented for p = 0 , p = 2 , p = 4 p = 0 , p = 2 , p = 4 p=0,p=2,p=4p=0, p=2, p=4p=0,p=2,p=4 and N = 4 , ε = 0.0001 N = 4 , ε = 0.0001 N=4,epsi=0.0001N=4, \varepsilon=0.0001N=4,ε=0.0001.
Figure 1:
Figure 2:
Figure 3:

Bibliography

[1] I. Daubechies, Orthonormal bases of compactly supported wavelets. Comm. pure Appl. Math. 41 (1998), pp. 909-996.
[2] R. Glowinski, W.N. Lawton, M. Ravachol, E. Tenenbaum, Wavelets solution of linear and nonlinear elliptic, parabolic and hyperbolic problems in one space dimension. In: R. Glowinski, A. Lichnewsky, ads., Computing Methods in Applied Sciences and Engineering, SIAM, Philadelphia (1990), pp. 55-120.
[3] P.W. Hemker, A numerical study of stiff two-point boundary problems, Amsterdam, 1997.
[4] J.-C. Xu, W.-C. Shann, Galerkin - wavelet methods for two point boundary value problems. Numer. Math., 63 (1992), pp. 123-142.
[5] H. Yserentant, On the multi - level splitting of finite element spaces. Numer. Math. 49 (1986), pp. 379-412.

Received: 15.10.1999

Universitatea de Nord Baia Mare Facultatea de Ştiinţe Catedra de Matematică şi Informatică Victoriei 76, 4800 Baia Mare ROMANIA imustata@icttp.math.ubbcluj.ro
  1. *Supportet by ANSTI Grant GR/4122 (1998)
1999

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