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A fast fully discrete mixed finite element scheme for fractional viscoelastic models of wave propagation

Hao Yuan, Xiaoping Xie

Abstract

Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time fractional derivative, our fast scheme reduces the memory complexity from $\mathcal O(N_sN) $ to $\mathcal O(N_sN_{exp})$ and the computation complexity from $\mathcal O(N_sN^2)$ to $\mathcal O(N_sN_{exp}N)$, where $N$ denotes the total number of temporal grid points, $N_{exp}$ the number of exponentials in SOE, and $N_s$ the complexity of memory and computation related to the spatial discretization. Numerical experiments confirm the theoretical results.

A fast fully discrete mixed finite element scheme for fractional viscoelastic models of wave propagation

Abstract

Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time fractional derivative, our fast scheme reduces the memory complexity from to and the computation complexity from to , where denotes the total number of temporal grid points, the number of exponentials in SOE, and the complexity of memory and computation related to the spatial discretization. Numerical experiments confirm the theoretical results.
Paper Structure (11 sections, 5 theorems, 92 equations, 3 figures, 5 tables, 2 algorithms)

This paper contains 11 sections, 5 theorems, 92 equations, 3 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Alternative form of the constitutive law and weak formulations
  4. Efficient SOE approximation of Mittag-Leffler function
  5. Gaussian quadrature approximation
  6. SOE approximation of $E_{\alpha}(-t^{\alpha})$
  7. Numerical schemes for the fractional viscoelastic model
  8. L1-Newmark mixed finite element scheme
  9. Fast numerical scheme with SOE approximation
  10. Numerical results
  11. Conclusions

Key Result

Lemma 2.1

(1) For $\alpha$, $\beta>0$ and $z\in\mathbb{C}$, there holds (2) For $\alpha$, $\beta>0$, $\lambda\geq0$ and $t>0$, there holds

Figures (3)

  • Figure 1: Results of SOE approximation error $R_{soe}(t)$ for $E_{\alpha}(-t^{\alpha})$.
  • Figure 2: Nodal degrees of freedom for Hu-Man-Zhang's element
  • Figure 3: A comparison of wall time and memory cost between the two algorithms for different time steps when $h=\frac{1}{8}$

Theorems & Definitions (16)

  • Lemma 2.1
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 6 more