2,794 research outputs found
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
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
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
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