Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation
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Hu Chen
Abstract
Time-fractional initial-boundary problems of parabolic type are considered. Previously, global error bounds for computed numerical solutions to such problems have been provided by Liao et al. (SIAM J. Numer. Anal. 2018, 2019) and Stynes et al. (SIAM J. Numer. Anal. 2017). In the present work we show how the concept of complete monotonicity can be combined with these older analyses to derive local error bounds (i.e., error bounds that are sharper than global bounds when one is not close to the initial time ). Furthermore, we show that the error analyses of the above papers are essentially the same – their key stability parameters, which seem superficially different from each other, become identical after a simple rescaling. Our new approach is used to bound the global and local errors in the numerical solution of a multi-term time-fractional diffusion equation, using the L1 scheme for the temporal discretisation of each fractional derivative. These error bounds are α-robust. Numerical results show they are sharp.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11801026
Award Identifier / Grant number: NSAF U1930402
Funding statement: The research of Hu Chen is supported in part by the National Natural Science Foundation of China young scientists fund Grant 11801026. The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grant NSAF U1930402.
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Articles in the same Issue
- Frontmatter
- Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity
- Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation
- Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers
- Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces
- A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space
- Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
- Dual System Least-Squares Finite Element Method for a Hyperbolic Problem
- Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes
- Numerical Analysis of a Finite Element Approximation to a Level Set Model for Free-Surface Flows
- Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials
- Convergence Theory for IETI-DP Solvers for Discontinuous Galerkin Isogeometric Analysis that is Explicit in ℎ and 𝑝
- Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary
Articles in the same Issue
- Frontmatter
- Robust Iterative Solvers for Gao Type Nonlinear Beam Models in Elasticity
- Using Complete Monotonicity to Deduce Local Error Estimates for Discretisations of a Multi-Term Time-Fractional Diffusion Equation
- Accurate Full-Vectorial Finite Element Method Combined with Exact Non-Reflecting Boundary Condition for Computing Guided Waves in Optical Fibers
- Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces
- A Totally Relaxed, Self-Adaptive Subgradient Extragradient Method for Variational Inequality and Fixed Point Problems in a Banach Space
- Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization
- Dual System Least-Squares Finite Element Method for a Hyperbolic Problem
- Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes
- Numerical Analysis of a Finite Element Approximation to a Level Set Model for Free-Surface Flows
- Inexact Inverse Subspace Iteration with Preconditioning Applied to Quadratic Matrix Polynomials
- Convergence Theory for IETI-DP Solvers for Discontinuous Galerkin Isogeometric Analysis that is Explicit in ℎ and 𝑝
- Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary
