Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

doi:10.1007/s42967-020-00081-7

Abstract

Abstract: The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L²-convergence of the scheme are proved for all variable-order α(t) ∈ (0, 1). The proposed method is of accuracy-order O(τ³ + h^k+1), where τ, h, and k are the temporal step size, the spatial step size, and the degree of piecewise P^k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

Key words: Variable-order derivative, Discontinuous Galerkin method, Stability, Error estimates

CLC Number:

65M06

Leilei Wei, Shuying Zhai, Xindong Zhang. Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 429-444.

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