An hp-local Discontinuous Galerkin Method for Parabolic Integro-Differential Equations

Abstract

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 L 2 -norm of the gradient and the L 2 -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.

Efficient parallel software system for solving Navier-Stokes equations by the discontinuous Galerkin method

“…The study of thermoelasticity problems, taking into account the interaction of deformation and temperature fields, began with [1][2][3]. This line of research was called the coupled thermoelasticity.…”

“…Analysis shows that in the overwhelming majority of studies in the solution of coupled thermoelasticity problems the finite element models of a fairly general purpose were developed, for example [7][8][9][10][11].The analytical methods for solving this class of problems did not become as widespread as the numerical ones. The results obtained with their help were summarized in [12]. Beginning with papers [13][14][15][16][17][18], scientists consider uncoupled problems of thermoelasticity about the sliding contact of a rigid body with an elastic coating, taking into account friction, heating of the coating from friction, and abrasive wear.…”

“…This method has a number of advantages inherent in both finite-element and finite-difference approximations. In particular, it provides a given order of accuracy, can be used for grids of arbitrary structure and is theoretically justified [1][2][3][4][5]. Discontinuous Galerkin Method, with all its merits, has a significant computational complexity, so the question of maximizing the use of all the possibilities of modern computer technology is very acute.…”

“…In the developed software package, a library of software implementations of these approximations has been created [8][9][10]. At the moment, various discrete approximations are used for both viscous [7,11,12] and non-viscous flow terms.…”

Local discontinuous Galerkin approximation of non-Fickian diffusion model in viscoelastic polymers

“…It has been widely applied to solve different partial differential equations, such as elliptic equations [8], Stokes equations [9], Oseen equations [10], and stationary convection diffusion equations [11], nonstationary equations [12], high-order time dependent partial differential equations [13], and integro-differential equations [14].…”