Convergence of a Discontinuous Galerkin Method on Bakhvalov-Type Meshes for Singularly Perturbed Volterra Integro-Differential Equations

Abstract

A discontinuous Galerkin (DG) method on Bakhvalov-type (B-type) meshes for singularly perturbed Volterra integro-differential equations (SPVIDEs) is proposed. We derive abstract error bounds of the DG method for the SPVIDEs in the $L^2$-norm. It is shown that the approximate solution generated by the DG method on B-type meshes has optimal convergence rate $k + 1$ in the $L^2$-norm, when using the piecewise polynomial space of degree $k.$ Numerical simulations demonstrate the validity of the theoretical results.