### Abstract

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945-964], error estimates in L^{2}(Ω)- and H^{1}(Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H^{2}(Ω) ∩ H^{1}
_{0} (Ω) and v ∈ H^{1}
_{0} (Ω) are established. For nonsmooth data, that is, v ∈ L^{2}(Ω), the optimal L^{2}(Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Super-convergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L^{∞}(Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Original language | English |
---|---|

Pages (from-to) | 773-801 |

Number of pages | 29 |

Journal | ESAIM: Mathematical Modelling and Numerical Analysis |

Volume | 52 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 1 2018 |

### Fingerprint

### Keywords

- Backward Euler and second-order backward difference methods
- Convolution quadrature
- Finite volume element method
- Fractional order evolution equation
- Laplace transform
- Optimal error estimate
- Smooth and nonsmooth data
- Subdiffusion

### ASJC Scopus subject areas

- Analysis
- Numerical Analysis
- Modelling and Simulation
- Applied Mathematics

### Cite this

**Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data.** / Karaa, Samir; Pani, Amiya K.

Research output: Contribution to journal › Article

*ESAIM: Mathematical Modelling and Numerical Analysis*, vol. 52, no. 2, pp. 773-801. https://doi.org/10.1051/m2an/2018029

}

TY - JOUR

T1 - Error analysis of a FVEM for fractional order evolution equations with nonsmooth initial data

AU - Karaa, Samir

AU - Pani, Amiya K.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945-964], error estimates in L2(Ω)- and H1(Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H1 0 (Ω) and v ∈ H1 0 (Ω) are established. For nonsmooth data, that is, v ∈ L2(Ω), the optimal L2(Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Super-convergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞(Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

AB - In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945-964], error estimates in L2(Ω)- and H1(Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H1 0 (Ω) and v ∈ H1 0 (Ω) are established. For nonsmooth data, that is, v ∈ L2(Ω), the optimal L2(Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Super-convergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞(Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

KW - Backward Euler and second-order backward difference methods

KW - Convolution quadrature

KW - Finite volume element method

KW - Fractional order evolution equation

KW - Laplace transform

KW - Optimal error estimate

KW - Smooth and nonsmooth data

KW - Subdiffusion

UR - http://www.scopus.com/inward/record.url?scp=85049005291&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85049005291&partnerID=8YFLogxK

U2 - 10.1051/m2an/2018029

DO - 10.1051/m2an/2018029

M3 - Article

VL - 52

SP - 773

EP - 801

JO - ESAIM: Mathematical Modelling and Numerical Analysis

JF - ESAIM: Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 2

ER -