525 research outputs found
Discrete maximal regularity of time-stepping schemes for fractional evolution equations
In this work, we establish the maximal -regularity for several time
stepping schemes for a fractional evolution model, which involves a fractional
derivative of order , , in time. These schemes
include convolution quadratures generated by backward Euler method and
second-order backward difference formula, the L1 scheme, explicit Euler method
and a fractional variant of the Crank-Nicolson method. The main tools for the
analysis include operator-valued Fourier multiplier theorem due to Weis [48]
and its discrete analogue due to Blunck [10]. These results generalize the
corresponding results for parabolic problems
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
Error Estimates for Approximations of Distributed Order Time Fractional Diffusion with Nonsmooth Data
In this work, we consider the numerical solution of an initial boundary value
problem for the distributed order time fractional diffusion equation. The model
arises in the mathematical modeling of ultra-slow diffusion processes observed
in some physical problems, whose solution decays only logarithmically as the
time tends to infinity. We develop a space semidiscrete scheme based on the
standard Galerkin finite element method, and establish error estimates optimal
with respect to data regularity in and norms for both smooth
and nonsmooth initial data. Further, we propose two fully discrete schemes,
based on the Laplace transform and convolution quadrature generated by the
backward Euler method, respectively, and provide optimal convergence rates in
the norm, which exhibits exponential convergence and first-order
convergence in time, respectively. Extensive numerical experiments are provided
to verify the error estimates for both smooth and nonsmooth initial data, and
to examine the asymptotic behavior of the solution.Comment: 25 pages, 2 figure
An Analysis of the Rayleigh-Stokes problem for a Generalized Second-Grade Fluid
We study the Rayleigh-Stokes problem for a generalized second-grade fluid
which involves a Riemann-Liouville fractional derivative in time, and present
an analysis of the problem in the continuous, space semidiscrete and fully
discrete formulations. We establish the Sobolev regularity of the homogeneous
problem for both smooth and nonsmooth initial data , including . A space semidiscrete Galerkin scheme using continuous piecewise
linear finite elements is developed, and optimal with respect to initial data
regularity error estimates for the finite element approximations are derived.
Further, two fully discrete schemes based on the backward Euler method and
second-order backward difference method and the related convolution quadrature
are developed, and optimal error estimates are derived for the fully discrete
approximations for both smooth and nonsmooth initial data. Numerical results
for one- and two-dimensional examples with smooth and nonsmooth initial data
are presented to illustrate the efficiency of the method, and to verify the
convergence theory.Comment: 23 pp, 4 figures. The error analysis of the fully discrete scheme is
shortene
A linear Galerkin numerical method for a quasilinear subdiffusion equation
We couple the L1 discretization for Caputo derivative in time with spectral
Galerkin method in space to devise a scheme that solves quasilinear
subdiffusion equations. Both the diffusivity and the source are allowed to be
nonlinear functions of the solution. We prove method's stability and
convergence with spectral accuracy in space. The temporal order depends on
solution's regularity in time. Further, we support our results with numerical
simulations that utilize parallelism for spatial discretization. Moreover, as a
side result we find asymptotic exact values of error constants along with their
remainders for discretizations of Caputo derivative and fractional integrals.
These constants are the smallest possible which improves the previously
established results from the literature.Comment: This is the accepted version of the manuscript published in Applied
Numerical Mathematic
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