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## ERROR ESTIMATES FOR A SEMIDISCRETE FINITE ELEMENT METHOD FOR FRACTIONAL ORDER PARABOLIC EQUATIONS

Citations: | 18 - 6 self |

### Citations

1087 |
Fractional differential equations
- Podlubny
- 1999
(Show Context)
Citation Context ... circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, and model of neurons in biology is provided in [5]; see also =-=[23]-=-. The capabilities of FPDEs to accurately model such processes has generated a considerable interest in derivation, analysis and testing of numerical methods for ∗ Department of Mathematics and Instit... |

499 |
Galerkin Finite Element Methods for Parabolic Problems
- Thomeé
- 1997
(Show Context)
Citation Context ... element method (FEM) for standard parabolic problems, α = 1. Here the error analysis is complete and various optimal with respect to the regularity of the solution estimates are available; see, e.g. =-=[26]-=-. The key inngredient of the analysis is the smoothing property of the parabolic operator and its discrete counterpart [26, Lemmas 3.2 and 2.5]. For the FPDE (1.1), such property has been established ... |

326 |
Theory and applications of fractional differential equations
- Kilbas, Srivastava, et al.
- 2006
(Show Context)
Citation Context ...t 0 (t− τ)−α d dτ u(τ) dτ, where Γ(·) is the Gamma function. Note that if the fractional order α tends to unity, the fractional derivative ∂αt u converges to the canonical first-order derivative dudt =-=[12]-=-, and thus the problem (1.1) reproduces the standard parabolic equation. The model (1.1) is known to capture well the dynamics of anomalous diffusion (also known as sub-diffusion) in which the mean sq... |

224 | Theory and practice of finite elements - Ern, Guermond - 2004 |

196 | Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications, Phys
- Bouchaud, Georges
- 1990
(Show Context)
Citation Context ...lic equation. The model (1.1) is known to capture well the dynamics of anomalous diffusion (also known as sub-diffusion) in which the mean square variance grows slower than that in a Gaussian process =-=[1]-=-, and has found a number of important practical applications. For example, it was introduced by Nigmatulin [22] to describe diffusion in media with fractal geometry. A comprehensive survey on fraction... |

99 | Theory and practice of finite elements, volume 159 of applied mathematical sciences - Ern, Guermond - 2004 |

95 | Superconvergence in Galerkin Finite Element Methods - Wahlbin - 1995 |

45 | Variational formulation for the stationary fractional advection dispersion equation
- Erwin, Roop
- 2006
(Show Context)
Citation Context ...ning the forward time centered space method and the Grunwald-Letnikov method, and provided a von Neumann type stability analysis. By exploiting the variational framework introduced by Ervin and Roop, =-=[9]-=-, Li and Xu [15] developed a spectral approximation method in both temporal and spatial variable, and established various a priori error estimates. Deng [6] analyzed the finite element method (FEM) fo... |

42 |
Finite difference/spectral approximations for the timefractional diffusion equation
- Lin, Xu
- 2007
(Show Context)
Citation Context ...izing the time interval, tn = nτ, n = 0, 1, . . . , with τ being the time step size, and then using a weighted finite difference approximation of the fractional derivative ∂ α t u(x, tn) developed in =-=[16]-=-: ∂ α t u(x, tn) = ≈ = 1 Γ(1 − α) n−1 ∫ tj+1 ∑ j=0 tj ∂u(x, s) (tn − s) ∂s −α ds n−1 1 ∑ u(x, tj+1) − u(x, tj) Γ(1 − α) τ j=0 n−1 1 ∑ Γ(2 − α) j=0 ∫ tj+1 u(x, tn−j) − u(x, tn−j−1) bj τ α , tj (tn − s)... |

37 |
An explicit finite difference method and a new von Neumann–type stability analysis for fractional diffusion equations
- Yuste, Acedo
- 2005
(Show Context)
Citation Context ...riving, analyzing and testing numerical methods for solving such problems. As a result, a number of numerical techniques were developed and their stability and convergence were investigated, see e.g. =-=[6, 14, 15, 19, 20, 27]-=-. Yuste and Acedo in [27] presented a numerical scheme by combining the forward time centered space method and the Grunwald-Letnikov method, and provided a von Neumann type stability analysis. By expl... |

35 |
The accuracy and stability of an implicit solution method for the fractional diffusion equation
- Langlands, Henry
- 2005
(Show Context)
Citation Context ... USA (btjin,lazarov,zzhou@math.tamu.edu) 1solving such problems. As a result a number of numerical techniques were developed and tested and their stability and convergence were investigated see e.g. =-=[6, 14, 15, 20, 21, 28]-=-. Yuste and Acedo in [28] presented a numerical scheme by combining the forward time centered space method and the Grunwald-Letnikov method, and provided a von Neumann type stability analysis. By expl... |

34 |
Finite difference methods for two-dimensional fractional dispersion equation
- Meerschaert, Scheffler, et al.
- 2006
(Show Context)
Citation Context ... USA (btjin,lazarov,zzhou@math.tamu.edu) 1solving such problems. As a result a number of numerical techniques were developed and tested and their stability and convergence were investigated see e.g. =-=[6, 14, 15, 20, 21, 28]-=-. Yuste and Acedo in [28] presented a numerical scheme by combining the forward time centered space method and the Grunwald-Letnikov method, and provided a von Neumann type stability analysis. By expl... |

27 |
Recent applications of fractional calculus to science and engineering
- Debnath
(Show Context)
Citation Context ...ry, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, and model of neurons in biology is provided in =-=[5]-=-; see also [23]. The capabilities of FPDEs to accurately model such processes has generated a considerable interest in derivation, analysis and testing of numerical methods for ∗ Department of Mathema... |

27 |
Finite element method for the space and time fractional Fokker-Planck equation
- Deng
(Show Context)
Citation Context ... USA (btjin,lazarov,zzhou@math.tamu.edu) 1solving such problems. As a result a number of numerical techniques were developed and tested and their stability and convergence were investigated see e.g. =-=[6, 14, 15, 20, 21, 28]-=-. Yuste and Acedo in [28] presented a numerical scheme by combining the forward time centered space method and the Grunwald-Letnikov method, and provided a von Neumann type stability analysis. By expl... |

26 |
A space-time spectral method for the time fractional diffusion equation
- Li, Xu
(Show Context)
Citation Context |

21 | Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
- Sakamoto, Yamamoto
(Show Context)
Citation Context ...sis is the smoothing property of the parabolic operator and its discrete counterpart [26, Lemmas 3.2 and 2.5]. For the FPDE (1.1), such property has been established recently by Sakamoto and Yamamoto =-=[24]-=-; see Theorem 2.2 below for details. The goal of this note is to develop an error analysis with optimal with respect to the regularity of the initial data estimates for the semidiscrete Galerkin and t... |

19 | Numerical identification of parameters in parabolic systems
- Keung, L
- 1998
(Show Context)
Citation Context ...dered. However, this is essential when due to the data the solution has limited regularity, a typical case for inverse problems related to this equation, see e.g., [3], [24, Problem (4.12)], and also =-=[10, 11]-=- for its parabolic counterpart. There are a few papers considering construction and analysis of numerical methods with optimal with respect to the regularity of the solution error estimates for the fo... |

19 | Nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term
- LUBICH, SLOAN, et al.
- 1996
(Show Context)
Citation Context ...ers considering construction and analysis of numerical methods with optimal with respect to the regularity of the solution error estimates for the following equation with a positive type memory terms =-=[17, 18, 19, 21]-=-): ∂tu − 1 Γ(α) ∫ t 0 (t − τ) α−1 ∆u(τ)dτ = f(x, t), t > 0, 0 < α < 1, (1.2) This equation is closely related, but different from (1.1). For example, McLean and Thomée in [18, 19] developed a numerica... |

17 |
The stability in Lp and W 1 p of the L2projection onto finite element function spaces
- Crouzeix, Thomée
- 1987
(Show Context)
Citation Context ...tability of Ph directly follows from the error bound (2.9) and the inverse inequality. However, for more general meshes such stability is valid only under some mild assumptions on the mesh; see, e.g. =-=[4]-=-. 3. Semidiscrete Galerkin FEM. In this section we derive error estimates for the standard semidiscrete Galerkin FEM. First we recall some basic known facts for the spatially semidiscrete standard Gal... |

15 |
On a global-superconvergence of the gradient of linear triangular elements
- Krizek, Neittaanmaki
- 1987
(Show Context)
Citation Context ...available for special meshes and solutions in H3 (Ω). Examples of special meshes exhibiting superconvergence property include triangulations in which every two adjacent triangles form a parallelogram =-=[13]-=-. To establish a super-convergent recovery of the gradient, Kˇriˇzek and Neittaanmäki in [13] introduced an operator of the averaged (recovered, postprocessed) gradient Gh(uh) (see [13, equation (2.2)... |

14 |
Numerical algorithm for calculating the generalized Mittag-Leffler function
- Seybold, Hilfer
(Show Context)
Citation Context ...lution for each example can be expressed by an infinite series involving the Mittag-Leffler function Eα,1(z). To accurately evaluate the Mittag-Leffler functions, we employ the algorithm developed in =-=[25]-=-, which is based on three different approximations of the function, i.e., Taylor series, integral representations and exponential asymptotics, in different regions of the domain. We divide the unit in... |

12 | Harmonic Analysis and Boundary Value - Djrbashian - 1993 |

9 |
Time discretization of an evolution equation via Laplace transforms
- McLean, Thomee
(Show Context)
Citation Context ...ers considering construction and analysis of numerical methods with optimal with respect to the regularity of the solution error estimates for the following equation with a positive type memory terms =-=[17, 18, 19, 21]-=-): ∂tu − 1 Γ(α) ∫ t 0 (t − τ) α−1 ∆u(τ)dτ = f(x, t), t > 0, 0 < α < 1, (1.2) This equation is closely related, but different from (1.1). For example, McLean and Thomée in [18, 19] developed a numerica... |

8 |
Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation
- Cheng, Nakagawa, et al.
(Show Context)
Citation Context ...nd the initial data v, was not considered. However, this is essential when due to the data the solution has limited regularity, a typical case for inverse problems related to this equation, see e.g., =-=[3]-=-, [24, Problem (4.12)], and also [10, 11] for its parabolic counterpart. There are a few papers considering construction and analysis of numerical methods with optimal with respect to the regularity o... |

5 |
An implicit finite-difference time-stepping method for a sub-diffusion equation with spatial discretization by finite elements
- Mustapha
- 2011
(Show Context)
Citation Context |

5 |
The realization of the generalized transfer equation in a medium with fractal geometry
- Nigmatulin
- 1986
(Show Context)
Citation Context ...diffusion) in which the mean square variance grows slower than that in a Gaussian process [1], and has found a number of important practical applications. For example, it was introduced by Nigmatulin =-=[22]-=- to describe diffusion in media with fractal geometry. A comprehensive survey on fractional order differential equations arising in dynamical systems in control theory, electrical circuits with fracta... |

5 |
Maximum-norm error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation
- McLean, Thomée
(Show Context)
Citation Context ...ers considering construction and analysis of numerical methods with optimal with respect to the regularity of the solution error estimates for the following equation with a positive type memory terms =-=[17, 18, 20]-=-): (1.2) ∂tu− 1 Γ(α) ∫ t 0 (t− τ)α−1∆u(τ)dτ = f(x, t), t > 0, 0 < α < 1, This equation is closely related, but different from (1.1). For example, McLean and Thomée in [17, 18] developed a numerical m... |

4 |
Numerical identification of a robin coefficient in parabolic problems
- Jin, Lu
- 2011
(Show Context)
Citation Context ...dered. However, this is essential when due to the data the solution has limited regularity, a typical case for inverse problems related to this equation, see e.g., [3], [24, Problem (4.12)], and also =-=[10, 11]-=- for its parabolic counterpart. There are a few papers considering construction and analysis of numerical methods with optimal with respect to the regularity of the solution error estimates for the fo... |

4 |
The stability in Lp and W p of the L2-projection onto finite element function spaces
- Crouzeix, Thomée
- 1987
(Show Context)
Citation Context ...tability of Ph directly follows from the error bound (2.9) and the inverse inequality. However, for more general meshes such stability is valid only under some mild assumptions on the mesh; see, e.g. =-=[4]-=-. 3. Semidiscrete Galerkin FEM In this section we derive error estimates for the standard semidiscrete Galerkin FEM. First we recall some basic known facts for the spatially semidiscrete standard Ga... |

3 | V.Thomée, Some error estimates for the lumped mass finite element method for a parabolic problem
- Chatzipantelidis
(Show Context)
Citation Context ... case of non-smooth data, v ∈ L2(Ω), for general quasi-uniform meshes, we were only able to establish a suboptimal L2-error bound of order O(hℓht −α ), see (4.15). Further, influenced by the study in =-=[2]-=-, we also consider special meshes. Namely, for a class of special triangulations satisfying the condition (4.16), which holds for meshes that are symmetric with respect to each internal vertex [2, Sec... |

1 |
error analysis of a numerical solution via Laplace transformation and quadrature of a fractional-order evolution equation
- Maximum-norm
(Show Context)
Citation Context ...ers considering construction and analysis of numerical methods with optimal with respect to the regularity of the solution error estimates for the following equation with a positive type memory terms =-=[17, 18, 19, 21]-=-): ∂tu − 1 Γ(α) ∫ t 0 (t − τ) α−1 ∆u(τ)dτ = f(x, t), t > 0, 0 < α < 1, (1.2) This equation is closely related, but different from (1.1). For example, McLean and Thomée in [18, 19] developed a numerica... |