Superconvergence analysis of interior penalty discontinuous Galerkin method for a class of time-fractional diffusion problems

TL;DR

This paper analyzes the superconvergence properties of a non-symmetric interior penalty Galerkin method combined with an L1-scheme for solving non-autonomous time-fractional PDEs, demonstrating stability and accuracy through theoretical and numerical results.

Contribution

It introduces a superconvergence analysis for the NIPG method applied to time-fractional PDEs, including linear and semilinear cases, with verified numerical experiments.

Findings

Discretely stable numerical scheme

Superconvergence of error estimates achieved

Effective linearization for semilinear problems

Abstract

In this study, we consider a class of non-autonomous time-fractional partial advection-diffusion-reaction (TF-ADR) equations with Caputo type fractional derivative. To obtain the numerical solution of the model problem, we apply the non-symmetric interior penalty Galerkin (NIPG) method in space on a uniform mesh and the L1-scheme in time on a graded mesh. It is demonstrated that the computed solution is discretely stable. Superconvergence of error estimates for the proposed method are obtained using the discrete energy-norm. Also, we have applied the proposed method to solve semilinear problems after linearizing by the Newton linearization process. The theoretical results are verified through numerical experiments.