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Well-posedness and time stepping adaptivity for a class of collocation discretisations of time-fractional subdiffusion equations

Sebastian Franz, Natalia Kopteva

Abstract

Time-fractional parabolic equations with a Caputo time derivative of order $α\in(0,1)$ are discretised in time using collocation methods, which assume that the Caputo derivative of the computed solution is piecewise-polynomial. For such discretisations of any order $m\ge 0$, with any choice of collocation points, we give sufficient conditions for existence and uniqueness of collocation solutions. Furthermore, we investigate the applicability and performance of such schemes in the context of the a-posteriori error estimation and adaptive time stepping algorithms.

Well-posedness and time stepping adaptivity for a class of collocation discretisations of time-fractional subdiffusion equations

Abstract

Time-fractional parabolic equations with a Caputo time derivative of order are discretised in time using collocation methods, which assume that the Caputo derivative of the computed solution is piecewise-polynomial. For such discretisations of any order , with any choice of collocation points, we give sufficient conditions for existence and uniqueness of collocation solutions. Furthermore, we investigate the applicability and performance of such schemes in the context of the a-posteriori error estimation and adaptive time stepping algorithms.
Paper Structure (11 sections, 7 theorems, 50 equations, 9 figures)

This paper contains 11 sections, 7 theorems, 50 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Well-posedness for discontinuous collocation schemes with $\theta_0>0$
  3. Matrix representation of the collocation scheme \ref{['eq:problemRescaled']}
  4. Well-posedness by means of the Lax-Milgram Theorem
  5. Well-posedness by means of an eigenvalue test
  6. A semi-computational approach
  7. Well-posedness for collocation schemes with $\theta_0=0$
  8. Residual-type a-posteriori error estimation and adaptive time stepping
  9. Residuals of the computed solution for collocation schemes \ref{['eq:coll_def']}
  10. Computationally stable implementation
  11. Numerical results

Key Result

Theorem 2.2

Suppose that the bilinear form $a(v,w):=\langle {\mathcal{L}} v, w\rangle$ on the space $H^1_0(\Omega)$ is bounded and coercive. Additionally, suppose that there exists a diagonal matrix $D\in \mathbb{R}^{m+1,m+1}$ with strictly positive diagonal elements such that the symmetric matrix $W^{\top}D WD

Figures (9)

  • Figure 1: Eigenvalues for collocation methods using equidistributed points for $m\in\{2,3,5,8\}$ (left to right) and real parts (top), imaginary parts (bottom)
  • Figure 2: Eigenvalues for collocation methods using Gauß-Legendre points for $m\in\{2,3,5,8\}$ (left to right) and real parts (top), imaginary parts (bottom)
  • Figure 3: Eigenvalues for collocation methods using Gauß-Lobatto points for $m\in\{2,3,5,8\}$ (left to right) and real parts (top), imaginary parts (bottom)
  • Figure 4: Adaptive time stepping algorithm for Test example \ref{['Ex1']}: $TOL$ and the corresponding $L_\infty(0,T;\, L_\infty(\Omega))$ errors vs. number of time intervals $M$ for $m=4$ and various choices of collocation points, $\alpha=0.4$, residual barrier ${\mathcal{R}}_0$ with $\lambda=\pi^2$ and $\omega=\lambda/8$.
  • Figure 5: Test example \ref{['Ex1']}, $\alpha=0.4$, adaptive computed solutions, $TOL=10^{-4}$, residual barrier ${\mathcal{R}}_0$ with $\lambda=\pi^2$ and $\omega=\lambda/8$: $\partial_t^\alpha u_\tau$ (left) and $u_\tau$ (center), as well as the collocation solution from FrK23 (right).
  • ...and 4 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Theorem 2.2: FrK24b
  • Corollary 2.3: Galerkin finite element discretisation
  • Lemma 2.4: FrK24b
  • proof
  • Corollary 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Theorem 4.1: a-posteriori error estimate
  • ...and 5 more