Abstract
In this work, a time-fractional diffusion problem with a time-space dependent diffusivity is considered. The solution of such a problem has a weak singularity at the initial time t=0. Based on the L1 scheme in time on a graded mesh and the conforming finite element method in space on a uniform mesh, the fully discrete L1 conforming finite element method (L1 FEM) of a time-fractional diffusion problem is investigated. The error analysis is based on a nonstandard discrete Gronwall inequality. The final superconvergence result shows that an optimal grading of the temporal mesh should be selected as rgeq (2-alpha )/alpha . Numerical results confirm that our analysis is sharp.
Highlights
1 Introduction During the past few decades, several physical models have been developed in the form of fractional differential equations
In order to obtain our optimal H1-norm convergence and superconvergence results given in Sect. 4, we introduce a time-dependent discrete Laplacian h(t) : V0h → V0h defined by h(t)v, w = – a(·, t)∇v, ∇w ∀v, w ∈ V0h, (7)
J., Li, H., Fang, Z., Liu, Y.: A mixed finite volume element method for time-fractional reaction-diffusion equations on triangular grids
Summary
During the past few decades, several physical models have been developed in the form of fractional differential equations. Mustapha [17] studied a semidiscrete Galerkin finite element method for time-fractional diffusion equations with time-space dependent diffusivity, and the optimal error bounds in spatial L2- and H1-norms were derived for smooth and nonsmooth initial data by using novel energy arguments. Zhang and Shi [45] proposed a fully discrete L1 mixed finite element method for time fractional diffusion equation with a smooth solution, and a novel result of the consistency error estimate with order O(h2) of the bilinear element was obtained. Zhao et al [47] presented a fully discrete L1 finite element method for multiterm time fractional diffusion equation with constant diffusivity, and a superconvergence result for H1-norm estimate was obtained. Where DαN unh, h(tn)unh and Phf n all lie in V0h are used
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