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An Extension of the Product Integration Method to [L.sub.1] with Applications in Astrophysics.

1 Introduction

We consider a Banach space X. Let T be the integral operator defined by

[for all]x [member of] X, Vs [member of] [a,b], Tx(s) := [[integral].sup.b.sub.a] L(s,t)H(s,t)x(t)dt, (1.1)

where (s,t) [??] H(s,t) is not smooth. For z in the resolvent set of T, re(T), and y in X we consider the Fredholm integral problem of the second kind

Find [phi] [member of] X s.t. (T - zI) [phi] = y, (1.2)

where I denotes the identity operator on X.

To approximate the solution of this equation, we define a finite rank approximation [T.sub.n] of T, so that the approximate equation ([T.sub.n] - zI)[[phi].sub.n] = y or ([T.sub.n] - zI)[[phi].sub.n] = [y.sub.n], where [y.sub.n] is an approximation of y, be uniquely solvable and the sequence of approximate solutions [[phi].sub.n] converges to the exact solution [phi] when n tends to + [infinity].

Among them, different classes of methods rely on a sequence of projections [[pi].sub.n] converging pointwise to the identity operator I. For example the Galerkin operator is defined by [T.sub.n] = [[pi].sub.n]T[[pi].sub.n], the projection operator by [T.sub.n] = [[pi].sub.n]T, the Sloan operator by [T.sub.n] = T[[pi].sub.n] and the Kulkarni operator by [T.sub.n] = T[[pi].sub.n] + [[pi].sub.n]T - [[pi].sub.n]T[[pi].sub.n] (see [5], [10]). These approximations of T are all v-convergent to T (see [2]). This property ensures existence and uniqueness of [[phi].sub.n], and convergence to [phi].

In the case of the space X := [C.sup.0]([a, b], C) methods based upon numerical quadrature have been proposed, such as Nystrom, truncated Nystrom and subtraction of the singularity approximations (see [4]).

In [C.sup.0]([a, b], C), we also encounter the so-called product integration method (see [5]). In this space, the assumptions are as follows:

(H1) L [member of] [C.sup.0]([a,b] x [a, b], C).

(H2) H verifies:

(H2.1) [mathematical expression not reproducible] is finite,

(H2.2) [mathematical expression not reproducible], where

[mathematical expression not reproducible].

Let [[DELTA].sub.n], defined by

a =: [t.sub.n,0] < [t.sub.n,1] < ... < [t.sub.n,n] := b (1.3)

be a uniform grid of [a, b]. If [h.sub.n] := (b - a)/n, then [t.sub.n,i] = a + i[h.sub.n], for i = 0,1, ..., n. For x [member of] [C.sup.0]([a, b], C) and s [member of] [a, b], the linear interpolation scheme is given by

[mathematical expression not reproducible]

for i = 1, ..., n and t [member of] [[t.sub.n,i - 1], [t.sub.n,i]].

[T.sub.n] is defined by replacing L(s,t)x(t) with [[L(s,t)x(t)].sub.n] in (1.1). In this method [T.sub.n] is a bounded finite rank linear operator defined in [C.sup.0]([a, b], C) and hence it is compact.

Under hypotheses (H1) and (H2), for z [member of] re(T) and for n large enough, [T.sub.n] - zI is invertible and its inverse is uniformly bounded, (see [5]).

In this paper we extend the product integration method to the space X := [L.sub.1]([a, b], C). It will appear that the properties of the method in [C.sup.0]([a, b], C) are preserved in [L.sub.1]([a, b], C). In Section 2, we present our method and we prove the existence and uniqueness of the approximate solution and its convergence to the exact solution. Section 3 is devoted to the numerical implementation of our algorithm. The choice of the integer n is limited by the capacity of the computer. The linear system to be solved is of the order of n. So, it is interesting to improve the accuracy of the approximate solution by applying some iterative refinement schemes. Section 4 is devoted to these schemes. In Section 5, we test our approximation with an academic example. In Section 6, we apply our method to a problem belonging to Astrophysics. Our method is compared with the projection method proposed by Titaud in [1] and [11].

2 The product integration method in [L.sub.1]([a,b], C)

We use the following notations: the norm in [L.sub.1]([a,b], C) is denoted by [parallel]x[parallel][sub.1] := [[integral].sup.b.sub.a] [absolute value of x(s)]ds. The subordinated operator norm is also denoted by [parallel]x[parallel][sub.1].

The oscillation of a function x in [L.sub.1]([a, b], C), relatively to a parameter h is defined by

[mathematical expression not reproducible], (2.1)

where x is extended by 0 outside [a, b].

The modulus of continuity of a continuous function on [a, b] is defined as

[mathematical expression not reproducible].

The modulus of continuity of a continuous function on [a, b] x [a, b] is defined as

[mathematical expression not reproducible].

If [mathematical expression not reproducible].

The aim of this section is to define the approximate operator [T.sub.n]. The approximate solution of (1.2) will be, if it exists and is unique, the solution of

([T.sub.n] - zI) [[phi].sub.n.] = y. (2.2)

[T.sub.n] is constructed so that [[phi].sub.n] [right arrow] [phi]. It is well known that a collectively compact convergence of [T.sub.n] towards T guarantees the convergence of [[phi].sub.n] towards [phi].

Let us recall the collectively compact convergence:

Definition 1. [T.sub.n] and T are bounded linear operators from X into X. The pointwise convergence, denoted by [mathematical expression not reproducible], means that

[for all]x [member of] X, [parallel] [T.sub.n]x - [T.sub.x] [parallel] [right arrow] 0.

The collectively compact convergence is denoted by [mathematical expression not reproducible] : if T is compact

[mathematical expression not reproducible]

and for some positive integer [n.sub.0] the set

[mathematical expression not reproducible]

is relatively compact in X.

We begin by proving that T is a compact bounded linear operator from [L.sub.1]([a, b], C) into itself. Then we propose an approximate operator [T.sub.n] which is a collectively compact convergent to T. Endly, we give an error estimation for the approximate solution in terms of the kernel, the norm of the exact solution, its oscillation in [L.sub.1]([a, b], C) and the mesh size.

The proof of the compactness in [L.sub.1]([a, b], C) relies on the Kolmogorov-Riesz-Frechet theorem which is recalled here below. As usual, if A is a set of functions, we define

A|[sub.[OMEGA]] := {f|[sub.[OMEGA]] : f [member of] A},

where f|[sub.[OMEGA]] is the restriction of f to the subdomain [OMEGA].

Theorem 1. (Kolmogorov-Riesz-Frechet Theorem) Let F be a bounded set in [L.sup.p]([R.sup.q], C), 1 [less than or equal to] p < [infinity]. If

[mathematical expression not reproducible] (2.3)

uniformly in f [member of] F, where [[tau].sub.h]f := f (. + h), then the closure of F|[sub.[OMEGA]] is compact in [L.sup.p]([OMEGA], C) for any measurable set [OMEGA] [subset] [R.sup.q] with finite measure.

Proof. See [7]. As one finds a lot of different versions of this theorem in the litterature, we propose a proof of it in the Appendix in the case q = 1, p = 1 and [OMEGA] = [a, b].

Now, the assumptions are as follows:

(P1) L [member of] [C.sup.0]([a, b] x [a, b], C). Let

[mathematical expression not reproducible]

(P2) H verifies:

(P2.1) [mathematical expression not reproducible] is finite.

(P2.2) [mathematical expression not reproducible],

where

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

Lemma 1.

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Proof. For h > 0,

[mathematical expression not reproducible].

According to the assumption (P2.2), [mathematical expression not reproducible].

This ends the proof.

Theorem 2. Under the assumptions (P1) and (P2), the operator T is linear from [L.sub.1]([a, b], C) into itself and compact in [L.sub.1]([a, b], C).

Proof. For all x [member of] [L.sub.1]([a, b], C),

[mathematical expression not reproducible],

so T is defined from [L.sub.1]([a, b], C) into itself.

The proof of the compactness of T relies on the Kolmogorov-Riesz-Frechet theorem where p = 1, q = 1 and [OMEGA] = [a, b]. We introduce the operator [??]:

[mathematical expression not reproducible].

Let A and S be the following subsets of [L.sub.1](R, C) and [L.sub.1]([a, b], C) respectively:

[mathematical expression not reproducible].

A is a bounded subset of [L.sub.1](R, C). Indeed

[mathematical expression not reproducible].

Let us prove that [mathematical expression not reproducible]. For h > 0,

[mathematical expression not reproducible].

Hence

[mathematical expression not reproducible]

and

[mathematical expression not reproducible].

So

[mathematical expression not reproducible]. (2.4)

For h < 0, we have similar bounds. Then [mathematical expression not reproducible] uniformly in f [member of] A. From the Kolmogorov-Riesz-Frechet theorem S = A|[sub.[a b]] is relatively compact so T is compact.

Let us define the approximate operator [T.sub.n]. Let [[DELTA].sub.n] be the partition defined by (1.3). For x [member of] [L.sub.1]([a, b], C), we define the operator

[mathematical expression not reproducible]

for i = 1,..., n and t [member of] [[t.sub.n,i-1], [t.sub.n,i]]. The approximate operator [T.sub.n] is given by:

[mathematical expression not reproducible], (2.5)

which can be rewritten as

[T.sub.n]x(s) = [n.summation over (i=1)] [c.sub.n,i] [w.sub.n,i](s),

where, for i = 1,..., n,

[mathematical expression not reproducible].

To prove that [mathematical expression not reproducible], the following lemmas are needed.

Lemma 2. For i = 1,..., n,

[mathematical expression not reproducible]. (2.6)

For h [member of] [R.sup.+],

[mathematical expression not reproducible], (2.7)

[mathematical expression not reproducible]. (2.8)

Proof. For t [member of] [[t.sub.n,i-1],[t.sub.n,i]],

[mathematical expression not reproducible].

Hence, by Fubini's theorem

[mathematical expression not reproducible].

Also

[mathematical expression not reproducible].

This ends the proof.

Lemma 3. For x [member of] [L.sub.1]([a,b], C),

[mathematical expression not reproducible],

where [w.sub.1](x,[h.sub.n]) is defined by (2.1). For t [member of] [a, b],

[absolute value of [Q.sub.n] (1,s,t) - L(s,t)] [less than or equal to] [w.sub.2](L, [h.sub.n]).

Proof. For i = 1, ..., n,

[mathematical expression not reproducible].

Hence

[mathematical expression not reproducible].

For i = 1,..., n and t [member of] [[t.sub.n,i-1], [t.sub.n,i]],

[mathematical expression not reproducible]

and the proof is complete.

Theorem 3. [T.sub.n] is a compact linear operator from [L.sub.1]([a, b], C) into itself and

[mathematical expression not reproducible].

Proof. Due to (2.6) in Lemma 2, for [mathematical expression not reproducible] is a linear bounded operator from [L.sub.1]([a, b], C) into itself. As [T.sub.n] is a linear bounded operator of finite rank, it is compact. Let us prove that [mathematical expression not reproducible]. Lemma 3 implies that

[mathematical expression not reproducible].

Hence

[mathematical expression not reproducible]. (2.9)

So we have [mathematical expression not reproducible]. To prove the relatively compactness of

[mathematical expression not reproducible]

we follow the same scheme as in the proof of the compactness of T. We define the operator

[mathematical expression not reproducible],

and [A.sub.n] as the following subset of [L.sub.1](R, C)

[mathematical expression not reproducible].

An is a bounded subset of [L.sub.1](R, C). Indeed,

[mathematical expression not reproducible].

Let us prove that [mathematical expression not reproducible]. For h> 0,

[mathematical expression not reproducible].

Hence, by (2.7) in Lemma 2,

[mathematical expression not reproducible]

and because of (2.8) in Lemma 2,

[mathematical expression not reproducible].

Hence

[mathematical expression not reproducible]. (2.10)

For h < 0, we have similar bounds. Then [parallel][[tau].sub.h] f - f [parallel] [sub.1] [right arrow] 0 as h [right arrow] 0 uniformly in f [member of] [A.sub.n]. From the Kolmogorov-Riesz-Frechet theorem, [A.sub.n]|[sub.[a,b]] is relatively compact so [mathematical expression not reproducible].

Proposition 1. Let z [member of] re(T). For n large enough, [T.sub.n] - zI is invertible and it exists a positive number [c.sub.z] > 0 such that

[[parallel] ([T.sub.n] - zI).sup.-1] [parallel][sub.1] [less than or equal to] [c.sub.z]. (2.11)

Proof. It is a consequence of the collectively compact convergence (see [3]).

Theorem 4. For z [member of] re(T) and under hypotheses (P1) and (P2), for n large enough, the approximate operator equation (2.2) has a unique solution [[phi].sub.n] satisfying the following error bound:

[mathematical expression not reproducible].

Proof. According to (2.9) in the proof of Theorem 3,

[mathematical expression not reproducible],

which ends the proof.

Remark 1. Often in practice, the kernel H is of convolution type. Let us fix a = 0 and b = 1. We suppose that there is a function g such that

H (s,t) = g([absolute value of s - t]),

where g is a weakly singular function defined on ]0,1]. This means that g satisfies the following properties:

[mathematical expression not reproducible].

Proposition 2. When the factor H in the kernel of the operator T is of weakly singular convolution type, then H verifies all the conditions imposed by the product integration methods.

Proof.

(H2.1) [for all]s [member of] [0,1], we have

[mathematical expression not reproducible].

(P2.1) is also valid because the variables s and t play symmetric roles.

(H2.2) Let us prove that, for h > 0,

[mathematical expression not reproducible].

Let [psi] be the function defined by [mathematical expression not reproducible]. Suppose that [tau] < s. It is easy to prove that [psi] has an axial symmetry with respect to [xi] = s + [tau]/2 over the interval [[tau], s]. Let G(t) := [[integral].sup.t.sub.0] g(s)ds. Then 0

[mathematical expression not reproducible],

hence,

[mathematical expression not reproducible].

(P2.2) Let us prove that, for h > 0,

[mathematical expression not reproducible].

For t [member of] [0,1],

[mathematical expression not reproducible],

so

[mathematical expression not reproducible],

which ends the proof.

3 Iterative refinement

Recall that z [not equal to] 0 because T is compact and z [member of] re(T). Consider that the solution of (1.2) is approximated by [G.sub.n](z)y, where [G.sub.n](z) is an approximate inverse of T - zI. The accuracy of [G.sub.n](z)y may be improved using the following iterative refinement schemes:

[mathematical expression not reproducible].

In [11], [G.sub.n](z) has been one of the following operators:

Scheme A (Atkinson):

[G.sub.n](z) := [R.sub.n](z) := ([T.sub.n] - zI)-1,

Scheme B (Brakhage):

[G.sub.n](z) := 1/z ([R.sub.n](z)T - I),

Scheme C (Titaud):

[G.sub.n](z) := 1/z (T[R.sub.n](z) - I).

Their convergence properties and error bounds have already been studied in terms of T, [T.sub.n] and [R.sub.n](z) (see [11] pp 40-41). If [phi] is the solution of (1.2), Scheme A (Atkinson):

[mathematical expression not reproducible],

Scheme B (Brakhage):

[mathematical expression not reproducible],

Scheme C (Titaud):

[mathematical expression not reproducible].

Let us state error estimations for these three refinement schemes for the approximate operator [T.sub.n] defined by (2.5) in this paper.

Theorem 5. For [T.sub.n] defined by (2.5), the following error bounds are satisfied: Scheme A (Atkinson):

[mathematical expression not reproducible],

Scheme B (Brakhage):

[mathematical expression not reproducible],

Scheme C (Titaud):

[mathematical expression not reproducible],

where

[mathematical expression not reproducible].

Proof. Using (2.9),

[mathematical expression not reproducible].

As

[mathematical expression not reproducible]

and due to (2.4),

[mathematical expression not reproducible].

As

[mathematical expression not reproducible].

* Scheme A. As

[mathematical expression not reproducible].

and according to (2.11),

[mathematical expression not reproducible].

We have

[mathematical expression not reproducible].

Then

[mathematical expression not reproducible],

so

[mathematical expression not reproducible].

* Scheme B. As

[mathematical expression not reproducible],

then

[mathematical expression not reproducible]

* Scheme C. As

[mathematical expression not reproducible],

so

[mathematical expression not reproducible].

This concludes the proof.

Remark 2. The upperbound of Scheme B appears to be the optimal one among the three error bounds. It improves slightly upon the one of Scheme C and is twice better than the one of Scheme A.

4 Numerical Implementations

The approximate equation is [T.sub.n][[phi].sub.n] - z[[phi].sub.n] = y, i.e.

[mathematical expression not reproducible].

By calculating the average over [[t.sub.n,i-1], [t.sub.n,i]] ,i = 1,..., n, of each member of the equation, we obtain a linear system of the form (A - zI)x = d, where

[mathematical expression not reproducible]. (4.1)

After solving the linear system, the approximate solution can be written as

[[phi].sub.n](s) = 1/z ([n.summation over (i=1)] [w.sub.n,j] (s) x (i) - y(s).

To measure the quality of the approximation we calculate the relative residual

[mathematical expression not reproducible].

In practice the evaluation of T is often not possible, so we replace it with [T.sub.m] where m [much less than] n and we caculate the average over [[t.sub.m,i-1], [t.sub.m,i]], i = 1,..., m, of (T - zI)[[phi].sub.n] - y and of y. We obtain two vectors of size m, and we calculate the vector norm in ([C.sup.m], [parallel] x [parallel][sub.1]).

5 Numerical Illustration

As an academic example we have taken

- [[integral].sup.1.sub.0] ln([absolute value of s - t]) [phi] (t)dt - [phi](s) = y(s),

with unique solution [phi](s) = [s.sup.2]. The estimations of the relative residual with m = 100 for two methods: the projection method proposed by Titaud in [11] and the [L.sub.1]([a, b], C) product integration method are shown in Table 1. We observe that the [L.sub.1]([a, b], C) product integration method is faster than the projection method.

Figure 1 shows the profile of the matrix A defined by (4.1). It is a full matrix.

In Figure 2 we chose n = 100, m = 1000 for a relative residual tolerance of [10.sup.-12]. We note that Scheme B is the fastest one to reach the tolerance.

The theoretical Remark 2 of Section 3 is confirmed by this numerical experiment.

6 An Application in Astrophysics

The radiative transfer problem is a system of differential equations coupled with a Fredholm integral equation of the second kind. It describes the energy conserved by a beam radiation traveling, such that a beam of radiation can lose or gain energy through absorbing, scattering and emitting medium. Let [[tau].sub.*] be the optical width of the medium, (see [8]). An example of this equation is

[mathematical expression not reproducible],

where [E.sub.1] is the first integral exponential function:

[mathematical expression not reproducible]

and the function [bar.w] describes the albedo. In our numerical example [[bar.[omega]](s) = 0.7 exp(-s) and

[mathematical expression not reproducible].

The singularity of that example is different from the Cauchy singularity treated by Beltram with the product integration method in [6].

Figure 3 shows the profile of the matrix A defined by (4.1). It is a sparse matrix.

The relative residual associated to the approximate solution [[phi].sub.n] obtained by the projection method and the product integration method proposed in this paper are shown in Table 2. We observe that the product integration method converges faster than the projection method.

For large values of n the computation of [[phi].sub.n] is prohibitively costly so that we will use the refinement schemes introduced in Section s:3 to compute the final approximate solution.

In Figure 4 we chose n = 100, m = 1000 for a relative residual tolerance of [10.sup.-12]. We note that Scheme C is the fastest one to reach the tolerance. This confirms the results obtained in [9].

Remark 3. In this application, Scheme C is apparently faster than Scheme B. This could be explained by the difference between the profiles of the corresponding auxiliary matrices A (see Figure 1 and Figure 3).

http://dx.doi.org/10.3846/13926292.2016.1243590

Acknowledgements

This research has been partially supported by the Indo French Center for Applied Mathematics (IFCAM).

References

[1] M. Ahues, F. D'Almeida, A. Largillier, O. Titaud and P. Vasconcelos. An [L.sub.1] refined projection approximate solution of the radiation transfer equation in stellar atmospheres. Journal of Computational and Applied Mathematics, 140(12):13-26, 2002. http://dx.doi.org/10.1016/S0377-0427(01)00403-4.

[2] M. Ahues, A. Largillier and B. Limaye. Spectral computations for bounded operators. CRC Press, 2001.

[3] P.M. Anselone. Collectively compact operator approximation theory and applications to integral equations. Prentice-Hall, Englewood Cliffs, NJ, 1971. Appendix by J. Davis

[4] P.M. Anselone. Singularity subtraction in the numerical solution of integral equations. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 22(04):408-418, 1981. http://dx.doi.org/10.1017/S0334270000002757.

[5] K.E. Atkinson. The numerical solution of integral equations of the second kind. Cambridge university press, New York, 1997.

[6] B. Bertram. On the product integration method for solving singular integral equations in scattering theory. Journal of computational and applied mathematics, 25(1):79-92, 1989. http://dx.doi.org/10.1016/0377-0427(89)90077-0.

[7] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Springer, New York, 2011. http://dx.doi.org/10.1007/978-0-387-70914-7.

[8] L. Chevallier, J. Pelkowski and B. Rutily. Exact results in modeling planetary atmospheres-I. gray atmospheres. Journal of Quantitative Spectroscopy and Radiative Transfer, 104(3):357-376, 2007. http://dx.doi.org/10.1016/j.jqsrt.2006.09.006.

[9] F. d'Almeida, O. Titaud and P.B. Vasconcelos. A numerical study of iterative refinement schemes for weakly singular integral equations. Applied mathematics letters, 18(5):571-576, 2005. http://dx.doi.org/10.1016/j.aml.2004.03.020.

[10] R.P. Kulkarni. A superconvergence result for solutions of compact operator equations. Bulletin of the Australian Mathematical Society, 68(03):517-528, 2003. http://dx.doi.org/10.1017/S0004972700037916.

[11] O. Titaud. Analyse et resolution numerique de l'equation de transferi. PhD thesis, Universite Jean Monnet, Saint Etienne, France, 2001.

Appendix

Proof of the Kolmogorov-Riesz-Frechet theorem. Without loss of generality we prove the theorem for the case p = 1, q = 1 and [OMEGA] = [a, b]. To simplify the notation, [parallel]x[parallel][sub.1] denotes the norm in [L.sub.1]([OMEGA],C) and also the norm in [L.sub.1](R, C). [parallel]x[parallel][sub.[infinity]] denotes the norm in [C.sup.0] ([OMEGA], C) and also the norm in [C.sup.0](R, C).

As [L.sub.1]([omega] a complete space, we just need to prove that F|[sub.[OMEGA]] precompact i.e.: For any [epsilon] > 0 there exist functions [f.sub.1],[f.sub.2],..., [f.sub.N] [member of] [L.sub.1]([OMEGA], C) such that

F|[sub.[OMEGA]] [subset] [[union].sup.N.sub.i=1] [B.sub.1]([f.sub.i], [epsilon]),

where [B.sub.1]([f.sub.i], [epsilon]) denotes the open ball in [L.sub.1]([OMEGA], C) centered in [f.sub.i] and of radius [epsilon]. The proof consists in constructing the functions [f.sub.i]. The main idea of the proof is to apply a convolution regularization process to deal with continuous functions and to be able to apply the Arzela-Ascoli theorem.

Step 1: Regularization process

Let us consider the regularizing sequence defined by

[[rho].sub.n] (x):= n[rho](nx),

where

[mathematical expression not reproducible],

0, otherwise,

and k is a constant such that [parallel][rho] [parallel][sub.1] = 1. For all n [member of] N, [[rho].sub.n] is infinitely differentiate. If * denotes the convolution product, and if f [member of] [L.sub.1](R, C), [[rho].sub.n] * f is a regularization of f in the sense that it is smooth: [[rho].sub.n] * f is infinitely differentiate. We know that [[rho].sub.n] * f [member of] [L.sub.1](R, C) and also [[rho].sub.n] * f [right arrow] f in [L.sub.1](R, C). We prove a stronger result under assumption (2.3):

[[rho].sub.n] * f [right arrow] f

uniformly in f [member of] F in [L.sub.1](R, C).

[mathematical expression not reproducible].

so that for all f [member of] F,

[mathematical expression not reproducible].

Hence for all f [member of] F,

[mathematical expression not reproducible].

According to assumption (2.3), for all [epsilon] > 0, [there exists] [N.sub.0] [member of] N :

[mathematical expression not reproducible].

Step 2: Application of Arzela-Ascoli theorem to [H.sub.n] := {[[rho].sub.n] * f : f [member of] F}|[sub.[OMEGA]]

Here n is fixed. Due to the regularization properties, [H.sub.n] is a subset of [C.sup.0]([OMEGA], C). Let us prove that [H.sub.n] is bounded in [C.sup.0]([OMEGA], C) equiped with the infinity norm [parallel]x[parallel][sub.[infinity]]. As F is bounded in [L.sub.1](R, C),

[mathematical expression not reproducible],

where M := [sup.sub.fmember of]F][parallel]f[parallel][sub.1]. Let us prove that [H.sub.n] is equicontinuous.

Let [x.sub.1], [x.sub.2] [member of] u.

[mathematical expression not reproducible],

where [mathematical expression not reproducible] is the gradient of [[rho].sub.n]. According to Arzela-Ascoli theorem, [H.sub.n] is relatively compact in [C.sup.0]([OMEGA], C) so it is precompact.

Step 3: Construction of the functions [f.sub.i]

As [H.sub.n] is precompact, for [epsilon] > 0 there exist functions [f.sub.i] [member of] [C.sup.0]([OMEGA], C), i = 1,..., N, such that [H.sub.n] [subset] [[union].sup.N.sub.i=1][B.sub.[infinity]] ([f.sub.i], [epsilon]), where [B.sub.[infinity]]([f.sub.i], [epsilon]) denotes the ball in [C.sup.0]([OMEGA], C) centered in [f.sub.i] and of radius [epsilon], i.e:

[mathematical expression not reproducible].

Step 4: Conclusion

Let us show that F|[sub.[OMEGA]] is precompact. Let [epsilon] > 0 and f [member of] F|[sub.[OMEGA]]. According to the step 1, [there exists][N.sub.0] [member of] N :

[mathematical expression not reproducible].

Let us fix n [greater than or equal to] [N.sub.0]. According to the step 3, there exists i [member of] {1,..., N}, such that [parallel][[rho].sub.n] * f - [f.sub.i][parallel][sub.[infinity]] < [epsilon]. We have

[mathematical expression not reproducible].

Hence

[parallel]f - [f.sub.i][parallel][sub.1] [less than or equal to] (1 + b - a)[epsilon].

So [mathematical expression not reproducible] is relatively compact.

Laurence Grammont, Mario Ahues and Hanane Kaboul

Institut Camille Jordan, UMR 5208, CNRS-Universite de Lyon 23 rue du Dr Paul Michelon, 42023 Saint-Etienne Cedex 2, France

E-mail(corresp.): hanane.kaboul@univ-st-etienne.fr

E-mail: laurence.grammont@univ-st-etienne.fr

E-mail: mario.ahues@univ-st-etienne.fr

Received December 10, 2015; revised September 27, 2016; published online November 15, 2016

Caption: Figure 1. Matrix A of the academic illustration.

Caption: Figure 2. Residual convergence with the three refinement schemes of the academic illustration.

Caption: Figure 3. Matrix A of the Astrophysics application.

Caption: Figure 4. Residual convergence with the three refinement schemes in the Astrophysics application.
Table 1. Relative residuals.

n    Projection method   Product integration method

10   0.0968              0.0246
20   0.0499              0.0087
50   0.0211              0.0018

Table 2. Relative residuals.

n    Projection method   Product integration method

10   0.0267              0.0172
20   0.0252              0.0145
50   0.0151              0.0075


Please Note: Illustration(s) are not available due to copyright restrictions.
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Author:Grammont, Laurence; Ahues, Mario; Kaboul, Hanane
Publication:Mathematical Modeling and Analysis
Article Type:Report
Date:Nov 1, 2016
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