Estimates for the accuracy of modelling boundary-value problems at the junction of domains with different limit dimensions
С. А. Назаров · Izvestiya Mathematics · 2004
This paper deals with the mixed boundary-value problem for the Poisson equation at the junction of thin rods and a massive body that have different stiffnesses. We suggest a new approach to the study of this singularly perturbed problem. Namely, we construct a model of the junction that gives an approximation to the solution of the original problem on the whole range of parameters and (the relative thickness and relative stiffness of the rods). The model contains ordinary differential equations on the line segments (the axes of the rods) and the Neumann problem on the domain , which are combined into a single problem by imposing asymptotic conjugation conditions at the points correlating the coefficients of the expansions of solutions on (as ) with those of solutions on (as ). We obtain estimates for the accuracy of the model that are asymptotically exact. The conjugation conditions preserve the parameters and but generate a regularly perturbed problem, and it is not difficult to obtain and justify asymptotics of its solutions and those of solutions of the original problem under any relation between and .