Grid Approximation for the Solution of the Singularly Perturbed Heat Equation with Concentrated CapacityU

Mixed boundary-value problems for the heat equation are considered. The diffusion coefficient is small, i.e., it is multiplied by a parameter « g 0, 1x. The coefficients in the equation are discontinuous at some points. We confine our attention to the case in which at these points heat capacity is concentrated. First, some properties of the solution regularity, large-time behavior, estimates for the derivatives. are studied. Then, we show that it is impossible to achieve uniform convergence on the discrete maximum norm of difference schemes on classical meshes. For the problems described we construct grid approximations based on standard weighted difference schemes and condensed Shishkin’s.mesh. We prove y2 2 mthe «-uniformly O N log Nqt s . order of convergence for the error in the discrete solution. Here N is the number of the nodes in the space mesh, t is the time step, and 1FmsF2 is a weight parameter