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 $\alpha\in(3/2, 2)$ 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 $L^2(D)$ and $H^1(D)$ 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

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 $\alpha\in(3/2, 2)$ 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 $L^2(D)$ and $H^1(D)$ 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

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 fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

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 $\alpha\in(0,1)$ 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