Adaptive least squares finite integration method for higher-dimensional singular perturbation problems with multiple boundary layers

摘要

Based on the recently developed finite integration method for solving one-dimensional partial differential equation, we extend in this paper the method by using the technique of least squares to tackle higher-dimensional singular perturbation problems with multiple boundary layers. Theoretical convergence and numerical stability tests indicate that, even with the most simple numerical trapezoidal integration rule, the proposed method provides a stable, efficient, and highly accurate approximate solutions to the singular perturbation problems. An adaptive scheme on the refinement of integration points is also devised to better capture the stiff boundary layers. Illustrative examples are given in both 1D and 2D with comparison among some existing numerical methods.