96 research outputs found
A Petrov-Galerkin Finite Element Method for Fractional Convection-Diffusion Equations
In this work, we develop variational formulations of Petrov-Galerkin type for
one-dimensional fractional boundary value problems involving either a
Riemann-Liouville or Caputo derivative of order in the
leading term and both convection and potential terms. They arise in the
mathematical modeling of asymmetric super-diffusion processes in heterogeneous
media. The well-posedness of the formulations and sharp regularity pickup of
the variational solutions are established. A novel finite element method is
developed, which employs continuous piecewise linear finite elements and
"shifted" fractional powers for the trial and test space, respectively. The new
approach has a number of distinct features: It allows deriving optimal error
estimates in both and norms; and on a uniform mesh, the
stiffness matrix of the leading term is diagonal and the resulting linear
system is well conditioned. Further, in the Riemann-Liouville case, an enriched
FEM is proposed to improve the convergence. Extensive numerical results are
presented to verify the theoretical analysis and robustness of the numerical
scheme.Comment: 23 p
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
Some error estimates for the lumped mass finite element method for a parabolic problem
We study the spatially semidiscrete lumped mass method for the model homogeneous heat equation with homogeneous Dirichlet boundary conditions. Improving earlier results we show that known optimal order smooth initial data error estimates for the standard Galerkin method carry over to the lumped mass method whereas nonsmooth initial data estimates require special assumptions on the triangulation. We also discuss the application to time discretization by the backward Euler and Crank-Nicolson methods
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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