\documentstyle[12pt]{article} \begin{document} \title{On the convergence of finite difference schemes for the heat equation with concentrated capacity} \author{Lubin G. Vulkov \\ %EndAName Center of Applied Mathematics and Informatics\\ University of Rousse, 7017 Rousse, Bulgaria\\ e-mail:vulkov@ami.ru.acad.bg} \date{} \maketitle \begin{abstract} \end{abstract} As a model problem we consider the initial boundary-value problem for the heat equation $\left( 1+C\delta \left( x-\xi \right) \right) \frac{\partial u}{\partial t}-% \frac{\partial ^2u}{\partial x^2}=f\left( x,t\right) ,$ in the domain $Q_T=\Omega \times \left( 0,T\right) ,\quad \Omega =\left( 0,1\right) ,\quad 0<\xi <1;$ $u\left( 0,t\right) =u\left( 1,t\right) =0,\quad t\in \left[ 0,T\right] ;\quad u\left( x,0\right) =u_0\left( x\right) ,\quad x\in \Omega .$ Here $C=$constant$>0$ is the magnitude of the concentrated capacity, $\delta \left( x-\xi \right)$ is the Dirac-delta (generalized) function. Stability of weighted difference schemes in integral norms is studied. Convergence rate estimates compatible with the smoothness of the generalized solution $% u\in H^{2,1}\left( Q_T^{*}\right) \cap C\left( Q_T^{*}\right) ,\quad \Omega ^{*}=\Omega \backslash \xi ,\quad Q_T^{*} =\Omega ^{*}\times \left( 0,T\right)$ are obtained. \end{document} 