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

TL;DR

This paper introduces a fast, memory-efficient numerical scheme for fractional viscoelastic wave models using Laplace transform, sum-of-exponentials approximation, mixed finite elements, and the Newmark scheme, significantly reducing computational costs.

Contribution

It develops a novel fast algorithm that combines Laplace transform, SOE approximation, and mixed finite elements to efficiently solve fractional wave equations with reduced memory and computation complexity.

Findings

The scheme reduces memory complexity from O(N_s N) to O(N_s N_{exp}).

The scheme reduces computation complexity from O(N_s N^2) to O(N_s N_{exp} N).

Numerical experiments confirm the theoretical efficiency and accuracy.

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.