49,774 research outputs found
Discontinuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity
An integro-differential equation, modeling dynamic fractional order
viscoelasticity, with a Mittag-Leffler type convolution kernel is considered. A
discontinuous Galerkin method, based on piecewise constant polynomials is
formulated for temporal semidiscretization of the problem. Stability estimates
of the discrete problem are proved, that are used to prove optimal order a
priori error estimates. The theory is illustrated by a numerical example.Comment: 16 pages, 2 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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