2,034 research outputs found
Time-stepping error bounds for fractional diffusion problems with non-smooth initial data
We apply the piecewise constant, discontinuous Galerkin method to discretize
a fractional diffusion equation with respect to time. Using Laplace transform
techniques, we show that the method is first order accurate at the \$n\$th time
level \$t_n\$, but the error bound includes a factor \$t_n^{-1}\$ if we assume
no smoothness of the initial data. We also show that for smoother initial data
the growth in the error bound as \$t_n\$ decreases is milder, and in some cases
absent altogether. Our error bounds generalize known results for the classical
heat equation and are illustrated for a model problem.Comment: 22 pages, 5 figure
Some time stepping methods for fractional diffusion problems with nonsmooth data
We consider error estimates for some time stepping methods for solving fractional diffusion problems with nonsmooth data in both homogeneous and inhomogeneous cases. McLean and Mustapha \cite{mclmus} (Time-stepping error bounds for fractional diffusion problems with non-smooth initial data, Journal of Computational Physics, 293(2015), 201-217) established an convergence rate for the piecewise constant discontinuous Galerkin method with nonsmooth initial data for the homogeneous problem when the linear operator is assumed to be self-adjoint, positive semidefinite and densely defined in a suitable Hilbert space, where denotes the time step size. In this paper, we approximate the Riemann-Liouville fractional derivative by Diethelm's method (or scheme) and obtain the same time discretisation scheme as in McLean and Mustapha \cite{mclmus}. We first prove that this scheme has also convergence rate with nonsmooth initial data for the homogeneous problem when is a closed, densely defined linear operator satisfying some certain resolvent estimates. We then introduce a new time discretization scheme for the homogeneous problem based on the convolution quadrature and prove that the convergence rate of this new scheme is with the nonsmooth initial data. Using this new time discretization scheme for the homogeneous problem, we define a time stepping method for the inhomogeneous problem and prove that the convergence rate of this method is with the nonsmooth data. Numerical examples are given to show that the numerical results are consistent with the theoretical results
Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview
Over the past few decades, there has been substantial interest in evolution
equations that involving a fractional-order derivative of order
in time, due to their many successful applications in
engineering, physics, biology and finance. Thus, it is of paramount importance
to develop and to analyze efficient and accurate numerical methods for reliably
simulating such models, and the literature on the topic is vast and fast
growing. The present paper gives a concise overview on numerical schemes for
the subdiffusion model with nonsmooth problem data, which are important for the
numerical analysis of many problems arising in optimal control, inverse
problems and stochastic analysis. We focus on the following aspects of the
subdiffusion model: regularity theory, Galerkin finite element discretization
in space, time-stepping schemes (including convolution quadrature and L1 type
schemes), and space-time variational formulations, and compare the results with
that for standard parabolic problems. Further, these aspects are showcased with
illustrative numerical experiments and complemented with perspectives and
pointers to relevant literature.Comment: 24 pages, 3 figure
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