231 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
An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion
In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion
model with a Caputo fractional derivative of order in time,
which is often used to describe anomalous diffusion processes in heterogeneous
media. The nonlocality of the fractional derivative requires storing all the
solutions from time zero. The proposed scheme is based on continuous piecewise
linear finite elements, L1 time stepping, and proper orthogonal decomposition
(POD). By constructing an effective reduced-order scheme using problem-adapted
basis functions, it can significantly reduce the computational complexity and
storage requirement. We shall provide a complete error analysis of the scheme
under realistic regularity assumptions by means of a novel energy argument.
Extensive numerical experiments are presented to verify the convergence
analysis and the efficiency of the proposed scheme.Comment: 25 pp, 5 figure
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