Superconvergence analysis of finite element method for time-fractional Thermistor problem

Shi, Dongyang; Yang, Huaijun

doi:10.1016/j.amc.2017.11.027

Cited by 13 publications

(8 citation statements)

References 33 publications

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“…In order to estimate A 4 and A 5 , similar to the technique used in previous studies, 13,14,31 we give the following induction hypothesis:…”

Section: Optimal Error Estimate For the Semidiscrete Schemementioning

confidence: 99%

“…Let u n and U n h be the solutions of (16) and (13), respectively, u, u t ∈ L ∞ (J; H 2 (Ω)), u tttt ∈ L ∞ (J; L 2 (Ω)), and u tt , u ttt ∈ L ∞ (J; H 1 (Ω)), for t ∈J, then we have…”

Section: Optimal Error Estimate For the Linearized Fully Discrete Schemementioning

confidence: 99%

“…Now, we estimate the terms on the right side of (17 Here, we give the following mathematical induction hypothesis to estimate B 4 and B 5 , and this technique can be found in previous studies 13,14,31,33,34 :…”

Section: Optimal Error Estimate For the Linearized Fully Discrete Schemementioning

confidence: 99%

“…Dond and Pani studied priori and posteriori estimates of the expanded mixed FEM with the piecewise constant and the lowest order Raviart‐Thomas element for the stationary case when

α false| u {false|}^{r - 1} u \equiv 0

. However, to our best knowledge, almost all of the previous analysis only concentrated on the conforming finite elements with the time‐dependent restrictions required in the error analysis because of the nonlinearity term, such as

τ = O false(h false)

in Peradze and

τ = O false(h^{2} false)

in previous studies, when the regularity assumption on the meshes, ie,

h_{K} false/ ρ_{K} \leq C

, where

h_{K}

and

ρ_{K}

denote the diameter and the radius of inscribed circle of the element

K

, respectively, is satisfied. Here and later,

C

(with or without subscripts) denotes a generic positive constant whose value may be different at different places but remains independent of

h

and

τ

.…”

Section: Introductionmentioning

confidence: 99%

“…with the time-dependent restrictions required in the error analysis because of the nonlinearity term, such as = O(h) in Peradze 8 and = O(h 2 ) in previous studies, [11][12][13][14] when the regularity assumption on the meshes, ie, h K ∕ K ≤ C, where h K and K denote the diameter and the radius of inscribed circle of the element K, respectively, is satisfied. Here and later, C (with or without subscripts) denotes a generic positive constant whose value may be different at different places but remains independent of h and .…”

Section: Introductionmentioning

confidence: 99%

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Nonconforming quadrilateral finite element method for nonlinear Kirchhoff‐type equation with damping

Shi

2019

Math Methods in App Sciences

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Nonconforming quadrilateral EQ1rot finite element method (FEM) of nonlinear Kirchhoff‐type equation with damping is studied on anisotropic meshes. Based on the property of the nonlocal term of this equation, unconditional optimal error estimates of Ofalse(hfalse) and Ofalse(h+τ2false) ( h, the spatial parameter, and τ, the time step) in the broken H1 norm are deduced for the semidiscrete and a linearized fully discrete schemes without any restrictions of τ through a distinct approach compared with the methods used for other partial differential equations, respectively. Besides, the damping term appearing in the Kirchhoff‐type equation is solved with a novel technique, which is the major difficulty in the theoretical analysis. Finally, some numerical results are provided to verify the theoretical analysis.

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“…In order to estimate A 4 and A 5 , similar to the technique used in previous studies, 13,14,31 we give the following induction hypothesis:…”

Section: Optimal Error Estimate For the Semidiscrete Schemementioning

confidence: 99%

Section: Optimal Error Estimate For the Linearized Fully Discrete Schemementioning

confidence: 99%

Section: Optimal Error Estimate For the Linearized Fully Discrete Schemementioning

confidence: 99%

“…Dond and Pani studied priori and posteriori estimates of the expanded mixed FEM with the piecewise constant and the lowest order Raviart‐Thomas element for the stationary case when

α false| u {false|}^{r - 1} u \equiv 0

τ = O false(h false)

in Peradze and

τ = O false(h^{2} false)

in previous studies, when the regularity assumption on the meshes, ie,

h_{K} false/ ρ_{K} \leq C

, where

h_{K}

and

ρ_{K}

denote the diameter and the radius of inscribed circle of the element

K

, respectively, is satisfied. Here and later,

C

(with or without subscripts) denotes a generic positive constant whose value may be different at different places but remains independent of

h

and

τ

.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Nonconforming quadrilateral finite element method for nonlinear Kirchhoff‐type equation with damping

Shi

2019

Math Methods in App Sciences

Self Cite

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Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

Peng

Zhao

et al. 2022

Numerical Methods Partial

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In this article, we propose and analyze several numerical methods for the nonlinear delay reaction-diffusion system with smooth and nonsmooth solutions, by using Quasi-Wilson nonconforming finite element methods in space and finite difference methods (including uniform and nonuniform L1 and L2-1 σ schemes) in time. The optimal convergence results in the senses of L 2 -norm and broken H 1 -norm, and H 1 -norm superclose results are derived by virtue of two types of fractional Grönwall inequalities. Then, the interpolation postprocessing technique is used to establish the superconvergence results. Moreover, to improve computational efficiency, fast algorithms by using sum-of-exponential technique are built for above proposed numerical schemes. Finally, we present some numerical experiments to confirm the theoretical correctness and show the effectiveness of the fast algorithms.

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Numerical simulations based on shifted second-order difference/finite element algorithms for the time fractional Maxwell’s system

Fan

Wang

Liu

et al. 2020

Engineering with Computers

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Superconvergence analysis of finite element method for time-fractional Thermistor problem

Cited by 13 publications

References 33 publications

Nonconforming quadrilateral finite element method for nonlinear Kirchhoff‐type equation with damping

Nonconforming quadrilateral finite element method for nonlinear Kirchhoff‐type equation with damping

Convergence and superconvergence analysis for nonlinear delay reaction–diffusion system with nonconforming finite element

Numerical simulations based on shifted second-order difference/finite element algorithms for the time fractional Maxwell’s system

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