Title of the talk: "Convergence Analysis of Higher-Order Approximation of Singularly Perturbed 2D Semilinear Parabolic PDEs with Non-homogeneous Boundary Conditions."

Date , Time & Venue: 21 August 2025  at 12 PM on Seminar Hall, Dept. of Mathematics

Abstract: This research talk focuses on developing and analyzing an efficient higher-order numerical approximation of singularly perturbed two-dimensional semilinear parabolic convection-diffusion problems with time-dependent boundary conditions. We approximate the governing nonlinear problem by an implicit fitted mesh method (FMM), which combines an alternating direction implicit scheme in the temporal direction together with a higher-order finite difference scheme in the spatial directions. Next, we established error analysis of the implicit-explicit (IMEX) fractional-step Euler scheme with the same higher-order finite difference scheme. Since the solution possesses exponential boundary layers, a Cartesian product of piecewise-uniform Shishkin meshes is used to discretize in space. To begin our analysis, we establish the stability corresponding to the continuous nonlinear problem, and obtain a-priori bounds for the solution derivatives. Thereafter, we pursue the stability analysis of  the discrete problem, and prove $\varepsilon$-uniform convergence in the maximum-norm. Next, for enhancement of the temporal accuracy,  we use the Richardson extrapolation technique solely in the temporal direction. In addition, we investigate the order reduction phenomenon naturally occurring due to the time-dependent boundary data and propose a suitable approximation to tackle this effect. Finally, we present the computational results to validate the theoretical estimates.

About the Speaker: Dr. Narendra Singh Yadav is currently serving as a regular Assistant Professor at the Indian Institute of Information Technology (IIIT), Sri City, Chittoor, Andhra Pradesh, India. He earned his B.Sc. in Mathematics from Kanpur University (2013), followed by an M.Sc. in Mathematics from the I.I.T. Delhi (2015). He completed his Ph.D. in Applied and Computational Mathematics at the Indian Institute of Space Science and Technology (IIST), Thiruvananthapuram in May 2022, and subsequently pursued postdoctoral research at IISER Thiruvananthapuram. His research focuses on the development and rigorous analysis of advanced numerical methods for time-dependent singularly perturbed partial differential equations (SPPDEs), along with their efficient numerical simulation and computational validation.