52 research outputs found
Two fully discrete schemes for fractional diffusion and diffusion-wave equations with nonsmooth data
We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.
Read More: http://epubs.siam.org/doi/10.1137/14097956
Correction of high-order BDF convolution quadrature for fractional evolution equations
We develop proper correction formulas at the starting steps to restore
the desired -order convergence rate of the -step BDF convolution
quadrature for discretizing evolution equations involving a fractional-order
derivative in time. The desired -order convergence rate can be
achieved even if the source term is not compatible with the initial data, which
is allowed to be nonsmooth. We provide complete error estimates for the
subdiffusion case , and sketch the proof for the
diffusion-wave case . Extensive numerical examples are provided
to illustrate the effectiveness of the proposed scheme.Comment: 22 pages, 3 figure
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