A superconvergent postprocessing technique for the continuous Galerkin method in solving delay differential equations with nonlinear vanishing delay

We introduce a novel postprocessing technique aimed at enhancing the overall accuracy of the continuous Galerkin method for solving linear delay differential equations with nonlinear vanishing delay. The fundamental idea behind this postprocessing approach is to augment the continuous Galerkin solution of degree k with a generalized Jacobi polynomial of degree \(k+1\) $k+1$ . We prove that the convergence rates of the \(L^{\infty }\) $L^{\infty }$ -error for the continuous Galerkin method with quasi-uniform time partitions are improved by one order. Several numerical examples are provided to validate the presented theoretical findings.