Analysis of a fast fully discrete finite element method for fractional viscoelastic wave propagation

TL;DR

This paper develops and analyzes a fast finite element method for simulating fractional viscoelastic wave propagation, providing theoretical error estimates and numerical validation for the approach.

Contribution

It introduces a novel fast scheme using SOE approximation for fractional wave equations and establishes rigorous error bounds.

Findings

Error estimates for semi-discrete and fully discrete schemes

Numerical results confirm theoretical convergence rates

Efficient computation of fractional convolution integrals

Abstract

This paper is devoted to a numerical analysis of a fractional viscoelastic wave propagation model that generalizes the fractional Maxwell model and the fractional Zener model. First, we convert the model problem into a velocity type integro-differential equation and establish existence, uniqueness and regularity of its solution. Then we consider a conforming linear/bilinear/trilinear finite element semi-discrete scheme and a fast scheme of backward Euler full discretization with a sum-of-exponentials (SOE) approximation for the convolution integral, and derive error estimates for the semi-discrete and fully discrete schemes. Finally, we provide several numerical examples to verify the theoretical results.