An hp-Local Discontinuous Galerkin method for Parabolic\ud Integro-Differential Equations

In this article, a priori error analysis is discussed for an hp-local discontinuous Galerkin (LDG) approximation to a parabolic integro-differential equation. It is shown that the L2 -norm of the gradient and the L2 -norm of the potential are optimal in the discretizing parameter h and suboptimal in the degree of polynomial p. Due to the presence of the integral term, an introduction of an expanded mixed type Ritz-Volterra projection helps to achieve optimal estimates. Further, it is observed that a negative norm estimate of the gradient plays a crucial role in our convergence analysis. As in the elliptic case, similar results on order of convergence are established for the semidiscrete method after suitably modifying the numerical fluxes. The optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

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

A posteriori error estimation for hp -version time-stepping methods for parabolic partial differential equations

The aim of this paper is to develop an hp-version a posteriori error analysis for the time discretization of parabolic problems by the continuous Galerkin (cG) and the discontinuous Galerkin (dG) time-stepping methods, respectively. The resulting error estimators are fully explicit with respect to the local time-steps and approximation orders. Their performance within an hp-adaptive refinement procedure is illustrated with a series of numerical experiment

Discontinuous Galerkin approximation of linear parabolic problems with dynamic boundary conditions

In this paper we propose and analyze a Discontinuous Galerkin method for a linear parabolic problem with dynamic boundary conditions. We present the formulation and prove stability and optimal a priori error estimates for the fully discrete scheme. More precisely, using polynomials of degree $p\geq 1$ on meshes with granularity $h$ along with a backward Euler time-stepping scheme with time-step $\Delta t$, we prove that the fully-discrete solution is bounded by the data and it converges, in a suitable (mesh-dependent) energy norm, to the exact solution with optimal order $h^p + \Delta t$. The sharpness of the theoretical estimates are verified through several numerical experiments

Optimal L2 estimates for semidiscrete Galerkin methods for\ud parabolic integro-differential equations with nonsmooth data

In this article, we discuss an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time dependent parabolic integro-differential equation with nonsmooth initial data. It is based on energy arguments and on a repeated use of time integration, but without using parabolic type duality technique. Optimal L2-error estimate is derived for the semidiscrete approximation, when the initial data is in L2

Artificial boundary conditions for parabolic Volterra integro-differential equations on unbounded two-dimensional domains

AbstractIn this paper we study the numerical solution of parabolic Volterra integro-differential equations on certain unbounded two-dimensional spatial domains. The method is based on the introduction of a feasible artificial boundary and the derivation of corresponding artificial (fully transparent) boundary conditions. Two examples illustrate the application and numerical performance of the method