42 research outputs found
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
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
High-order splitting finite element methods for the subdiffusion equation with limited smoothing property
In contrast with the diffusion equation which smoothens the initial data to
for (away from the corners/edges of the domain), the
subdiffusion equation only exhibits limited spatial regularity. As a result,
one generally cannot expect high-order accuracy in space in solving the
subdiffusion equation with nonsmooth initial data. In this paper, a new
splitting of the solution is constructed for high-order finite element
approximations to the subdiffusion equation with nonsmooth initial data. The
method is constructed by splitting the solution into two parts, i.e., a
time-dependent smooth part and a time-independent nonsmooth part, and then
approximating the two parts via different strategies. The time-dependent smooth
part is approximated by using high-order finite element method in space and
convolution quadrature in time, while the steady nonsmooth part could be
approximated by using smaller mesh size or other methods that could yield
high-order accuracy. Several examples are presented to show how to accurately
approximate the steady nonsmooth part, including piecewise smooth initial data,
Dirac--Delta point initial data, and Dirac measure concentrated on an
interface. The argument could be directly extended to subdiffusion equations
with nonsmooth source data. Extensive numerical experiments are presented to
support the theoretical analysis and to illustrate the performance of the
proposed high-order splitting finite element methods.Comment: 25 page