Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order

Li, Buyang; Wang, Hong; Wang, Jilu

doi:10.1051/m2an/2020072

Cited by 14 publications

(4 citation statements)

References 55 publications

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“…We point out some recent comprehensive and interesting studies for the theory of timefractional gradient flows. Li et al [20] present the well-posedness and regularity of solutions to a fractional diffusion porous media equation with variable fractional order, and analyze the convergence of a linearly implicit convolution quadrature method. Li and Ma [19] propose an exponential convolution quadrature method for the nonlinear subdiffusion equation with nonsmooth initial data, that has high-order convergence in time.…”

Section: Introductionmentioning

confidence: 99%

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

Quan,

null,

Yang

2025

JCM

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The numerical integration of phase-field equations is a delicate task which needs to recover at the discrete level intrinsic properties of the solution such as energy dissipation and maximum principle. Although the theory of energy dissipation for classical phase field models is well established, the corresponding theory for time-fractional phase-field models is still incomplete. In this article, we study certain nonlocal-in-time energies using the first-order stabilized semi-implicit L1 scheme. In particular, we will establish a discrete fractional energy law and a discrete weighted energy law. The extension for a (2 − α)-order L1 scalar auxiliary variable scheme will be investigated. Moreover, we demonstrate that the energy bound is preserved for the L1 schemes with nonuniform time steps. Several numerical experiments are carried to verify our theoretical analysis.

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Section: Introductionmentioning

confidence: 99%

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

Quan,

null,

Yang

2025

JCM

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Section: Introduction Time-fractional Diffusion Equations (Tfdes)mentioning

confidence: 99%

“…There exists some recent work on the mathematical and numerical analysis for the model (1.2) with a variable order depending on time and/or space [15,18,32,37,38,39]. In particular, the semidiscrete-in-time schemes were analyzed for the model (1.2) with spacetime dependent variable order in [18,39]. Without discretization in space, the difficulties in the numerical analysis arising from the coupling of the space-dependent variable order and the spatial inner product of the FEM are avoided (cf.…”

Section: Introduction Time-fractional Diffusion Equations (Tfdes)mentioning

confidence: 99%

A time-stepping finite element method for a time-fractional partial differential equation of hidden-memory space-time variable order

Zheng¹,

Wang²

2022

etna

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We analyze a time-stepping finite element method for a time-fractional partial differential equation with hidden-memory space-time variable order. Due to the coupling of the space-dependent variable order with the finite element formulation and the hidden memory, the variable fractional order cannot be split from the space and destroys the monotonicity in the time-stepping discretization. Because of these difficulties, the numerical analysis of a fully-discrete finite element method of the proposed model remained untreated in the literature. We develop an alternative analysis to address these issues and to prove an optimal-order error estimate of the fully-discrete finite element scheme without any assumption on the regularity of the true solution and perform numerical experiments to substantiate the theoretical findings.

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“…For many kinds of subdiffusion equations, it is difficult to obtain their analytical solutions. It is worth noting that there has been some research on numerical methods for subdiffusion equations (see [14,25,29]). For example, finite difference methods (see [5,33]), FEMs (see [2,27,28,35]), spectral methods (see [20]), meshless methods (see [21]), Petrov-Galerkin methods (see [23]), and so on.…”

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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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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Well-posedness and numerical approximation of a fractional diffusion equation with a nonlinear variable order

Cited by 14 publications

References 55 publications

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

Numerical Energy Dissipation for Time-Fractional Phase-Field Equations

A time-stepping finite element method for a time-fractional partial differential equation of hidden-memory space-time variable order

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

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