Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem
type: Item
instance of: scholarly article
title: Optimal \(L^\infty (L^2)\) error analysis of a direct discontinuous Galerkin method for a time-fractional reaction-diffusion problem
zbMATH Open document ID: 1433.65212
author: Chaobao Huang
author: Martin Stynes
author: Na An
published in: BIT
publication date: 2018-09-20T00:00:00Z
review text: A time-fractional reaction-diffusion initial-boundary value problem with a Caputo time derivative of order \(\alpha\in (0,1)\) is considered. New pointwise bounds on certain derivatives of the solution are derived. Prior work on the analysis of the DDG method considered periodic boundary conditions; in the present work, homogeneous Dirchlet boundary conditions are used instead (which in turn renders the analysis more complicated). An \(L1\) finite difference scheme in time and a direct discontinuous Galerkin (DDG) FEM in space are considered. The mesh is uniform in space and graded in time. A discrete \(L2\)-stability result for the considered scheme is derived. The convergence results provide guidance on how to choose an optimal grading of the temporal mesh. Numerical experiments demonstrate that the theoretical convergence results are sharp; the reported numerical errors agree with the theoretical rate of convergence.
reviewed by: Andreas Mang
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 6940691
zbMATH Keywords: numerical analysis
zbMATH Keywords: DDG method
zbMATH Keywords: Caputo derivative
zbMATH Keywords: time-fractional reaction-diffusion equation
MaRDI profile type: Publication
full work available at URL: s10543-018-0707-z
OpenAlex ID: W2804149282
cites work: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations
cites work: The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
cites work: Q5823354
cites work: The analysis of fractional differential equations. An application-oriented exposition using differential operators of Caputo type
cites work: Convergence Outside the Initial Layer for a Numerical Method for the Time-Fractional Heat Equation
cites work: Geometric theory of semilinear parabolic equations
cites work: Error analysis of semidiscrete finite element methods for inhomogeneous time-fractional diffusion
cites work: Error Estimates for a Semidiscrete Finite Element Method for Fractional Order Parabolic Equations
cites work: Optimal error estimates of the direct discontinuous Galerkin method for convection-diffusion equations
cites work: The Direct Discontinuous Galerkin (DDG) Method for Diffusion with Interface Corrections
cites work: Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation
cites work: Fractional differential equations. An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications
cites work: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems
cites work: Q3141895
cites work: Spectral Methods
cites work: Error Analysis of a Finite Difference Method on Graded Meshes for a Time-Fractional Diffusion Equation
cites work: Analysis of a local discontinuous Galerkin method for time‐fractional advection‐diffusion equations
DOI: 10.1007/S10543-018-0707-Z
description: scientific article; zbMATH DE number 6940691