Spectral Approximation to Advection—Diffusion Problems by the Fictitious Interface Method

In this work we face the numerical approximation by spectral methods of advection-diffusion equations for convective dominated regimes. For either boundary and internal layer problems, it has been recently pointed out that effective methods can be based on dropping the viscous terms far from the thin layer. This yields a problem that couples two different model equations (one of hyperbolic type, the other one of parabolic type) through suitable matching conditions at subdomain interfaces. An extensive theory has been developed and effective algorithms have been derived. Here we apply this theory to spectral approximations to two-dimensional steady problems. In particular, we investigate the issues of stability and convergence, and we propose effective algebraic solvers to face both the hyperbolic and elliptic problems.