# Finite volume element method for two-dimensional fractional subdiffusion problems

Samir Karaa, Kassem Mustapha, Amiya K. Pani

Research output: Contribution to journalArticle

19 Citations (Scopus)

### Abstract

In this article, a semidiscrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ? (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L2)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞ )-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h2 + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

Original language English 945-964 20 IMA Journal of Numerical Analysis 37 2 https://doi.org/10.1093/imanum/drw010 Published - Apr 1 2017

### Fingerprint

Finite Volume Element Method
Subdiffusion
Finite volume method
Finite Volume Method
Fractional
Discretization
Norm
Optimal Convergence Rate
Optimal Error Estimates
Superconvergence
Discontinuous Galerkin Method
Convex Domain
Order of Convergence
Fractional Derivative
Galerkin methods
Piecewise Linear
Anomalous
Convergence Rate
Regularity
Numerical Experiment

### Keywords

• discontinuous Galerkin method
• error analysis.
• finite volume element
• fractional diffusion

### ASJC Scopus subject areas

• Mathematics(all)
• Computational Mathematics
• Applied Mathematics

### Cite this

Finite volume element method for two-dimensional fractional subdiffusion problems. / Karaa, Samir; Mustapha, Kassem; Pani, Amiya K.

In: IMA Journal of Numerical Analysis, Vol. 37, No. 2, 01.04.2017, p. 945-964.

Research output: Contribution to journalArticle

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AB - In this article, a semidiscrete spatial finite volume (FV) method is proposed and analyzed for approximating solutions of anomalous subdiffusion equations involving a temporal fractional derivative of order α ? (0, 1) in a two-dimensional convex domain. An optimal error estimate in the L∞ (L2)-norm is shown to hold. A superconvergence result is proved, and as a consequence, it is proved that a quasi-optimal order of convergence in the L∞ (L∞ )-norm holds. We also consider a fully discrete scheme that employs a FV method in space and a piecewise linear discontinuous Galerkin method to discretize in time. It is further shown that the convergence rate is of order h2 + k1+α, where h denotes the spatial discretization parameter and k represents the temporal discretization parameter. Numerical experiments indicate optimal convergence rates in both time and space, and also illustrate that the imposed regularity assumptions are pessimistic.

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