Spectral methods for singular perturbation problems
This is a study on spectral discretization methods (where the discretization error tends to zero faster than any power of the grid size) for singular perturbation problems, i.e. homogeneous Dirichlet problems for inhomogeneous second order differential equations with a small parameter at the second-order term in one and two dimensions, e.g. appearing as advection-diffusion equations. First, stabilization of the spectral method is discussed: The introduction of artificial viscosity stabilizes the scheme but makes the method only first-order accurate. To avoid the loss of the high spectral accuracy, the author proposes a simultaneous change of the (smooth) right-hand side, similarly to the second-order accurate ``Mehrstellen''- method due to \textit{H. Yserentant} [Numer. Math. 34, 171-187 (1980; Zbl 0413.65072)]. Numerical examples substantiate the improvement for mesh sizes with sufficiently high spectral accuracy. Second, boundary layer effects are treated by domain decomposition. The interface relaxation of the Dirichlet data follows mainly the pattern of \textit{D. Funaro}, \textit{A. Quarteroni} and \textit{P. Zanolli} [SIAM J. Numer. Anal. 25, No. 6, 1213-1236 (1988; Zbl 0678.65082)], especially in the automatic selection of relaxation parameters. The convergence properties are analyzed for one and two dimensions. Finally, an iterative solver is proposed, getting components for a multigrid method in this way.
scientific article; zbMATH DE number 704269
scientific article