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

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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 ≥ 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.

Keywords

tempered fractional derivative / discontinuous Galerkin method / graded mesh / stability / error estimates

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