Estimates for the accuracy of modelling boundary-value problems at the

S. A. Nazarov

Izvestiya. Mathematics , Tome 68 (2004) no. 6, pp. 1179-1215

Résumé

This paper deals with the mixed boundary-value problem for the Poisson equation at the junction of thin rods and a massive body $\Omega$ 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 $h\in(0,h_0]$ and $\gamma\in(0,+\infty)$ (the relative thickness and relative stiffness of the rods). The model contains ordinary differential equations on the line segments $\Upsilon_j$ (the axes of the rods) and the Neumann problem on the domain $\Omega$, which are combined into a single problem by imposing asymptotic conjugation conditions at the points $P^j=\overline\Upsilon_j\cap\overline\Omega$ correlating the coefficients of the expansions of solutions on $\Upsilon_j$ (as $\Upsilon_j\ni z^j\rightarrow P^j$) with those of solutions on $\Omega$ (as $\Omega\ni x\rightarrow P^j$). We obtain estimates for the accuracy of the model that are asymptotically exact. The conjugation conditions preserve the parameters $h$ and $\gamma$ 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 $\gamma$ and $h$.