El taller como implementación de las pedagogías de la memoria y cimiento para el empoderamiento de los jóvenes como sujetos políticos

We assume that the mesh functionf(p, v) satisfies the two-sided constraint

0< c∗ ≤fu ≤c∗, c∗, c∗= const. (1.20)

We select an initial mesh function v(0) _{on Ω}h _{such that it satisfies the boundary condi-}

tion g(p) on ∂Ωh. The monotone iterative method is given by the following recurrence formulae:

(L+c∗)z(n)(p) =−R(p, v(n−1)), p∈Ωh, (1.21) z(n)(p) = 0, p∈∂Ωh, v(n)(p) =z(n)(p) +v(n−1)(p), p∈Ωh,

R(p, v(n−1)_{) =Lv}(n−1)_{(p) +f(v}(n−1)_{, p).}

Remark 1. For simplicity, we require our initial solution satisfies the boundary condition g(p) on ∂Ωh, however this is not essential. Instead, we can only require that

In this case, z(n)(p) is now defined by

z(1)(p) =g(p)−v(0)(p), orz(1)(p) =g(p)−v(0)(p), p∈∂Ωh, z(n)(p) = 0, n≥2, p∈∂Ωh.

Using this definition, we see that

v(1)(p) =v(0)(p) +z(1)(p) =v(0)(p) + (g(p)−v(0)(p)) =g(p), p∈∂Ωh.

Theorem 1. Let v(0), v(0) be upper and lower solutions of (1.15). Assume that the dif- ference operator L and function f satisfy (1.16) and (1.20), respectively, and the mesh

Ωh is connected. Then the upper and lower sequences {v(n)}and {v(n)} generated by the iterative method (1.21) converge monotonically from, respectively, above and below, to the unique solution v of (1.15):

v(n−1)(p)≤v(n)(p)≤v(p)≤v(n)(p)≤v(n−1)(p), p∈Ωh, n≥1.

Proof. We only consider the case of upper solutions. The case of lower solutions may be shown in a similar manner.

Assume thatv(0)(p) is an upper solution which satisfies the boundary condition. Then from (1.21) withn= 1, we have

(L+c∗)z(1)(p) =−R(p, v(0))≤0, p∈Ωh, z(1)(p) = 0, p∈∂Ωh.

By the maximum principle from Lemma 1, we conclude that z(1)_{(p) ≤0, p∈Ω}h_{. Using}

the recurrence formula (1.21) and the mean value theorem, we have

R(p, v(1)) = R(p, v(0)+z(1)) = Lv(0)(p) +Lz(1)(p) +f(p, v(0)+z(1)) = Lv(0)(p) +f(p, v(0)) +Lz(1)(p) +f(p, v(0)+z(1))−f(p, v(0)) = R(p, v(0)) +Lz(1)(p) +fv(1)(p)z(1)(p) = −(L+c∗)z(1)(p) +Lz(1)(p) +fv(1)(p)z(1)(p) = −(c∗−fv(1)(p))z(1)(p),

wherefv(1)(p) =fv[p, v(0)(p) + Θ(1)(p)z(1)(p)],0<Θ(1)(p)<1. From here and (1.20), we

obtain

and conclude that v(1)(p) is an upper solution. By induction on n, we can prove that z(n)_{(p) ≤ 0, p ∈ Ω}h_{, and v}(n)_{(p) is an upper solution for all n ≥ 0. Thus, {v}(n)_{} is a}

monotonically decreasing sequence of upper solutions. This sequence is bounded below by v, wherev is any lower solution (1.19). We conclude that the mesh function defined by

v(p) = lim

n→∞v

(n)_{(p), p∈Ω}h_,

exists.

We now prove that v(p) is the exact solution to (1.15). From (1.21), it follows that lim n→∞(L+c ∗_)z(n)_{(p) = lim} n→∞−R(p, v (n−1)_). As limn→∞z(n)(p)→0,p∈Ωh, lim n→∞−R(p, v (n−1)_{) =−R(p, v) = 0.}

Thus,Lv(p) +f(p, v) = 0,p∈Ωh, and taking into account thatv(p) =g(p),p∈∂Ωh, we prove that v(p) is the exact solution of (1.15).

We now prove thatv(p) is the unique solutions of (1.15). Suppose there exists another solution _bv(p) of (1.15). Letς(p) =v(p)−_bv(p),p∈Ωh. By the mean value theorem,

R(p, v(p))− R(p,_bv(p)) = Lv(p) +f(p, v(p))− L_bv(p)−f(p,v(p))_b = L(v(p)−_bv(p)) +f(p, v(p))−f(p,bv(p))

= Lς(p) +f_v∗ς(p) = 0, p∈Ωh,

where f_v∗ = fv(p, v∗(p)), v∗(p) lies between v(p) and v(p). Sinceb ς(p) = 0, p ∈ ∂Ω

h_{, by}

the maximum principle from Lemma 1, we conclude that ς(p) =v(p)−_bv(p) = 0, p∈Ωh, and prove the uniqueness of the solutionv(p) and the theorem.

Remark 2. An advantage of the monotone iterative method is that we can provide a simple method of finding initial upper or lower solutions without any prior knowledge of the solution. Let t(p) be an arbitrary mesh function defined on Ωh which satisfies the boundary condition t(p) =g(p) on ∂Ωh. We consider the difference problems

Lzν(p) +c∗zν(p) =ν|R(t, q)|, p∈Ωh, zν = 0, p∈∂Ωh, ν= 1,−1, (1.22)

where R(t, q) = Lt(p) +f(t, q). Then v(0)(p) = t(p) +z1(p), v(0)(p) = t(p) +z−1(p),

Proof. We consider only the case for the initial upper solution, as the lower solution case may be shown in a similar manner. By the maximum principle from Lemma 1, we conclude that z1(p)≥0, p ∈Ω

h

. From here and (1.22), by the mean value theorem, we have Lv(0)(p) +f(p, v(0)) = L(t(p) +z1(p)) +f(p, t+z1) = Lt(p) +f(p, t+z1) +Lz1(p)±c∗z1(p) = Lt(p) +f(p, t+z1)±f(t, p) +|Lt(p) +f(t, p)| −c∗z1(p) = Lt(p) +f(p, t) +|Lt(p) +f(t, p)| +f_v∗z1(p)−c∗z1(p) = Lt(p) +f(p, t) +|Lt(p) +f(t, p)| +(f_v∗−c∗)z1(p)≥0,

wheref_v∗=fv(p, v∗(p)), v∗(p) lies betweent(p) andt(p) +z1(p). From here, we conclude

One dimensional

convection-diffusion problem

This chapter deals with a uniform (in a perturbation parameter) convergent difference scheme for solving a nonlinear singularly perturbed two-point boundary value problem of the convection-diffusion type with discontinuous data. Construction of the difference scheme is based on locally exact schemes or on local Green’s functions. Uniform con- vergence of the proposed difference scheme on arbitrary meshes is proven. A monotone iterative method, which is based on the method of upper and lower solutions, is applied to computing the nonlinear difference scheme. Numerical experiments are presented.

2.1

Introduction

In this chapter, we are interested in the semilinear two-point boundary-value problem with a convective dominated term and discontinuous data

−εu00+b(x)u0+c(x, u) +f(x) = 0, x∈Ω = (0,1), (2.1)

u(0) = 0, u(1) = 0, b(x)≥b∗= const>0, cu≥0, (cu ≡∂c/∂u),

whereεis a small positive parameter. Suppose that the function c is sufficiently smooth and b,f are piecewise smooth functions, i.e.

b(x), f(x)∈Qn_p(Ω), n≥0. 23

We say that v(x) ∈ Qn_p(Ω) if it is defined on Ω and has derivatives up to order n, the function itself and its derivatives may only have jump discontinuities at a finite set of pointsp={p1, . . . , pJ},0< pj < pj+1, j = 1, . . . , J −1,i.e., Qnp(Ω) =Cn(Ω\p).

The solution to (2.1) is a function with a continuous first derivative, which satisfies the boundary conditions and the equation everywhere, with the exception of the points inp. The problem (2.1) has a unique solution [70]

u(x)∈C1(Ω)∩Qn_p+2(Ω).

Linear versions of problem (2.1) with discontinuous data are investigated in [19], [33]. The solution of the linear problem possesses a strong boundary layer at x= 1 and weak interior layers at the points of discontinuity p. The boundary layer is strong in the sense that the solution is bounded, but the magnitude of its first derivative at x = 1, grows unboundedly as ε → 0. The interior layers at p are weak: the solution and the first derivative are bounded but the magnitude of the second derivative grows unboundedly as ε→0. We show (see Lemma 3) that problem (2.1) possesses a strong boundary layer at x= 1, and the solution and the first derivative are bounded at the points of discontinuity p.

Our goal is to construct an ε-uniform numerical method (1.1) for solving problem (2.1). In [19], [33], for solving the linear version of problem (2.1), the uniform numerical methods are constructed by using the integral-difference method (or the method of locally exact schemes) on arbitrary nonuniform meshes [19], and by using the standard upwind finite difference method on the piecewise uniform mesh, which is fitted to boundary and interior layers [33].

In Section 2.2, we establish somea prioriestimates of the solution and its first deriva- tive. In Section 2.3, we construct a numerical method by applying the integral-difference approach. Note that in the constructed numerical method, a difference operator corre- sponding to the linear differential operator−εd2/dx2+bd/dxis equivalent to the upwind finite volume method from [44], [79]. In Section 2.4, we prove uniform convergence of the numerical method on arbitrary nonuniform meshes by extending in a natural way the proof of the main theoretical result from [13] (the difference scheme in the case of problem (2.1) with smooth data converges ε-uniformly). In Section 2.5, we construct a monotone iterative method for solving the nonlinear difference scheme and prove that the iterates

converge ε-uniformly to the solution of problem (2.1). In Section 2.6, numerical results are presented, which are in agreement with the theoretical results.

Most of the content of this chapter is published in [25].