Spectral approximation to advection-diffusion problems by the fictitious interface method
This work originates from numerical matching techniques for problems involving boundary layers. In particular, a spatial region \(\Omega\) is considered that can be partitioned into two regions, \(\Omega_ 1\) and \(\Omega_ 2\), such that an advection-diffusion equation \(-\epsilon\Delta u \text{div }(bu) b_ 0 u= f\), is satisfied in \(\Omega_ 1\), and an advection equation (obtained by dropping the viscous term), \(\text{div}(bu) b_ 0 u= f\), is satisfied in \(\Omega_ 2\). Here, \(b_ 0\) and \(f\) are given scalar functions, and \(b\) is the given advection vector. The difficulty in such formulations lies in the specification and numerical treatment of the boundary conditions at the interface of \(\Omega_ 1\) and \(\Omega_ 2\). The authors describe discretization of these equations in a 2D rectangular region, and in particular, the boundary conditions at the interface. These discretizations arise from spectral approximations to the equations based upon Gauss-Lobatto-Legendre knots. The resulting system of equations is sparse, and numerical examples are presented showing the results of preconditioned GMRES solution techniques. Reference is made to applications involving time-dependent and nonlinear advection-diffusion systems.
preconditioned GMRES solution techniques
scientific article; zbMATH DE number 428546