Discontinuous Galerkin Methods for Nonlinear Parabolic Delay-Equations of Nonmonotone Type

In this paper, we study two discontinuous Galerkin (DG) methods: the Symmetric Interior Penalty Galerkin (SIPG) method and the Non-symmetric Interior Penalty Galerkin (NIPG) method. These methods are applied to nonlinear parabolic delay problems of nonmonotone type considering the locally Lipschitz type nonlinearity and using the backward finite difference method for time discretization. By employing Brouwer’s fixed point theorem and by showing Lipschitz’s continuity of a discrete map, we first prove the existence of a discrete solution and then establish the uniqueness of the discrete solution. After this, we derive the stability of the backward scheme in \(L^2\) $L^2$ norm and energy norm. We also discuss error bounds for both versions of DG methods (SIPG and NIPG) under various hypotheses on the regularity of the solution. Finally, we validate our theoretical results by finding numerical solutions for different population models. We also study the behavior of the matured population for different initial functions in the Nicholson Blowflies equation with time.