509 research outputs found
Space-Time Petrov–Galerkin FEM for Fractional Diffusion Problems
We present and analyze a space-time Petrov-Galerkin finite element
method for a time-fractional diffusion equation involving a Riemann-Liouville
fractional derivative of order α ∈ (0, 1) in time and zero initial data. We derive a
proper weak formulation involving different solution and test spaces and show
the inf-sup condition for the bilinear form and thus its well-posedness. Further,
we develop a novel finite element formulation, show the well-posedness of the
discrete problem, and derive error bounds in both energy and L
2 norms for the
finite element solution. In the proof of the discrete inf-sup condition, a certain
nonstandard L
2
stability property of the L
2 projection operator plays a key role.
We provide extensive numerical examples to verify the convergence analysis
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
An advanced meshless method for time fractional diffusion equation
Recently, because of the new developments in sustainable engineering and renewable energy, which are usually governed by a series of fractional partial differential equations (FPDEs), the numerical modelling and simulation for fractional calculus are attracting more and more attention from researchers. The current dominant numerical method for modeling FPDE is Finite Difference Method (FDM), which is based on a pre-defined grid leading to inherited issues or shortcomings including difficulty in simulation of problems with the complex problem domain and in using irregularly distributed nodes. Because of its distinguished advantages, the meshless method has good potential in simulation of FPDEs. This paper aims to develop an implicit meshless collocation technique for FPDE. The discrete system of FPDEs is obtained by using the meshless shape functions and the meshless collocation formulation. The stability and convergence of this meshless approach are investigated theoretically and numerically. The numerical examples with regular and irregular nodal distributions are used to validate and investigate accuracy and efficiency of the newly developed meshless formulation. It is concluded that the present meshless formulation is very effective for the modeling and simulation of fractional partial differential equations
FIC/FEM formulation with matrix stabilizing terms for incompressible flows at low and high Reynolds numbers
The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-006-0060-yWe present a general formulation for incompressible fluid flow analysis using the finite element method. The necessary stabilization for dealing with convective effects and the incompressibility condition are introduced via the Finite Calculus method using a matrix form of the stabilization parameters. This allows to model a wide range of fluid flow problems for low and high Reynolds numbers flows without introducing a turbulence model. Examples of application to the analysis of incompressible flows with moderate and large Reynolds numbers are presented.Peer ReviewedPostprint (author's final draft
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 superdiffusion 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 (FEM) 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 the derivation of optimal error estimates in both the 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
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