Superconvergence of a Finite Element Method for the Multi-term Time-Fractional Diffusion Problem

This paper analyzes a class of two-dimensional (2-D) time fractional reaction-subdiffusion equations with variable coefficients. The high-order L2-1 𝜎 time-stepping scheme on graded meshes is presented to deal with the weak singularity at the initial time t = 0, and the bilinear finite element method (FEM) on anisotropic meshes is used for spatial discretization. Using the modified discrete fractional Grönwall inequality, and combining the interpolation operator and the projection operator, the L 2 -norm error estimation and H 1 -norm superclose results are rigorously proved. The superconvergence result in the H 1 -norm is derived by applying the interpolation postprocessing technique. Finally, numerical examples are presented to verify the validation of our theoretical analysis.

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“…Theorem 5.1 Suppose 𝜎 = 1−𝛼/ 2. Let {U n } N n=0 be the solutions of the fully discrete scheme (8) on graded meshes and u be the solution of (1)…”

Section: Convergence Superclose and Superconvergence Analysismentioning

confidence: 99%

“…The above convergence results of [17,24] were all derived in the maximum L 2 -norm. The maximum H 1 -norm convergence and superconvergence results of the FEMs were given in [7,8], respectively. In [1], Alikhanov constructed the L2-1 𝜎 time-stepping scheme by piecewise approximation of the fractional derivative.…”

mentioning

confidence: 99%

On the convergence and superconvergence for a class of two‐dimensional time fractional reaction‐subdiffusion equations

Wei

Zhao

Chen

et al. 2022

Numerical Methods Partial

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“…For example, Ref. [33] provided L ∞ (H 1 ) error estimates and superconvergence results for the multi-term time-fractional diffusion problem utilizing the L1-FEM on graded meshes. Moreover, by combining the L2-1 σ scheme on graded meshes and the nonconforming Wilson FEM, the superconvergence analysis of the time-fractional diffusion equation was demonstrated in [34].…”

Section: Introductionmentioning

confidence: 99%

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

et al. 2022

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In this paper, based on the L2-1σ scheme and nonconforming EQ1rot finite element method (FEM), a numerical approximation is presented for a class of two-dimensional time-fractional diffusion equations involving variable coefficients. A novel and detailed analysis of the equations with an initial singularity is described on anisotropic meshes. The fully discrete scheme is shown to be unconditionally stable, and optimal second-order accuracy for convergence and superconvergence can be achieved in both time and space directions. Finally, the obtained numerical results are compared with the theoretical analysis, which verifies the accuracy of the proposed method.

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“…In recent years, the superconvergence of FEMs and mixed FEMs for FPDEs were investigated in [35] and [36], respectively. In [37], the superconvergence of nonconforming FEMs for FPDEs was obtained.…”

Section: Introductionmentioning

confidence: 99%

Error analysis of interpolated coefficient finite elements for nonlinear fractional parabolic equations

2021

J. Nonlinear Funct. Anal.

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Superconvergence of a Finite Element Method for the Multi-term Time-Fractional Diffusion Problem

Cited by 37 publications

References 11 publications

On the convergence and superconvergence for a class of two‐dimensional time fractional reaction‐subdiffusion equations

On the convergence and superconvergence for a class of two‐dimensional time fractional reaction‐subdiffusion equations

Optimal H1-Norm Estimation of Nonconforming FEM for Time-Fractional Diffusion Equation on Anisotropic Meshes

Error analysis of interpolated coefficient finite elements for nonlinear fractional parabolic equations

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