Particular Solutions of Singularly Perturbed Partial Differential Equations with Constant Coefficients in Rectangular Domains, Part II: Computational Aspects

This is a continued study on the solution of convection-diffusion equations with boundary layers. In our previous work, Part I, the solutions of the homogenenous singularly perturbed differential equations with Dirichlet boundary conditions are expressed in a series of particular solutions have been proposed, based on the separation of variables. In this study we extend our previous results to the homogeneous equations with anisotropic diffusion coefficients, and to non-homogeneous equations. Moreover, we will discuss in detail computational aspects of the methods for three different models of fast convergence. Among them, the model with waterfalls solutions is especially interesting, because it presents an intrinsic nature of singular boundary layers. A new convergence analysis for a model of the non-homogeneous equation is also performed in this work. Numerical results for 2 = 10−7 will be presented for all models, where 2 denotes the singular perturbation parameter. The numerical solutions for 2 = 0.1 are also illustrated to give a clear view of the solutions of the singularly perturbed partial differential equations. Finally, we propose a collocation Trefftz method using the particular solutions obtained in the work. Results from numerical experiments on the collocation Trefftz methods will be presented.