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Spectral element methods for boundary-value problems of functional differential equations

Alessia andò, Jan Sieber

Abstract

We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution approximates the true solution with accuracy of order $\mathrm{e}^{-ηm}$ for some $η>0$ and increasing degree $m$ of the polynomials for a case that is common in applications: differential equations where the right-hand side depends on a finite number of delayed arguments with parametric delays and real analytic coefficients. For state-dependent delays the spectral element method also converges under mild regularity assumptions, but the geometric convergence of the collocation solution depends on the properties of the true solution, which may in general not be real analytic even for analytic coefficients. However, in those cases the convergence rate is still higher than all finite orders.

Spectral element methods for boundary-value problems of functional differential equations

Abstract

We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution approximates the true solution with accuracy of order for some and increasing degree of the polynomials for a case that is common in applications: differential equations where the right-hand side depends on a finite number of delayed arguments with parametric delays and real analytic coefficients. For state-dependent delays the spectral element method also converges under mild regularity assumptions, but the geometric convergence of the collocation solution depends on the properties of the true solution, which may in general not be real analytic even for analytic coefficients. However, in those cases the convergence rate is still higher than all finite orders.

Paper Structure

This paper contains 20 sections, 10 theorems, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Periodic BVPs for FDEs
  3. Collocation discretization
  4. Existing convergence results
  5. Convergence of SEM strategy
  6. Main results
  7. Periodic BVPs for FDEs and mild differentiability
  8. Convergence of solution of discretized BVP
  9. Fixed-point problem equivalent to periodic BVP and relevant notation
  10. Infinite-dimensional fixed point problem
  11. Discretized fixed-point problem
  12. Numerical representation of the piecewise polynomials
  13. Regularity of right-hand sides and solutions of BVP
  14. Accuracy of interpolation for analytic functions
  15. Analyticity of periodic orbits in DDEs
  16. ...and 5 more sections

Key Result

Proposition 2.4

If the nodes $(t_{\mathrm{c},j})_{j=1}^m$ have a slowly diverging Lebesgue constant $\Lambda_m$ according to Assumption ass:lebesgue and $v$ is analytic on a complex neighborhood of $[0,1]$, then there exist constants $\eta>0$ and $C_\mathrm{sp}>0$ such that where $v^m$ is the unique polynomial of degree $m-1$ satisfying $v^m(t_{\mathrm{c},j})=v(t_{\mathrm{c},j})$ for all $j=1,\ldots,m$.

Figures (2)

  • Figure 7.1: Left: periodic solution computed with $L=11$ and $m=4$, rescaled to period $1$. Right: residual $\mathop{\mathrm{Err}}\nolimits(y,T,\tau)$ for \ref{['mackeyglass']}, computed at $10001$ equidistant points in $[0,1]$ for different values of $m$ and $L$, in linear-log scale.
  • Figure 7.2: Left: Periodic solution of BVP defined by \ref{['eq:quadratic']} at $\tau = 0.95$ (blue) and $\tau = 1.1$ (orange) computed with $L=12$ and $m=5$, rescaled to period $1$. Middle: $\mathop{\mathrm{Err}}\nolimits(y,T,\tau)$ computed at $10001$ equidistant points in $[0,1]$ for $L=10$ (solid) and $L=20$ (dashed), in linear-log scale. Right: first iterates of the circle map \ref{['circle_map']} modulo $[0,1]$. The five fixed points of the fifth iterate for $\tau=0.95$ are shown in magenta.

Theorems & Definitions (17)

  • Definition 2.1: Mild differentiability and extended local Lipschitz continuity
  • Proposition 2.4: Exponential accuracy of interpolation
  • Theorem 2.5: Convergence of discretization
  • Corollary 3.1: Convergence of interpolation projection with finite mesh
  • Lemma 4.1: Mild differentiability with time shift
  • Corollary 4.4: Smoothness of solution of fixed-point problem \ref{['Phi1']}
  • Definition 4.6: Extended local Lipschitz condition, Definition \ref{['def:mild']} \ref{['def:extlip:eq']}, formulated for periodic functions
  • Lemma 6.1: Bounds on deviations of $g$
  • proof
  • Lemma 6.2
  • ...and 7 more