Postprocessing technique of the discontinuous Galerkin method for solving delay differential equations

We introduce an innovative postprocessing technique aimed at refining the accuracy of the discontinuous Galerkin method for solving linear delay differential equations (DDEs) with vanishing delays. The fundamental idea behind this postprocessing technique is to enhance the discontinuous Galerkin solution of degree k by incorporating a generalized Jacobi polynomial of degree \(k+1\). We demonstrate that this postprocessing step enhances convergence by one order under the \(L^\infty \)-norm. Moreover, we apply this technique to both nonlinear DDEs with vanishing delays and linear DDEs with non-vanishing delays. We further validated the theoretical results through a series of numerical examples.