Efficient Galerkin finite element methods for a time-fractional Cattaneo equation

An Chen,Lijuan Nong

doi:10.1186/s13662-020-03009-w

Abstract

In this paper, we develop two efficient fully discrete schemes for solving the time-fractional Cattaneo equation, where the fractional derivative is in the Caputo sense with order in (1, 2]. The schemes are based on the Galerkin finite element method in space and convolution quadrature in time generated by the backward Euler and the second-order backward difference methods. Error estimates are established with respect to data regularity. We further compare our schemes with the L2-1_{sigma } scheme. Numerical examples are provided to show the efficiency of the schemes.

Highlights

Let ⊂ R2 be a bounded convex domain with a boundary ∂, and T > 0 be a fixed time
We aim to develop the robust and efficient Galerkin finite element method for the fractional model (1.1) by employing the convolution quadrature method in time and derive error estimates that are expressed by problem data
We focus on the two cases: the backward Euler (BE) and the second-order backward difference methods (SBD)

Summary

Introduction

Let ⊂ R2 be a bounded convex domain with a boundary ∂ , and T > 0 be a fixed time. The time-fractional Cattaneo equation considered in this paper is described as∂tu(x, t) + κCDα0,tu(x, t) = μ u(x, t) + f (x, t), in × (0, T) (1.1)with a homogeneous Dirichlet boundary condition u(x, t) = 0 on ∂ × (0, T], and initial conditions u(x, 0) = v(x), ∂tu(x, 0) = b(x) in. Let ⊂ R2 be a bounded convex domain with a boundary ∂ , and T > 0 be a fixed time. The time-fractional Cattaneo equation considered in this paper is described as. ∂tu(x, t) + κCDα0,tu(x, t) = μ u(x, t) + f (x, t), in × (0, T) (1.1). With a homogeneous Dirichlet boundary condition u(x, t) = 0 on ∂ × (0, T], and initial conditions u(x, 0) = v(x), ∂tu(x, 0) = b(x) in. The parameter κ is some fixed positive constant related to the relaxation time and μ is a diffusion constant. The source term f and the initial conditions v and b are given functions. ∂2 ∂t2 u(·, t), where the Riemann–Liouville (R–L) integral RLD–0,νt with ν > 0 is given by RLD–0,νt u(·, t) =

Objectives

Conclusion