Convergence analysis of a Petrov-Galerkin method for fractional wave problems with nonsmooth data

TL;DR

This work analyzes a time fractional wave equation with 1<\alpha<2 using a Petrov–Galerkin discretization in time and space and develops a robust weak-solution framework via variational methods and the transposition technique. It proves existence, uniqueness, and regularity of the weak solution, then derives optimal convergence estimates for the discrete method under nonsmooth data, supported by numerical experiments in one dimension that confirm the theoretical rates. The convergence results quantify how temporal and spatial discretizations interact with fractional regularity, showing optimal behavior for 1<\alpha\leq 3/2 and near-optimal behavior for 3/2<\alpha<2, with practical guidance on grid choices. Overall, the paper provides a rigorous foundation for reliable numerical approximation of fractional wave problems with low-regularity data and demonstrates the method’s effectiveness in validating the theory.

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.

Figures (4)

• Figure 1: Spatial errors of Example 1, $\tau=2^{-17}$.
• Figure 2: Temporal errors of Example 1, $h=2^{-10}$.
• Figure 3: Spatial errors of Example 2, $\tau=2^{-17}$.
• Figure 4: Temporal errors of Example 2, $h=2^{-10}$.

Theorems & Definitions (64)

• Lemma 2.1
• Remark 2.1
• Lemma 2.2
• proof
• Definition 3.1
• Lemma 3.1: Samko1993
• Lemma 3.2: Kilbas2006
• Lemma 3.3
• proof
• Remark 3.1