Superconvergence Analysis of the Non-Symmetric Interior Penalty Galerkin (NIPG) Method for Singularly Perturbed Differential Equations

Abstract: This talk will provide efficient numerical methods for solving singular perturbation problems (SPPs) of convection-diffusion and reaction-diffusion types with boundary layers. A differential equation is called singularly perturbed when a small parameter is multiplied with the highest-order derivative and this small parameter is called the perturbation parameter. Because of the presence of the parameter in the solution of the differential equation, steep, thin layers occur at the boundaries or/and interior of the domain. Solution of singularly perturbed problems normally has smooth and singular components and its singular component is called the boundary layer function which varies very rapidly in the boundary layer region and behaves smoothly in the outer region. Due to this layer phenomena, it is a very difficult and challenging task to provide parameter-uniform numerical methods for solving SPPs. Parameter-uniform numerical methods means those numerical methods in which the approximate solution converges to the corresponding exact solution of SPPs independently with respect to the perturbation parameter(s).

It is well-known that uniform meshes with classical schemes fail to converge uniformly with respect to the singular perturbation parameter. It is desirable to develop methods which converges uniformly. In this talk we will develop and analyze superconvergence properties of the NIPG method for solving SPPs in one-and two-dimensions on layer-adapted meshes.