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
“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
About this paper
Journal
Buletinul ştiinţific al Universitatii Baia Mare,
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Sinus Association
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12221201
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[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.
Paper (preprint) in HTML form
A Galerkin Method for a Singularly Perturbed Bilocal Problem*
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){[-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.
where 0 < epsi≪1,a(x) > alpha > 0,x in[0,1]0<\varepsilon \ll 1, a(x)>\alpha>0, x \in[0,1] and functions aa and ff are sufficiently smooth.
The exact solution of problem (P)(P) has a boundary layer in x=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) (in the space H_(0)^(1)[0,1]H_{0}^{1}[0,1] ). First, the space H_(0)^(1)[0,1]H_{0}^{1}[0,1] turns "conveniently" into the space GH_(0)^(1)[0,1]G H_{0}^{1}[0,1], which is the image of H_(0)^(1)[0,1]H_{0}^{1}[0,1] by Gu:=uog,u inH_(0)^(1)[0,1]G u:=u o g, u \in H_{0}^{1}[0,1] and g:[0,1]- > [0,1]g:[0,1]->[0,1] with g(0)=0,g(1)=1g(0)=0, g(1)=1 and EE M > 0\exists M>0, such that 0 <= g^(')(y) <= M,y in[0,1]0 \leq g^{\prime}(y) \leq M, y \in[0,1]. Problem ( PP ) is transcribed in GH_(0)^(1)[0,1]G H_{0}^{1}[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 GH_(0)^(1)[0,1]G H_{0}^{1}[0,1] will have a Galerkin solution with attenuated oscillations in the area of the boundary layer. Getting back to problem ( PP ), 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]) < 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\},
norm on L^(2)[0,1]L^{2}[0,1] being:
||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}}
Let
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\}
with norm
||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}}
and seminorm
|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}}
and subspace
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\} .
Seminorm |*|_(1)|\cdot|_{1} is norm (equivalent to ||*||_(1)\|\cdot\|_{1} ) on the space H_(0)^(1)[0,1]H_{0}^{1}[0,1].
Let g:[0,1]- > [0,1]g:[0,1]->[0,1] a differe. atiable function on [0,1][0,1] so that:
Let g:[0,1]- > [0,1]g:[0,1]->[0,1] a differe. atiable function on [0,1][0,1] so that:
{:[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}
We note J(y)=g^(')(y)J(y)=g^{\prime}(y) and we define GH_(0)^(1)[0,1]G H_{0}^{1}[0,1] to be the image of space H_(0)^(1)[0,1]H_{0}^{1}[0,1] by transformation Gu:=u@gG u:=u \circ g.
Lemma 1
||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}}
is norm on the space GH_(0)^(1)G H_{\mathbf{0}}^{1} and the following inequality takes place :
||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}} for AA v in CH_(0)^(1)\forall v \in C 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}} for AA v in CH_(0)^(1)\forall v \in C H_{0}^{1}.
Proof.
Because for 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) so that v(y)=u(g(y))v(y)=u(g(y)) we have
||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}
so ||.||_(GH_(0)^(1)(0,1))\|.\|_{G H_{0}^{1}(0,1)} is norm on space GH_(0)^(1)G H_{0}^{1}.
Let v in GH_(0)^(1)(0,1)Longrightarrow vv \in G H_{0}^{1}(0,1) \Longrightarrow v is absolutely continuous, and v(0)=0v(0)=0, thus.
Let v in GH_(0)^(1)(0,1)Longrightarrow vv \in G H_{0}^{1}(0,1) \Longrightarrow v is absolutely continuous, and v(0)=0v(0)=0, thus.
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]
By use of Cauchy-Schwartz inequality, we obtain
|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. }
For the sake of simplicity, we consider a(x)-=1a(x) \equiv 1 in ( PP ) and consequently we have the following singularly perturbed problem :
(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.
Making the change of variable x=g(y)x=g(y) we obtain:
(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.
where F(y)=f(g(y))F(y)=f(g(y)), and v(y)=u(g(y))v(y)=u(g(y)).
Problem (P1) has one solution,
(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.
and problem ( P2P 2 ) will have one solution,
(2) {[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. (see [3]).
Problem (P1) has one solution,
(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.
and problem ( P2P 2 ) will have one solution,
(2) {[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. (see [3]).
We will describe further the approximation scheme of the solution of pröblem (2).
Let L > 0L>0 fixed natural number, and Pi^(L):0=y_(0) < y_(1) < dots < y_(N)=1\Pi^{L}: 0=y_{0}<y_{1}<\ldots<y_{N}=1 a uniform division of interval [0,1][0,1], where N=2^(L)N=2^{L} and y_(j)=(j)/(2^(L)),0 <= j <= 2^(L)y_{j}=\frac{j}{2^{L}}, 0 \leq j \leq 2^{L}. We define set V^(L)V^{L} as being the subspace of GH_(0)^(1)G H_{0}^{1}, which contains those functions ww satisfying,
(3) ((1)/(J(y))w^(')(y))^(')=0\left(\frac{1}{J(y)} w^{\prime}(y)\right)^{\prime}=0 where y_(j) < y < y_(j+1)y_{j}<y<y_{j+1} for each 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-1.
(3) ((1)/(J(y))w^(')(y))^(')=0\left(\frac{1}{J(y)} w^{\prime}(y)\right)^{\prime}=0 where y_(j) < y < y_(j+1)y_{j}<y<y_{j+1} for each 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-1.
In V^(L)V^{L} we consider the system of functions:
" (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}
for 1 <= k <= 2^(L)-11 \leq k \leq 2^{L}-1, where chi_(L,k)\chi_{L, k} is characteristic function of interval [(k)/(2^(L)),(k+1)/(2^(L))]\left[\frac{k}{2^{L}}, \frac{k+1}{2^{L}}\right]. Obviously Phi_(L,k)inV^(L)\Phi_{L, k} \in V^{L} for 1 <= k <= 2^(L)-11 \leq k \leq 2^{L}-1 and Phi_(L,k)(0)=Phi_(L,k)(1)=0\Phi_{L, k}(0)=\Phi_{L, k}(1)=0.
Because in addition,
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}
very function g inV^(L)g \in V^{L} can be represented using base {Phi_(L,k)}_(k=1)^(2^(L)-1)\left\{\Phi_{L, k}\right\}_{k=1}^{2^{L}-1} thus
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) .
onsequently {Phi_(L,k)},1 <= k <= 2^(L)-1\left\{\Phi_{L, k}\right\}, 1 \leq k \leq 2^{L}-1 form a base in V^(L)V^{L}.
For v in GH_(0)^(1)(0,1)v \in G H_{0}^{1}(0,1), we define the interpolant w_(L)^(**)w_{L}^{*} from V^(L)V^{L} as being the nique element from V^(L)V^{L} which satisfies
For v in GH_(0)^(1)(0,1)v \in G H_{0}^{1}(0,1), we define the interpolant w_(L)^(**)w_{L}^{*} from V^(L)V^{L} as being the nique element from V^(L)V^{L} which satisfies
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} .
emma 2 Let vv be the solution of problem ( P2P 2 ) and be w_(L)^(**)w_{L}^{*} the unique V^(L)-V^{L}- nterpolant . Then,
||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
here 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) and x_(j)=g(y_(j))x_{j}=g\left(y_{j}\right) for 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-1.
Proof.
Consider a certain interval [y_(j),y_(j+1)],0 <= j <= 2^(L)-1\left[y_{j}, y_{j+1}\right], 0 \leq j \leq 2^{L}-1. Because w_(L)^(**)-vw_{L}^{*}-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)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=
(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=
ecause 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)=0 and 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) so that 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]
=-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=
Proof.
Consider a certain interval [y_(j),y_(j+1)],0 <= j <= 2^(L)-1\left[y_{j}, y_{j+1}\right], 0 \leq j \leq 2^{L}-1. Because w_(L)^(**)-vw_{L}^{*}-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)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=
(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=
ecause 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)=0 and 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) so that 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]
=-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=
{:(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*}
For 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, we have :
{:[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}
Using this and inequality (5), we obtain:
||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.
where 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) and x_(j)=g(y_(j))x_{j}=g\left(y_{j}\right) for 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-1.
Let w_(L)(g(y))=w_(L)^(**)(y)w_{L}(g(y))=w_{L}^{*}(y) for AA y in[y_(j),y_(j+1)]\forall y \in\left[y_{j}, y_{j+1}\right]; then, we have
||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.
where 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) and x_(j)=g(y_(j))x_{j}=g\left(y_{j}\right) for 0 <= j <= 2^(L)-10 \leq j \leq 2^{L}-1.
Let w_(L)(g(y))=w_(L)^(**)(y)w_{L}(g(y))=w_{L}^{*}(y) for AA y in[y_(j),y_(j+1)]\forall y \in\left[y_{j}, y_{j+1}\right]; then, we have
{:[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}
and, therefore,
||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. }
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} - interpolant of solution of problem (P2).
Theorem 3 Let vv be the solution of problem (P2), let w_(L)^(#)w_{L}^{\#} be the Galerkin approximation of it, and w_(L)^(**)w_{L}^{*} its interpolant in space V^(L)V^{L}. Then, the following inequalities
||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]}
holds.
Proof. We will use the fact that on every interval (y_(j),y_(j+1)),0 <= j <= 2^(L)-1\left(y_{j}, y_{j+1}\right), 0 \leq j \leq 2^{L}-1, the functions from V^(L)V^{L} satisfy the differential equation
Proof. We will use the fact that on every interval (y_(j),y_(j+1)),0 <= j <= 2^(L)-1\left(y_{j}, y_{j+1}\right), 0 \leq j \leq 2^{L}-1, the functions from V^(L)V^{L} satisfy the differential equation
((1)/(J(y))w_(L)^(**')(y))^(')=0\left(\frac{1}{J(y)} w_{L}^{* \prime}(y)\right)^{\prime}=0
Let 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}. -1 . Because
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}-
it results that
(6) 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 y.
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}-
it results that
(6) 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 y.
Similarly, we have
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
thus
(7) 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 y.
(7) 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 y.
We also have
{:[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}
and using this in (6), we obtain
" (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
Considering 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}, and 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}, from (8) we obtain :
{:[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}
ıd similarly, using (7)
" (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}=
e find
{:[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}
From (8) and (9) it results :
{:[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}
{:[=(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}
We have
{:[||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}
We similarly define space V_(L-1)V_{L-1}, and within this space {Phi_(L-1,k)}_(k=1)^(2^(L-1)-1)\left\{\Phi_{L-1, k}\right\}_{k=1}^{2^{L-1}-1} is วase, where
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}
We have :
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)
where
Obviously V^(L-1)subV^(L)V^{L-1} \subset V^{L}.
Let 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} and
Let 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} and
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\} .
Functions Psi_(L-1,k)(y)\Psi_{L-1, k}(y) have as support the interval [(2k-2)/(2^(L)),(2k)/(2^(L))]\left[\frac{2 k-2}{2^{L}}, \frac{2 k}{2^{L}}\right] and obviously suppPsi_(L-1,k)nnn suppPsi_(L-1,l)=Phi\operatorname{supp} \Psi_{L-1, k} \bigcap \operatorname{supp} \Psi_{L-1, l}=\Phi if k!=lk \neq l. System {Psi_(L-1,k)}_(k=1)^(2L-1)\left\{\Psi_{L-1, k}\right\}_{k=1}^{2 L-1} forms a base of space W^(L-1)W^{L-1} and we also have W^(L-1)subV^(L-1)W^{L-1} \subset V^{L-1}.
We will show that V^(L-1)o+W^(L-1)=V^(L)V^{L-1} \oplus W^{L-1}=V^{L}. The fact that V^(L-1)+W^(L-1)=I^(-1)V^{L-1}+W^{L-1}=I^{-1}. is obvious; we will further prove that
V^(L-1)_|_W^(L-1).V^{L-1} \perp W^{L-1} .
We have
{:[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}
Similarly,
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
Consequently, subspaces V^(L-1)V^{L-1} and W^(L-1)W^{L-1} are orthogonal related to the scalar product,
(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
which means that the sum is direct..
The procedure can continue, leading finally to decomposition of space V^(L)V^{L}, so that :
The procedure can continue, leading finally to decomposition of space V^(L)V^{L}, so that :
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}
This decomposition allows the consideration of another base in space V^(L)V^{L}, namely :
{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}}
where Psi_(jk)(y)=Phi_(j+1,2k-1)(y)\Psi_{j k}(y)=\Phi_{j+1,2 k-1}(y).
We also have
We also have
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.
where 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) and psi\psi is Haar's function
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.
Numerical example.
We consider the following problem :
{:[-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}
which have the solution 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-1.
We do the change of variable x=g(y)x=g(y) where 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=4p=0, p=2, p=4 and N=4,epsi=0.0001N=4, \varepsilon=0.0001.
We do the change of variable x=g(y)x=g(y) where 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=4p=0, p=2, p=4 and N=4,epsi=0.0001N=4, \varepsilon=0.0001.
Figure 1:
Figure 2:
Figure 3:
Bibliography
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[3] P.W. Hemker, A numerical study of stiff two-point boundary problems, Amsterdam, 1997.
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[5] H. Yserentant, On the multi - level splitting of finite element spaces. Numer. Math. 49 (1986), pp. 379-412.
[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
- *Supportet by ANSTI Grant GR/4122 (1998)