HDG method for linear parabolic integro-Differential equations

Abstract

This paper develops the hybridizable discontinuous Galerkin (HDG) method for a linear parabolic integro-differential equation and analyzes uniform in time apriori error bounds. To handle the integral term, an extended Ritz-Volterra projection is introduced, which helps in achieving optimal order convergence of O(hk+1) for the semi-discrete problem when polynomials of degree k≥0 are used to approximate both the solution and the flux variables. Further, element-by-element post-processing is proposed, and it is established that it achieves convergence of the order O(hk+2) for k≥1. Using the backward Euler method in temporal direction and quadrature rule to discretize the integral term, a fully discrete scheme is derived along with its error estimates. Finally, with the help of numerical examples in two-dimensional domains, computational results are obtained, which verify our results.