Convergence Analysis of a Petrov–Galerkin Method for Fractional Wave Problems with Nonsmooth Data

Abstract

This paper analyzes the convergence of a Petrov-Galerkin method for time fractional wave problems with nonsmooth data. Well-posedness and regularity of the weak solution to the time fractional wave problem are firstly established. Then an optimal convergence analysis with nonsmooth data is derived. Moreover, several numerical experiments are presented to validate the theoretical results.

Discontinuous Galerkin method for a distributed optimal control problem governed by a time fractional diffusion equation

“…which proves (20). Furthermore, (21) follows from ( 24) and (26), and (22) follows from (25) and (27). This completes the proof.…”

Analysis of the L1 scheme for fractional wave equations with nonsmooth data

“…Using the convolution quadratures in [22] and the techniques in [23], Jin et al [13] developed first-and second-order time-stepping methods for fractional wave equations, and derived optimal error estimates with nonsmooth initial data. In [24], Luo et al derived the convergence in the H 1 0 (Ω)-norm with nonsmooth source term for a loworder Petrov-Galerkin method. We note that, as pointed out in Remark 3.1, the low-order Petrov-Galerkin method in [24] is identical to the L1 scheme.…”

“…In [24], Luo et al derived the convergence in the H 1 0 (Ω)-norm with nonsmooth source term for a loworder Petrov-Galerkin method. We note that, as pointed out in Remark 3.1, the low-order Petrov-Galerkin method in [24] is identical to the L1 scheme.…”

A symmetric fractional-order reduction method for direct nonuniform approximations of semilinear diffusion-wave equations

“…The convolution quadrature methods generated by backward difference formulas are rigorously discussed in [10], where the first-and second-order temporal convergence rates are obtained under proper assumptions of the given data, and their discrete maximal regularities are further studied by Jin, Li and Zhou [11]. Lately, for the problem with nonsmooth data, a Petrov-Galerkin method and a time-stepping discontinuous Galerkin method are proposed in [22] (Luo, Li and Xie) and [14] (Li, Wang and Xie), where the temporal convergence rate is (3 − α)/2-order and about first-order respectively. Numerical schemes with classical L1 approximation in time and the standard P1-element in space are also implemented in [13] to have the temporal accuracies of O(τ 3−α ) and O(τ 2 ) provided the ratio τ α /h 2 min is uniformly bounded.…”

“…Numerical schemes with classical L1 approximation in time and the standard P1-element in space are also implemented in [13] to have the temporal accuracies of O(τ 3−α ) and O(τ 2 ) provided the ratio τ α /h 2 min is uniformly bounded. We note that the numerical methods in the above works [10,11,13,14,22] are implemented on uniform temporal steps. On the other hand, Mustapha & McLean [29] and Mustapha & Schötzau [30] considered the time-stepping discontinuous Galerkin methods on nonuniform temporal meshes to solve the following kind of fractional wave equation: u t + I β Au(t) = f (t), for β ∈ (0, 1) and t ∈ (0, T ], (1.2) where A is a self-adjoint linear elliptic spatial operator.…”