A direct higher order discretization in singular perturbations via domain split -- a computational approach
The authors consider a specified form of a singularly perturbed two-point boundary value problem. As a means of solving the equation considered, two subproblems are generated over suitably chosen subintervals and then solved numerically. This then allows for the use of standard methods on each of the subintervals. The novelty of the work is that the authors do not treat the coupling iteratively, but rather introduce a linear equation which approximates the coupling condition to be solved exactly. Two different types of couplings similar to domain decomposition are studied: one based upon the notion of non-overlapping the domain, and the other on an overlapping domain. In each of the subintervals that are obtained in both the non-overlapping and overlapping domain they find amendable conditions for applying standard discretization techniques in each subinterval separately. A fourth-order compact finite difference method generating a linear discrete problem is applied. In splitting the domain into two subdomains the discretization is applied on a uniform grid separately in each domain. Given that approximations of the first order derivatives are required, the coupling of the different domains is performed via discrete approximations of the derivatives in the case of the non-overlapping domain split. In the case of the overlapping domain split the value at the one boundary is required and hence the coupling is devised via coinciding function values. Theoretical results are provided with regards to these two cases. Numerical experiments are also conducted, establishing the effect of certain chosen parameter values, associated computational errors and which technique used over a chosen type of domain produces the best numerical results. In comparison to the implementation of direct discretization on Shishkin meshes over the full interval, the authors verify that the separate treatment of the boundary value problems generated on the related subintervals produces a significant reduction of the natural ill-conditioning of the discrete schemes induced by the singular perturbation character of the problem.
scientific article; zbMATH DE number 5913742
scientific article