Unconditional Convergence of a Fast Two-Level Linearized Algorithm for Semilinear Subdiffusion Equations

A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algor...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 80; no. 1; pp. 1 - 25
Main Authors: Liao, Hong-lin, Yan, Yonggui, Zhang, Jiwei
Format: Journal Article
Language:English
Published: New York Springer US 01.07.2019
Springer Nature B.V
Subjects:
Algorithms
Approximation
Boundary value problems
Computational efficiency
Computational Mathematics and Numerical Analysis
Energy methods
Error analysis
Finite element method
Formulas (mathematics)
Linearization
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Numerical analysis
Theoretical
Semilinear subdiffusion equation
energy method
Discrete
Discrete fractional Grönwall inequality
Two-level L1 formula
Unconditional convergence
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:A fast two-level linearized scheme with nonuniform time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations or small time-steps because it significantly reduces the computational cost O ( M N 2 ) and storage O ( MN ) for the standard L1 formula to O ( M N log N ) and O ( M log N ) , respectively, for M grid points in space and N levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time t = 0 , and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a recently developed discrete fractional Grönwall inequality, a global consistency analysis and a discrete H 2 energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-019-00927-0