Local discontinuous Galerkin methods for fractional ordinary differential equations

This paper discusses the upwinded local discontinuous Galerkin methods for the one-term/multi-term fractional ordinary differential equations (FODEs). The natural upwind choice of the numerical fluxes for the initial value problem for FODEs ensures stability of the methods. The solution can be computed element by element with optimal order of convergence $k+1$ in the $L^2$ norm and superconvergence of order $k+1+\min\{k,\alpha\}$ at the downwind point of each element. Here $k$ is the degree of the approximation polynomial used in an element and $\alpha$ ($\alpha\in (0,1]$) represents the order of the one-term FODEs. A generalization of this includes problems with classic $m$'th-term FODEs, yielding superconvergence order at downwind point as $k+1+\min\{k,\max\{\alpha,m\}\}$. The underlying mechanism of the superconvergence is discussed and the analysis confirmed through examples, including a discussion of how to use the scheme as an efficient way to evaluate the generalized Mittag-Leffler function and solutions to more generalized FODE's.Comment: 17 pages, 7 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

A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems

We consider a two-point boundary value problem involving a Riemann-Liouville fractional derivative of order \al\in (1,2) in the leading term on the unit interval $(0,1)$. Generally the standard Galerkin finite element method can only give a low-order convergence even if the source term is very smooth due to the presence of the singularity term x^{\al-1} in the solution representation. In order to enhance the convergence, we develop a simple singularity reconstruction strategy by splitting the solution into a singular part and a regular part, where the former captures explicitly the singularity. We derive a new variational formulation for the regular part, and establish that the Galerkin approximation of the regular part can achieve a better convergence order in the $L^2(0,1)$, H^{\al/2}(0,1) and $L^\infty(0,1)$-norms than the standard Galerkin approach, with a convergence rate for the recovered singularity strength identical with the $L^2(0,1)$ error estimate. The reconstruction approach is very flexible in handling explicit singularity, and it is further extended to the case of a Neumann type boundary condition on the left end point, which involves a strong singularity x^{\al-2}. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea

Discontinuous Galerkin methods for convection-diffusion equations and applications in petroleum engineering

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in Linfinity(0, T; L2) norm for the solution and the auxiliary variables will be derived. Then, high-order bound-preserving (BP) discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacements on triangular meshes will be developed. There are three main difficulties to make the concentration of each component between 0 and 1. Firstly, the concentration of each component did not satisfy a maximum-principle. Secondly, the first-order numerical flux was difficult to construct. Thirdly, the classical slope limiter could not be applied to the concentration of each component. To conquer these three obstacles, we first construct special techniques to preserve two bounds without using the maximum-principle-preserving technique. The time derivative of the pressure was treated as a source of the concentration equation. Next, we apply the flux limiter to obtain high-order accuracy using the second-order flux as the lower order one instead of using the first-order flux. Finally, L2-projection of the porosity and constructed special limiters that are suitable for multi-component fluid mixtures were used. Lastly, a new LDG method for convection-diffusion equations on overlapping mesh introduced in [J. Du, Y. Yang and E. Chung, Stability analysis and error estimates of local discontinuous Galerkin method for convection-diffusion equations on overlapping meshes, BIT Numerical Mathematics (2019)] showed that the convergence rates cannot be improved if the dual mesh is constructed by using the midpoint of the primitive mesh. They provided several ways to gain optimal convergence rates but the reason for accuracy degeneration is still unclear. We will use Fourier analysis to analyze the scheme for linear parabolic equations with periodic boundary conditions in one space dimension. To investigate the reason for the accuracy degeneration, we explicitly write out the error between the numerical and exact solutions. Moreover, some superconvergence points that may depend on the perturbation constant in the construction of the dual mesh were also found out