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Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite-dimensional boundary conditions

Corresponding Author

Alessia Andò

[email protected]

orcid.org/0000-0002-4051-2049

CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

Correspondence Alessia Andò, CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle scienze 206, 33100 Udine, Italy.

Email: [email protected]

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Alessia Andò,

Corresponding Author

Alessia Andò

[email protected]

orcid.org/0000-0002-4051-2049

CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, Udine, Italy

Correspondence Alessia Andò, CDLab - Computational Dynamics Laboratory, Department of Mathematics, Computer Science and Physics, University of Udine, via delle scienze 206, 33100 Udine, Italy.

Email: [email protected]

Search for more papers by this author

First published: 08 September 2021

https://doi.org/10.1002/cmm4.1190

Citations: 3

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Abstract

The problem of computing periodic solutions can be expressed as a boundary value problem and solved numerically via piecewise collocation. Here, we extend to renewal equations the corresponding method for retarded functional differential equations in (K. Engelborghs et al., SIAM J Sci Comput., 22 (2001), pp. 1593–1609). The theoretical proof of the convergence of the method has been recently provided in (A. Andò and D. Breda, SIAM J Numer Anal., 58 (2020), pp. 3010–3039) for retarded functional differential equations and in (A. Andò and D. Breda, submitted in 2021) for renewal equations and consists in both cases in applying the abstract framework in (S. Maset, Numer Math., 133 (2016), pp. 525–555) to a reformulation of the boundary value problem featuring an infinite-dimensional boundary condition. We show that, in the renewal case, the proof can also be carried out and even simplified when considering the standard formulation, defined by boundary conditions of finite dimension.

1 INTRODUCTION

Renewal equations (REs), also known as Volterra integral equations (see, e.g., Reference 1 for an extensive treatment), constitute an important class of delay equations which do not involve derivatives. In other words, they specify the value of the unknown function at the present time in terms of its past values. Such equations appear in a wide range of applications, from the theory of industrial replacement to that of population dynamics, where they arise from the presence of some physiological structure (e.g., age) of the relevant population.^2-5

A general RE can be written as

x (t) = F (x_{t})

()

where

F : X \to ℝ^{d}

is an autonomous, usually nonlinear function, the state space

X

is defined as

X : = L^{1} ([- τ, 0], ℝ^{d})

⁶ for some delay

τ > 0

, and the state or history

x_{t} \in X

at time t associated to (1) is defined as

x_{t} (θ) : = x (t + θ), θ \in [- τ, 0] .

The infinite dimension of the state space represents a main reason why the dynamical analysis for REs is harder than that for ordinary differential equations. An important aspect of the dynamical analysis is the study of invariant sets; in particular, we are interested in periodic orbits. In this respect, the theory for REs has been developed only recently. In Reference 7 an extension of the Floquet theory to REs was proposed, allowing the study of local asymptotic stability of periodic solutions, while⁸ provides a theoretical analysis of convergence of the piecewise collocation method described in Reference 9 to actually compute the periodic solutions. Such analysis is based on the abstract approach in Reference 10 for general boundary value problems (BVPs).

The problem of finding a periodic solution of (1) with period

ω > 0

can be expressed as a BVP in two different but equivalent ways (see Reference 9 (sect.2) for a comparison of the two formulations in the case of Retarded Functional Differential Equation, RFDEs).⁸ considers a BVP of the form

\{\begin{array}{ll} x (t) = F (x_{t}), t \in [0, ω], \\ x_{0} = x_{ω} \\ p (x |_{[0, ω]}) = 0, \end{array}

()

whose solution represents the restriction to two periods (say

[- ω, ω]

without loss of generality) of the original solution. The boundary condition includes the periodicity condition, stating that the solution states at the extrema of the period are equal, plus an extra phase condition, defined by a scalar linear function p and necessary in order to remove translational invariance. The convergence proof in Reference 8 is based on reformulating (2) as a fixed-point problem, similarly to what was done in Reference 11 for RFDEs.

The aim of the present article is to provide a simplification of the proof of convergence given in Reference 8 by considering the alternative periodic BVP formulation, which features a finite-dimensional boundary condition, is the one which is most commonly considered in the literature on RFDEs (e.g., References 12-15) and, most importantly, leads to the same numerical method (the reader can refer to Reference 8 (sect. 4) for numerical tests to support the validity of the method). Section 2 will describe both the BVP formulation and the relevant numerical method. Section 3 will go through the steps of the convergence analysis, by stating the propositions that ensure the validity of both the theoretical assumptions (Subsection 3.1) and the numerical ones (Subsection 3.2) required in Reference 10. As it will be shown, some parts of the proofs of the latter can be notably simplified by resorting to the BVP with finite-dimensional boundary condition.

2 PIECEWISE ORTHOGONAL COLLOCATION FOR PERIODIC BVPS

The problem of finding a periodic solution of (1) with period

ω > 0

*can be expressed as a BVP whose solution represents the restriction to one period (say

[0, ω]

without loss of generality) of the original solution. The boundary condition includes part of the periodicity condition, stating that the solution values at the extrema of the period are equal. The rest of the periodicity condition is, in a certain sense, included within the relevant RE: indeed, whenever one needs to evaluate the solution x at points that fall off the interval

[0, ω]

due to the delay, one exploits the periodicity to bring back the evaluation to the domain

[0, ω]

. More rigorously, this corresponds to defining, for

t \in [0, ω]

, a periodic state

{\overline{x}}_{t} \in X

{\overline{x}}_{t} (θ) = \{\begin{array}{ll} x (t + θ), & t + θ \in [0, ω], \\ x (t + θ + ω), & t + θ \in [- ω, 0), \end{array}

for

θ \in [- τ, 0] \subset [- ω, 0]

, allowing in turn to define the periodic BVP as

\{\begin{aligned} x (t) & = F ({\overline{x}}_{t}), t \in [0, ω], \\ x (0) & = x (ω) \\ p (x) & = 0 . \end{aligned}

()

Examples of phase conditions are the trivial one, of the form

x_{k} (0) = \hat{x}

for some

k \in {1, \dots, d}

for a fixed

\hat{x}

and the integral one, of the form

\int_{0}^{ω} < x (t), {\tilde{x}}^{'} (t) > d t = 0,

where

\tilde{x}

is (an approximation of) a given reference solution.¹⁶ Since the method described here is typically used within a numerical continuation framework, such a reference solution could be, for example, the (piecewise polynomial approximation of) solution obtained at the previous continuation step.

The following description of the numerical method to solve (3) assumes, for the time being, that the function F defining the right-hand side can be exactly computed. The first step (see, e.g., Reference 9) is to reformulate (3) so that the interval of definition of the RE remains fixed and no longer depends on the unknown

ω

. This is achieved by defining the map

s_{ω} : ℝ \to ℝ

s_{ω} (t) : = \frac{t}{ω},

which allows to write (3) equivalently as

\{\begin{aligned} x (t) & = F ({\overline{x}}_{t} \circ s_{ω}), t \in [0, 1] \\ x (0) & = x (1) \\ p (x) & = 0 . \end{aligned}

()

As explained in Reference 9 and then reported in Reference 8, (4) can be solved through piecewise orthogonal collocation, which is now a standard approach (originally developed for ODEs, see MatCont¹⁷). For L and m positive integers, the numerical solution u in

[0, 1]

is a piecewise continuous polynomial obtained by solving the following system having dimension

(1 + L m) \times d + 1

\{\begin{aligned} u (t_{i, j}) & = F ({\overline{u}}_{t_{i, j}} \circ s_{w}), j \in {1, \dots, m}, i \in {1, \dots, L} \\ u (0) & = u (1) \\ p (u) & = 0 \end{aligned}

for a given mesh

0 = t_{0} < \dots < t_{L} = 1

and collocation points

t_{i - 1} < t_{i, 1} < \dots < t_{i, m} < t_{i}

for all

i \in {1, \dots, L}

. The unknowns are, other than w approximating

ω

, those of the form

u_{i, j} : = u (t_{i, j})

for

(i, j) = (1, 0)

and

i \in {1, \dots, L}

j \in {1, \dots, m}

. In the context of piecewise collocation, one typically refers to the corresponding finite element method (FEM), which consists in considering a fixed value for m while the mesh size L increases, in view of convergence. This is indeed the case for the rest of the present article.

In realistic applications, right-hand sides F usually feature an integral, therefore cannot be exactly computed in general. For instance, F can be of the form

F (φ) : = \int_{- τ}^{0} K (σ, φ (σ)) d σ

()

for some integration kernel

K : [- τ, 0] \times ℝ^{d} \to ℝ^{d}

, or

F (φ) : = f (\int_{0}^{τ} k (σ) φ (- σ) d σ),

()

for some integration kernel

k : [0, τ] \to ℝ^{d}

and some function

f : ℝ^{d} \to ℝ^{d}

. The analysis that follows focuses on (5), but one can apply the same ideas to extend it to (6). By rescaling time via the function

s_{ω}

one obtains

F (x_{t} \circ s_{ω}) = \int_{- \frac{τ}{ω}}^{0} ω K (ω θ, x_{t} (s_{ω} (ω θ)) d θ = \int_{- \frac{τ}{ω}}^{0} ω K (ω θ, x (t + θ)) d θ .

()

If the integrand can be exactly computed, as it is usually the case in applications, then (7) is approximated as

F_{M} (x_{t} \circ s_{ω}) : = ω \sum_{i = 0}^{M} w_{i} K (ω α_{i}, x (t + α_{i})),

where M is a given approximation level and the quadrature nodes

- \frac{τ}{ω} = α_{0} < \dots < α_{M} = 0

are allowed to be completely independent of the collocation nodes mentioned earlier. Indeed, such nodes and the corresponding weights

w_{0}, \dots, w_{M}

are meant to define a quadrature formula which exploits the possible irregularities in K.

3 CONVERGENCE ANALYSIS

The convergence analysis of the numerical method described in Section 2 follows the abstract approach,¹⁰ intended for neutral functional differential equations. Note that REs can be treated as equations of this kind by interpreting the relevant solutions as derivatives of other functions. This may recall the approach used in Reference 18, which involves the integration of the relevant REs.

The general fixed-point problem described in Reference 10 consists in finding

(v^{*}, β^{*}) \in 𝕍 \times 𝔹

with

v^{*} : = 𝒢 (u^{*}, α^{*})

and

(u^{*}, α^{*}, β^{*}) \in 𝕌 \times 𝔸 \times 𝔹

such that

(u^{*}, α^{*}, β^{*}) = Φ (u^{*}, α^{*}, β^{*})

for

Φ : 𝕌 \times 𝔸 \times 𝔹 \to 𝕌 \times 𝔸 \times 𝔹

given by

Φ (u, α, β) : = (\begin{matrix} ℱ (𝒢 (u, α), u, β) \\ (α, β) - ℬ (𝒢 (u, α), u, β) \end{matrix}),

()

where

ℱ : 𝕍 \times 𝕌 \times 𝔹 \to 𝕌

in the first line defines the right-hand side of the functional equation of the relevant BVP and

ℬ : 𝕍 \times 𝕌 \times 𝔹 \to 𝔸 \times 𝔹

in the second one represents the boundary conditions. The solution

v = 𝒢 (u, α)

lies in a normed space of functions

𝕍

and is obtained through the operator

𝒢 : 𝕌 \times 𝔸 \to 𝕍

which reconstructs it given a function u from a Banach space

𝕌

and an initial value/state

α

from a Banach space

𝔸

β

is a vector of possible parameters, living in a Banach space

𝔹

of finite dimension.

In the case of REs, the right-hand side prescribes the values of the solution, rather than those of its derivative. This means that no further step is required to retrieve the solution $v^{*}$ , once one has the solution $(u^{*}, α^{*}, β^{*})$ of (8). Thus, in principle, one could choose to work with $𝔸 = \emptyset$ - instead of with an infinite-dimensional $𝔸$ , as it is needed in Reference 8-and $𝒢 (u) = u$ . This is indeed the case, as it will be shown in the rest of this section.

In order to prove that the BVP (4) is amenable of the analysis in Reference 10, the first step is to observe that it can be rewritten as a fixed point problem of the form (8). The only parameter involved is the period

ω

, therefore

𝔹 = ℝ

. In order to work with the space

𝔸 = \emptyset

, the boundary conditions need to be scalar and limited to the phase condition, unlike in the formulation (2) considered in Reference 8, where infinite-dimensional boundary conditions need to be imposed. As a consequence, the periodicity condition must be moved from the BVP to the definition of the space, that is,

𝕌

can only contain functions satisfying the periodicity condition. By choosing

𝕌_{} \subseteq {u : [0, 1] \to ℝ^{d} | u (0) = u (1)}

, the solutions of (4) are exactly the pairs

(u^{*}, ω^{*}) \in 𝕌_{} \times ℝ

with

(u^{*}, ω^{*}) \in 𝕌_{} \times (0, + \infty)

the fixed point of the map

Φ_{} : 𝕌_{} \times ℝ \to 𝕌_{} \times ℝ

defined by

Φ_{} (u, ω) : = (\begin{matrix} F ({\overline{u}}_{(\cdot)} \circ s_{ω}) \\ ω - p (u) \end{matrix}) .

()

Consequently, (4) leads to an instance of (8) with

𝔸 = \emptyset

ℱ : 𝕌_{} \times ℝ \to 𝕌_{}

and

ℬ : 𝕌_{} \times ℝ \to ℝ

given respectively by

ℱ_{} (u, ω) : = F ({\overline{u}}_{(\cdot)} \circ s_{ω})

()

and

ℬ_{} (u, ω) : = p (u) .

The operator

ℬ

is linear and independent of

ω

. Provided that

𝕌

only contains function satisfying the periodicity condition, and does actually contain the sought solution of (4), its precise definition does not play a role up to this point. However, the fixed point problem (9) needs to satisfy several assumptions in order to apply the general framework in Reference 10, and the validity of such assumptions depend on both

𝕌

and the regularity of the right-hand side F, given in turn by the regularity of the integrand K in (5). Unlike the theoretical assumptions, the numerical ones also depend on the discretization of the problem. Subsections 3.1 and 3.2 contain the propositions stating the validity of the theoretical and numerical assumptions respectively. As it is eventually remarked in Subsection 3.3, the validity of both the theoretical and the numerical assumptions implies that the relevant numerical method is convergent.

3.1 Theoretical assumptions

The hypotheses on the regularity of the problem (9) that are needed to satisfy the theoretical assumptions are listed below.

B^{\infty}

denotes the set of measurable and bounded functions and it has been chosen—instead of the classical

L^{1}

space—in order to deal with collocation, which requires to evaluate the relevant functions pointwise. X denotes the state space relevant to the rescaled BVP (4). Note that, with respect to the similar hypotheses needed in Reference 8, the only difference is in (T2), concerning the specific choices of the spaces

𝕌

and

𝔸

(T1)
$X = B^{\infty} ([- τ, 0], ℝ^{d})$ , $X = B^{\infty} ([- 1, 0], ℝ^{d})$ .
(T2)
$𝕌_{} = {u \in B^{\infty} ([0, 1], ℝ^{d}) | u (0) = u (1)}$ , $𝔸_{} = \emptyset$ .
(T3)
$K : [- τ, 0] \times ℝ^{d} \to ℝ^{d}$ is continuous and has partial derivatives $D_{1} K, D_{2} K$ .
(T4)
$D_{2} K : ℝ^{d} \to ℝ^{d}$ is continuous.
(T5)
There exist $r > 0$ and $κ \geq 0$ such that
$‖ D K (ω \cdot, v_{t}) - D K (ω^{*} \cdot, v_{t}^{*}) ‖_{ℝ^{d} \leftarrow ℒ (ℝ \times ℝ^{d}, ℝ^{d})} \leq κ ‖ (v_{t}, ω) - (v_{t}^{*}, ω^{*}) ‖_{X \times ℝ}$
for every $(v_{t}, ω) \in \overline{B} ((v_{t}^{*}, ω^{*}), r)$ , uniformly with respect to $t \in [0, 1]$ .

The first theoretical assumption (A $F B$ ,¹⁰^{(p. 534)}) states that the operators $ℱ$ and $ℬ$ appearing in (8) are Fréchet-differentiable. In our case, $ℬ$ is given by a linear function, thus the assumption is a consequence of the following proposition, which can be proved by simply following the steps in the proof of Reference 8 (proposition 3.2).

Proposition 1.Under (T1), (T2), and (T3), $ℱ_{}$ in (10) is Fréchet-differentiable at every $(û, \hat{ω}) \in 𝕌 \times (0, + \infty)$ , and

D ℱ (û, \hat{ω}) (u, ω) = L (\cdot; û, \hat{ω}) [{\overline{u}}_{(\cdot)} \circ s_{\hat{ω}}] + ω M (\cdot; û, \hat{ω})

()

for

(u, ω) \in 𝕌 \times (0, + \infty)

, where, for

t \in [0, 1]

L (t; û, \hat{ω}) [u_{(\cdot)} \circ s_{\hat{ω}}] : = \hat{ω} \int_{- \frac{τ}{\hat{ω}}}^{0} D_{2} K (\hat{ω} θ, \overline{û} (t + θ)) u (t + θ) d θ

()

and

M (t; u, ω) : = \int_{- \frac{τ}{ω}}^{0} K (ω θ, \overline{u} (t + θ)) d θ - \frac{τ}{ω} K (- τ, \overline{u} (t - \frac{τ}{ω})) + ω \int_{- \frac{τ}{ω}}^{0} D_{1} K (ω θ, \overline{u} (t + θ)) θ d θ .

The second theoretical assumption (A $G$ ,¹⁰^{(p. 534)}) concerns the boundedness of the Green operator $𝒢$ appearing in (8). However, in the case of (9), such operator is not involved, as explained at the beginning of Section 3. Thus, formulation (9) allows us to skip the corresponding technical step in the convergence proof.

The third theoretical assumption (A $x^{*} 1$ ,¹⁰^{(p. 536)}) states that the Fréchet derivative of $Φ$ in (8) is locally Lipschitz continuous at its fixed points. From this point on, $(u^{*}, ω^{*})$ denotes a fixed point of $Φ$ in (9). The validity of the assumption is a consequence of the following proposition, whose proof goes as the one of Reference 8 (proposition 3.4).

Proposition 2.Under (T1), (T2), (T3), and (T5), there exist $r_{} \in (0, ω^{*})$ and $κ_{} \geq 0$ such that

‖ D Φ (u, ω) - D Φ_{} (u^{*}, ω^{*}) ‖_{𝕌_{} \times ℝ \leftarrow 𝕌 \times (0, + \infty)} \leq κ ‖ (u, ω) - (u^{*}, ω^{*}) ‖_{𝕌_{} \times ℝ}

for all

(u, ω) \in ℬ ((u^{*}, ω^{*}), r_{})

In order to prove the validity of the last theoretical assumption (A

x^{*} 2

,¹⁰^{(p. 536)}), one needs an additional requirement, other than the hypotheses stated at the beginning of this subsection. Such requirement is anyway a consequence of the hyperbolicity of the periodic solution of interest, which is in turn a standard assumption in the context of application of the principle of linearized stability (see Reference 7). In particular, if the periodic solution is hyperbolic and

L

in (12) is continuous, then the initial value problem associated to the linear homogeneous RE

x (t) = L (t; u^{*}, ω^{*}) [x_{t} \circ s_{ω^{*}}]

()

is well-posed.

Proposition 3.Let $T^{*} (t, s) : X \to X$ be the evolution operator for the linear homogeneous RE (13). Under (T1), (T2), (T3), and (T4), if 1 is a simple eigenvalue of $T_{}^{*} (1, 0)$ , then the linear bounded operator $I_{𝕌_{} \times ℝ} - D Φ (u^{*}, ω^{*})$ is invertible, that is, for all $(u_{0}, ω_{0}) \in 𝕌 \times ℝ$ there exists a unique $(u, ω) \in 𝕌 \times ℝ$ such that

\{\begin{cases} u = L^{*} [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] + ω M^{*} + u_{0} \\ p (u) = ω_{0} . \end{cases}

()

Proof.The proof is based on treating (14) as a boundary value problem for v the periodic extension of u to $[- 1, 1]$ , that is,

\{\begin{array}{l} v (t) = L^{*} (t) [v_{t} \circ s_{ω^{*}}] + ω M^{*} (t) + u_{0} (t) \\ v_{1} = v_{0} \\ p (v |_{[0, 1]}) = ω_{0} . \end{array}

()

for

t \in [0, 1]

. (15) is, in turn, an instance of the boundary value problem⁸^{((19) in proposition 4)} with

φ_{0} = 0

. By Reference 8 (proposition 4), there is a unique solution

(v, ω)

to (15), which means that

(v |_{[0, 1]}, ω) \in 𝕌 \times ℝ

is the unique solution to (14).

3.2 Numerical assumptions

The present subsection deals with the numerical assumptions required by Reference 10, that is, those concerning the discretization scheme used to reduce (9) to a finite-dimensional problem. Before stating the hypotheses needed to prove such numerical assumptions, we describe below the discretization scheme, given by both the primary and the secondary discretization. The former concerns the approximation of the Banach spaces involved, and is defined as in Reference 8 for the space $𝕌$ , except for the fact that in the present case it needs to take into account the boundary condition defining $𝕌$ in (T2). The latter, on the other hand, concerns the approximation of the right-hand side (10) whenever such an approximation is needed, such as in the case (5).

The primary discretization consists in reducing the space

𝕌

to finite-dimensional space

𝕌_{L}

given a level of discretization L. This happens by means of a restriction operator

ρ_{L}^{+} : 𝕌 \to 𝕌_{L}

and a prolongation operator

π_{L}^{+} : 𝕌_{L} \to 𝕌

, which extend respectively to

R_{L} : 𝕌 \times ℝ \to 𝕌_{L} \times ℝ, R_{L} (u, ω) : = (ρ_{L}^{+} u, ω)

and

P_{L} : 𝕌_{L} \times ℝ \to 𝕌 \times ℝ, P_{L} (u_{L}, ω) : = (π_{L}^{+} u_{L}, ω) .

All of them are linear and bounded. In the following we describe the specific choices we make in this context, based on piecewise polynomial interpolation.

Recall that

𝕌

concerns the interval

[0, 1]

. We choose the uniform outer mesh

Ω_{L}^{+} : = {t_{i}^{+} = i h : i = 0, 1, \dots, L, h = 1 / L} \subset [0, 1],

()

and inner meshes

Ω_{L, i}^{+} : = {t_{i, j}^{+} : = t_{i - 1}^{+} + c_{j} h : j = 0, \dots, m} \subset [t_{i - 1}^{+}, t_{i}^{+}], i = 1, \dots, L,

()

where

0 = c_{0} < \dots < c_{m} = 1

are given abscissae for m a positive integer and can be chosen arbitrarily for the purposes of the present proof. Correspondingly, we define

𝕌_{L} : = ℝ^{L m \times d},

()

whose elements

u_{L}

are indexed as

u_{L} : = {(u_{1, 1}, \dots, u_{1, m}, \dots, u_{L, 1} \dots, u_{L, m})}^{T}

()

with components in

ℝ^{d}

. Note that

𝕌_{L}

(as well as its elements

u_{L}

and the relevant restriction and prolongation operators) depend indeed on both L and m. However, only L is being used for indexing, in order to recall that L is in fact what determines the discretization level, since m remains constant when using FEM. Finally, we define, for

u \in 𝕌

ρ_{L}^{+} u : = {(u (t_{1, 1}^{+}), \dots, u (t_{1, m}^{+}), \dots, u (t_{L, 1}^{+}) \dots, u (t_{L, m}^{+}))}^{T} \in 𝕌_{L}

()

and, for

u_{L} \in 𝕌_{L}

π_{L}^{+} u_{L} \in 𝕌

as the unique element of the space

Π_{L, m}^{+} : = {p \in C ([0, 1], ℝ^{d}) : p |_{[t_{i - 1}^{+}, t_{i}^{+}]} \in Π_{m}, i = 1, \dots, L}

()

such that

π_{L}^{+} u_{L} (0) = u_{L, m}, π_{L}^{+} u_{L} (t_{i, j}^{+}) = u_{i, j}, j = 1, \dots, m, i = 1, \dots, L .

()

Above

Π_{m}

is the space of

ℝ^{d}

-valued polynomials having degree m and, when needed, we represent

p \in Π_{L, m}^{+}

through its pieces as

p |_{[t_{i - 1}^{+}, t_{i}^{+}]} (t) = \sum_{j = 0}^{m} ℓ_{m, i, j} (t) p (t_{i, j}^{+}), t \in [0, 1],

()

where, for ease of notation, we implicitly set

t_{i, 0}^{+} : = t_{i - 1}^{+}, i = 1, \dots, L,

and

{ℓ_{m, i, 0}, ℓ_{m, i, 1}, \dots, ℓ_{m, i, m}}

is the Lagrange basis relevant to the nodes

{t_{i, 0}^{+}} \cup Ω_{L, i}^{+}

. Observe that the latter is invariant with respect to i as long as we fix the abscissae

c_{j}

j = 1, \dots, m

, defining the inner meshes (17). Indeed, for every

i = 1, \dots, L

ℓ_{m, i, j} (t) = ℓ_{m, j} (\frac{t - t_{i - 1}^{+}}{h}), t \in [t_{i - 1}^{+}, t_{i}^{+}],

where

{ℓ_{m, 0}, ℓ_{m, 1}, \dots, ℓ_{m, m}}

is the Lagrange basis in

[0, 1]

relevant to the abscissae

c_{0}, c_{1}, \dots, c_{m}

with

c_{0} : = 0

The secondary discretization is defined as in Reference 8 and reported below for convenience. Such discretization is only needed whenever

ℱ

in the first of (8) cannot be computed exactly. In this case, it is approximated with an operator

ℱ_{M}

that can be computed, for a given level of discretization M. In our case, given by (5), the secondary discretization basically consists in performing a quadrature. Thus, we define

ℱ_{M} (u, ω) = F_{M} ({\overline{u}}_{(\cdot)} \circ s_{ω}) : = ω \sum_{i = 0}^{M} w_{i} K (ω α_{i}, {\overline{u}}_{α_{i}}),

()

where

- \frac{τ}{ω} = α_{0} < \dots < α_{M} = 0

, as done in Reference 8. The operator

ℬ

in the second of (8) is not subject to a secondary discretization, since in our case it is entirely defined by p in the second of (9), which can typically be evaluated exactly in

π_{L}^{+} 𝕌_{L}

. Thus, by setting

Φ_{M} (u, ω) : = (\begin{matrix} F_{M} ({\overline{u}}_{(\cdot)} \circ s_{ω}) \\ ω - p (u) \end{matrix}),

()

the discrete version

Φ_{L, M} : = R_{L} Φ_{M} P_{L} : 𝕌_{L} \times ℝ \to 𝕌_{L} \times ℝ

of the fixed point operator

Φ

in (9) is defined as

Φ_{L, M} (u_{L}, ω) : = (\begin{matrix} ρ_{L}^{+} F_{M} ({\overline{π_{L}^{+} u_{L}}}_{(\cdot)} \circ s_{ω}) \\ ω - p (π_{L}^{+} u_{L}) \end{matrix}) .

We are now able to state the further hypotheses on the regularity of the right-hand side and on the chosen discretization scheme which are needed to prove the validity of the numerical assumptions.

(N1)
The primary discretization of the space $𝕌$ is based on the choices (16)–(22).
(N2)
The nodes $α_{0}, \dots, α_{M}$ , together with the weights $w_{0}, \dots, w_{M}$ chosen for the secondary discretization as in (24) define an interpolatory quadrature formula which is convergent in $B^{\infty} ([0, 1], ℝ^{d})$ .

The validity of the first numerical assumption (A $F_{K} B_{K}$ ,¹⁰^{(p. 535)}) can be proven as done in Reference 8, thanks to the following proposition.

Proposition 4.Under (T1), (T2), and (T3) $ℱ_{M}$ is Fréchet-differentiable, from the right with respect to $ω$ , at every point $(û, \hat{ω}) \in 𝕌 \times (0, + \infty)$ and

D ℱ_{M} (û, \hat{ω}) (u, ω) = L_{M} (\cdot; û, \hat{ω}) [{\overline{u}}_{(\cdot)} \circ s_{\hat{ω}}] + ω M_{M} (\cdot; û, \hat{ω})

for

(u, ω) \in 𝕌 \times (0, + \infty)

, where, for

t \in [0, 1]

L_{M} (t; û, \hat{ω}) [u_{(\cdot)} \circ s_{\hat{ω}}] : = \hat{ω} \sum_{i = 0}^{M} w_{i} D_{2} K (\hat{ω} α_{i}, \overline{û} (t + α_{i})) u (t + α_{i})

and

M_{M} (t; u, ω) : = \sum_{i = 0}^{M} w_{i} K (ω α_{i}, \overline{u} (t + α_{i})) - \frac{τ}{ω} K (- τ, \overline{u} (t - \frac{τ}{ω})) + ω \sum_{i = 0}^{M} w_{i} D_{1} K (ω α_{i}, \overline{u} (t + α_{i})) α_{i} .

For the sequel it is useful to define

Ψ, Ψ_{L, M} : 𝕌 \times ℝ \to 𝕌 \times ℝ

Ψ : = I_{𝕌 \times ℝ} - Φ, Ψ_{L, M} : = I_{𝕌 \times ℝ} - P_{L} R_{L} Φ_{M} .

()

Indeed, the second numerical assumption (CS1,¹⁰^{(p. 536)}) concerns the Lipschitz continuity of

Ψ_{L, M}

in (26), and is a consequence of the following proposition, which in turn can be proved as Reference 8 (proposition 3.8).

Proposition 5.Under (T1), (T2), (T3), (T5), and (N1), there exist $r_{1} \in (0, ω^{*})$ and $κ \geq 0$ such that

‖ D Ψ_{L, M} (u, ω) - D Ψ_{L, M} (u^{*}, ω^{*}) ‖_{𝕌 \times ℝ \leftarrow 𝕌 \times (0, + \infty)} \leq κ ‖ (u, ω) - (u^{*}, ω^{*}) ‖_{𝕌 \times ℝ}

for all

(u, ω) \in \overline{B} ((u^{*}, ω^{*}), r_{1})

and for all positive integers L and M.

Finally, the last numerical assumption (CS2¹⁰^{(p. 537)}) ensures in particular that the chosen discretization scheme is both stable and consistent. The stability part is given by the next proposition, which can be proved much more easily than in Reference 8 due to the possibility to apply Banach's perturbation lemma more directly. This represents, in particular, the greatest simplification in the proof of the convergence analysis with respect to that presented in Reference 8. The proof of such proposition makes use of some standard results on the primary discretization, stated below as a lemma for convenience.

Lemma 1.Under (N1), let $ρ_{L}^{+}$ , $π_{L}^{+}$ be defined respectively in (20), (22) and let $Λ_{m}$ be the Lebesgue constant corresponding to the nodes used for discretization in a single mesh interval. Then, under (T2),

‖ π_{L}^{+} ρ_{L}^{+} ‖_{𝕌 \leftarrow 𝕌} \leq Λ_{m}

()

holds for all positive integers L and

\lim_{L \to \infty} ‖ π_{L}^{+} ρ_{L}^{+} u - u ‖_{𝕌} = 0

()

holds for all

u \in C ([0, 1], ℝ^{d})

Proof.By (23),

π_{L}^{+} ρ_{L}^{+} u (t) = \sum_{j = 0}^{m} ℓ_{m, i, j} (t) u (t_{i, j}^{+})

holds for

u \in 𝕌

and

t \in [t_{i - 1}^{+}, t_{i}^{+}]

i = 1, \dots, L

. Then (27) follows from

‖ π_{L}^{+} ρ_{L}^{+} u ‖_{𝕌} \leq \max_{i = 1, \dots, L} \max_{t \in [t_{i - 1}^{+}, t_{i}^{+}]} \sum_{j = 0}^{m} | ℓ_{m, i, j} (t) | ‖ u ‖_{𝕌} = Λ_{m} ‖ u ‖_{𝕌} .

As for (28),

\begin{align} π_{L}^{+} ρ_{L}^{+} u (t) - u (t) & = \sum_{j = 0}^{m} ℓ_{m, i, j} (t) u (t_{i, j}^{+}) - \sum_{j = 0}^{m} ℓ_{m, i, j} (t) u (t) + \sum_{j = 0}^{m} ℓ_{m, i, j} (t) u (t) - u (t) \\ = \sum_{j = 0}^{m} ℓ_{m, i, j} (t) [u (t_{i, j}^{+}) - u (t)] + (\sum_{j = 0}^{m} ℓ_{m, i, j} (t) - 1) u (t) \\ = \sum_{j = 0}^{m} ℓ_{m, i, j} (t) [u (t_{i, j}^{+}) - u (t)] \end{align}

holds always for

t \in [t_{i - 1}^{+}, t_{i}^{+}]

i = 1, \dots, L

. Therefore

‖ π_{L}^{+} ρ_{L}^{+} u - u ‖_{𝕌} \leq Λ_{m} ω (u; h)

, where

ω

denotes the modulus of continuity. The latter vanishes as

h \to 0

only if u is at least continuous.

Proposition 6.Under (T1), (T2), (T4), (N1), and (N2), $D Ψ_{L, M} (u^{*}, ω^{*})$ is invertible and its inverse is uniformly bounded with respect to both L and M.

Proof.The result follows from the Banach's perturbation lemma, once we show that

\lim_{L, M \to \infty} ‖ D Ψ_{L, M} (u^{*}, ω^{*}) - D Ψ (u^{*}, ω^{*}) ‖_{𝕌 \times ℝ \leftarrow 𝕌 \times (0, + \infty)} = 0 .

Since the phase condition is not affected by the discretization scheme, we are left to prove that

\lim_{L, M \to \infty} ‖ π_{L}^{+} ρ_{L}^{+} L_{M}^{*} [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] - L^{*} [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] ‖_{𝕌} = 0

()

for all

u \in 𝕌

, and that

\lim_{L, M \to \infty} ‖ π_{L}^{+} ρ_{L}^{+} M_{M}^{*} - M^{*} ‖_{𝕌} = 0 .

()

As for (29), one gets

‖ (π_{L}^{+} ρ_{L}^{+} L_{M}^{*} - L^{*}) [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] ‖_{𝕌} \leq ‖ π_{L}^{+} ρ_{L}^{+} (L_{M}^{*} - L^{*,}) [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] ‖_{𝕌} + ‖ (π_{L}^{+} ρ_{L}^{+} - I_{𝕌}) L^{*} [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] ‖_{𝕌} .

The first addend in the right-hand side above vanishes thanks to (N2), (27) in Lemma 1 as well as standard results such as Reference 19 (corollary of theorem Ia). The second addend vanishes as well thanks to (28) of Lemma 1, given that

L^{*} [{\overline{u}}_{(\cdot)} \circ s_{ω^{*}}] \in C ([0, 1], ℝ^{d})

for all

u \in 𝕌

, as it follows through the definition of

L^{*}

in (12) under (T3). Similar arguments hold for (30).

Eventually, to prove the validity of CS2, one can prove the following proposition, similarly to Reference 8 (proposition 3.9).

Proposition 7.Under (T1), (T2), (T4), (N1), and (N2),

\lim_{L, M \to \infty} \frac{1}{r_{2} (L, M)} ‖ {[D Ψ_{L, M} (u^{*}, ω^{*})]}^{- 1} ‖_{𝕌 \times ℝ \leftarrow 𝕌 \times ℝ} \cdot ‖ Ψ_{L, M} (u^{*}, ω^{*}) ‖_{𝕌 \times ℝ} = 0,

where

r_{2} (L, M) : = \min \{r_{1}, \frac{1}{2 κ ‖ {[D Ψ_{L, M} (u^{*}, ω^{*})]}^{- 1} ‖_{𝕌 \times ℝ \leftarrow 𝕌 \times ℝ}}\}

with

r_{1}

and

κ

as in Proposition 5.

3.3 Convergence results

As anticipated, the results of the present section on the theoretical and numerical assumptions imply that the FEM for the fixed point problem (9) converges, according to Reference 10. Note that such method corresponds exactly to that analyzed in Reference 8, despite the formulation of the fixed point problem being formally different. Theorem 3.1 therein can be reformulated for (9) as follows.

Theorem 1.10 (theorem 2, p. 539]) Under (T1), (T2), (T4), (N1), and (N2), there exists a positive integer $\hat{N}$ such that, for all $L, M \geq \hat{N}$ , the operator $R_{L} Φ_{M} P_{L}$ has a fixed point $(u_{L, M}^{*}, ω_{L, M}^{*})$ and

‖ (u_{L, M}^{*}, ω_{L, M}^{*}) - (u^{*}, ω^{*}) ‖_{𝕌 \times ℝ} \leq 2 ‖ {[D Ψ_{L, M} (u^{*}, ω^{*})]}^{- 1} ‖_{𝕌 \times ℝ \leftarrow 𝕌 \times ℝ} \cdot ‖ Ψ_{L, M} (u^{*} ω^{*}) ‖_{𝕌 \times ℝ} .

The consequent results on the order of convergence in Reference 8 (subsection 3.3) remain unchanged. Concerning the primary discretization error

ε_{L} : = ‖ (I_{𝕌 \times ℝ} - P_{L} R_{L}) (u^{*}, ω^{*}) ‖_{𝕌 \times ℝ}

the following holds.

Theorem 2. 8 (theorem 3.11) Let $K \in 𝒞^{p} ([- τ, 0] \times ℝ^{d}, ℝ^{d})$ for some integer $p \geq 0$ . Then, Under (T1), (T2), (T4), (N1), and (N2), it holds that $u^{*} \in C^{p + 1} ([0, 1], ℝ^{d})$ , $ψ^{*} \in C^{p + 1} ([- 1, 0], ℝ^{d})$ , $v^{*} \in C^{p + 1} ([- 1, 1], ℝ^{d})$ and

ε_{L} = O (h^{\min {m, p}}) .

Observe that $ϵ_{L}$ determines entirely the final order of convergence provided that one chooses a suitable quadrature formula, as explained in Reference 8 (subsection 3.3). Thus, the final order m of convergence is guaranteed if K is sufficiently smooth.

Remark 1.As stated in Reference 8, the entire convergence analysis can as well be carried out for right-hand sides of the form (6). In this case, the different theoretical and numerical assumptions read

(T3)
$k \in 𝒞 (ℝ, ℝ^{d})$ .
(T4)
$f \in 𝒞^{1} (ℝ^{d}, ℝ^{d})$ .
(T5)
There exist $r > 0$ and $κ \geq 0$ such that
$\begin{align} ‖ f^{'} (ω \int_{0}^{\frac{τ}{ω}} k (θ) v (t - θ) d θ) & - f^{'} (ω^{*} \int_{0}^{\frac{τ}{ω^{*}}} k (θ) v^{*} (t - θ) d θ) ‖_{ℝ^{d}} \leq κ {||(v_{t}, ω) - (v_{t}^{*}, ω^{*})||}_{X \times ℝ} \end{align}$
for every $(v_{t}, ω) \in \overline{B} ((v_{t}^{*}, ω^{*}), r)$ , uniformly with respect to $t \in [0, 1]$ .

Moreover, the above can be easily further generalized to the case

F (ψ) = f (\int_{0}^{τ_{1}} k_{1} (σ) ψ (- σ) d σ, \dots, \int_{0}^{τ_{n}} k_{n} (σ) ψ (- σ) d σ) .

4 CONCLUSIONS

The recent work⁸ provided the first complete theoretical proof of the convergence of the piecewise collocation method to compute periodic solutions of REs. Such proof is based on formulating the problem as a BVP of the form (2), in view of applying the abstract approach in Reference 10 for general BVPs. The aim of the present article is to describe how the proof can be shortened and simplified when one applies the approach in Reference 10 to the BVP (3), which is equivalent to (2) while being defined by a finite-dimensional boundary condition. In particular, as it turns out, the proof of the stability of the method can be notably simplified.

As observed in Reference 11, the periodic BVP corresponding to (3) for RFDEs is not amenable of the analysis in Reference 10. However, one could in principle try to further reformulate the problem as a different—albeit equivalent—BVP featuring a finite-dimensional boundary condition (e.g., as the one considered in Reference 20). The considerations above suggest that doing so might be worth the effort, especially in view of extending the convergence analysis to classes of more complex equations, for example, neutral equations or differential equations defined by non-constant delays.

ACKNOWLEDGMENT

We thank Dimitri Breda (University of Udine) for his careful review of the manuscript and valuable advice.

CONFLICT OF INTEREST

The authors declare no potential conflict of interests.

Biography

Alessia Andò obtained her MSc in Mathematics at University of Udine (Italy) in 2014. After that, she obtained a Master in High Performance Computing, organized by SISSA and ICTP (Trieste, Italy) in 2016. She completed her PhD in Computer Science, Mathematics and Physics at University of Udine in 2020. She is currently a postdoctoral researcher at its Department of Mathematics, Computer Science and Physics, where she has also been a member of the Computational Dynamics Laboratory (CDLab) since 2017.

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* One can always consider $ω \geq τ$ without loss of generality, since a solution with period $ω$ is also a solution with period $k ω$ for any positive integer k.

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Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite-dimensional boundary conditions

Abstract

1 INTRODUCTION

2 PIECEWISE ORTHOGONAL COLLOCATION FOR PERIODIC BVPS

3 CONVERGENCE ANALYSIS

3.1 Theoretical assumptions

3.2 Numerical assumptions

3.3 Convergence results

4 CONCLUSIONS

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biography

REFERENCES

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Convergence of collocation methods for solving periodic boundary value problems for renewal equations defined through finite-dimensional boundary conditions

Abstract

1 INTRODUCTION

2 PIECEWISE ORTHOGONAL COLLOCATION FOR PERIODIC BVPS

3 CONVERGENCE ANALYSIS

3.1 Theoretical assumptions

3.2 Numerical assumptions

3.3 Convergence results

4 CONCLUSIONS

ACKNOWLEDGMENT

CONFLICT OF INTEREST

Biography

REFERENCES

Citing Literature

References

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