A Novel Adaptive Mesh Strategy for Singularly Perturbed Parabolic Convection Diffusion Problems

Abstract

In this article, we study a novel adaptive mesh strategy for singularly perturbed problems (SPPs) of the parabolic convection-diffusion type that exhibit regular boundary layers. Our central insight is that the introduction of an auxiliary inequality for an entropy-like variable also serves as a remarkably effective adaptation indicator. The primary novelty of this method is that, unlike extant methods used for layer adapted meshes (in which enough mesh points exist in the layer region for well resolved numerical solution), the current method requires no a priori knowledge of the location and width of the boundary layers. Further, the current method, [(which is an extension of the methodology from Kumar and Srinivasan (Appl Math Model 39:2081–2091, 2015)] is completely independent of the perturbation parameter and results in accurate solutions for a wide range of problems. We include some preliminary error estimates and also the results of several numerical experiments including the Black–Scholes equation. The results exhibit the promise of the proposed strategy to generate efficient adaptive meshes for time dependent convection-diffusion problems.