355 research outputs found
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
Galerkin FEM for fractional order parabolic equations with initial data in
We investigate semi-discrete numerical schemes based on the standard Galerkin
and lumped mass Galerkin finite element methods for an initial-boundary value
problem for homogeneous fractional diffusion problems with non-smooth initial
data. We assume that , is a convex
polygonal (polyhedral) domain. We theoretically justify optimal order error
estimates in - and -norms for initial data in . We confirm our theoretical findings with a number of numerical tests
that include initial data being a Dirac -function supported on a
-dimensional manifold.Comment: 13 pages, 3 figure
The Galerkin Finite Element Method for A Multi-term Time-Fractional Diffusion equation
We consider the initial/boundary value problem for a diffusion equation
involving multiple time-fractional derivatives on a bounded convex polyhedral
domain. We analyze a space semidiscrete scheme based on the standard Galerkin
finite element method using continuous piecewise linear functions. Nearly
optimal error estimates for both cases of initial data and inhomogeneous term
are derived, which cover both smooth and nonsmooth data. Further we develop a
fully discrete scheme based on a finite difference discretization of the
time-fractional derivatives, and discuss its stability and error estimate.
Extensive numerical experiments for one and two-dimension problems confirm the
convergence rates of the theoretical results.Comment: 22 pages, 4 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
Numerical analysis of nonlinear subdiffusion equations
We present a general framework for the rigorous numerical analysis of
time-fractional nonlinear parabolic partial differential equations, with a
fractional derivative of order in time. The framework relies
on three technical tools: a fractional version of the discrete Gr\"onwall-type
inequality, discrete maximal regularity, and regularity theory of nonlinear
equations. We establish a general criterion for showing the fractional discrete
Gr\"onwall inequality, and verify it for the L1 scheme and convolution
quadrature generated by BDFs. Further, we provide a complete solution theory,
e.g., existence, uniqueness and regularity, for a time-fractional diffusion
equation with a Lipschitz nonlinear source term. Together with the known
results of discrete maximal regularity, we derive pointwise norm
error estimates for semidiscrete Galerkin finite element solutions and fully
discrete solutions, which are of order (up to a logarithmic factor)
and , respectively, without any extra regularity assumption on
the solution or compatibility condition on the problem data. The sharpness of
the convergence rates is supported by the numerical experiments
Error Analysis of Semidiscrete Finite Element Methods for Inhomogeneous Time-Fractional Diffusion
We consider the initial boundary value problem for the inhomogeneous
time-fractional diffusion equation with a homogeneous Dirichlet boundary
condition and a nonsmooth right hand side data in a bounded convex polyhedral
domain. We analyze two semidiscrete schemes based on the standard Galerkin and
lumped mass finite element methods. Almost optimal error estimates are obtained
for right hand side data , , for both semidiscrete schemes. For lumped mass method, the optimal
-norm error estimate requires symmetric meshes. Finally, numerical
experiments for one- and two-dimensional examples are presented to verify our
theoretical results.Comment: 21 pages, 4 figure
- …