A novel numerical method on a posteriori mesh for a singularly perturbed convection-diffusion problem with mixed type boundary conditions

This article introduces a novel numerical method for solving a singularly-perturbed convection-diffusion problem with mixed type boundary conditions. The approach begins by transforming the second-order boundary value problem into a system of first-order differential equations. The stability result of its corresponding differential operator vector is obtained although the system of equations is in a special form. The transformed system is approximated through a hybrid difference method, followed by a posteriori error analysis that relies on the stability result of the differential operator vector. A posteriori mesh and approximation solution are derived by designing a solution-adaptive algorithm derived from the a posteriori error bound. Finally, the second-order uniform convergence of the novel numerical method is corroborated by numerical experiments.