Unconditional stability and convergence of a discontinuous Galerkin method on graded meshes for the tempered fractional advection-diffusion equation with initial weak singularity

Huidi Jia; Lijie Liu; Huidi Jia; Lijie Liu

doi:10.3934/nhm.2026088

Networks and Heterogeneous Media

2026, Volume 21, Issue 6: 2368-2382. doi: 10.3934/nhm.2026088

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Research article

Unconditional stability and convergence of a discontinuous Galerkin method on graded meshes for the tempered fractional advection-diffusion equation with initial weak singularity

Huidi Jia ^{1
,
,},
Lijie Liu ²

1.
School of General Education, Zhengzhou Business University, Zhengzhou 450046, China
2.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China

Received: 29 August 2026 Revised: 17 October 2026 Accepted: 28 October 2026 Published: 04 November 2026

This work develops and examines a fully discrete local discontinuous Galerkin (LDG) approach on graded temporal partitions for the tempered fractional advection-diffusion problem. The tempered fractional derivative is approximated via an L1-type formula over a nonuniform time mesh, which efficiently resolves the weak singularity near the initial instant. In space, an LDG discretization with purely alternating fluxes is employed, and its stability is established through an energy estimate. It is demonstrated that the resulting scheme is unconditionally stable and converges with order $ O(h^{k+1}+N^{-\min\{2-\alpha, r\alpha\}}) $, where $ h $ denotes the spatial step, $ N $ is the number of time intervals, $ k $ stands for the polynomial degree, and $ r\ge 1 $ is the grading parameter. Several numerical experiments are performed to validate the theoretical findings and to illustrate the benefit of graded meshes over uniform ones.
- tempered fractional derivative,
- discontinuous Galerkin method,
- graded mesh,
- stability,
- error estimates
Citation: Huidi Jia, Lijie Liu. Unconditional stability and convergence of a discontinuous Galerkin method on graded meshes for the tempered fractional advection-diffusion equation with initial weak singularity[J]. Networks and Heterogeneous Media, 2026, 21(6): 2368-2382. doi: 10.3934/nhm.2026088

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Abstract

This work develops and examines a fully discrete local discontinuous Galerkin (LDG) approach on graded temporal partitions for the tempered fractional advection-diffusion problem. The tempered fractional derivative is approximated via an L1-type formula over a nonuniform time mesh, which efficiently resolves the weak singularity near the initial instant. In space, an LDG discretization with purely alternating fluxes is employed, and its stability is established through an energy estimate. It is demonstrated that the resulting scheme is unconditionally stable and converges with order $ O(h^{k+1}+N^{-\min\{2-\alpha, r\alpha\}}) $, where $ h $ denotes the spatial step, $ N $ is the number of time intervals, $ k $ stands for the polynomial degree, and $ r\ge 1 $ is the grading parameter. Several numerical experiments are performed to validate the theoretical findings and to illustrate the benefit of graded meshes over uniform ones.

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